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}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "P ower series" }{TEXT 258 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 46 "Maclaurin series are examples of power series." }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 259 12 "power series" }{TEXT -1 14 " in powers of " }{TEXT 275 1 "x" }{TEXT -1 38 " with real coefficients has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[n]*x^n,n = 0 .. infinity) = a[0]+a[1]*x+a[ 2]*x^2+a[3]*x^3+` . . . `+a[n]*x^n+` . . . `" "6#/-%$SumG6$*&&%\"aG6#% \"nG\"\"\")%\"xGF+F,/F+;\"\"!%)infinityG,0&F)6#F1F,*&&F)6#F,F,F.F,F,*& &F)6#\"\"#F,*$F.FF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The Taylor series expan sion of a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " about " }{XPPEDIT 18 0 "x = c" "6#/%\"xG%\"cG" }{TEXT -1 15 " h as this form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Radius and interval of convergence of power series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "A power series: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(a[n]*x^n,n = 0 .. infinity) = a[0]+a[1]*x+a[2]*x^2+ a[3]*x^3+` . . . `+a[n]*x^n+` . . . `" "6#/-%$SumG6$*&&%\"aG6#%\"nG\" \"\")%\"xGF+F,/F+;\"\"!%)infinityG,0&F)6#F1F,*&&F)6#F,F,F.F,F,*&&F)6# \"\"#F,*$F.F=1 " "6#1\"\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 23 "interval of convergence" }{TEXT -1 14 " is \+ therefore " }{XPPEDIT 18 0 "-1 " 0 "" {MPLTEXT 1 0 256 "f := x -> 1/(1-x) :\n'f(x)'=f(x);\np := x -> 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"-11 " "6#2\"\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "The series also clearly diverges when " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 10 " and when " }{XPPEDIT 18 0 "x = -1" "6# /%\"xG,$\"\"\"!\"\"" }{TEXT -1 27 ", because, in these cases, " } {XPPEDIT 18 0 "Limit(u[n],n=infinity)<>0" "6#0-%&LimitG6$&%\"uG6#%\"nG /F*%)infinityG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 23 "interval of convergence" }{TEXT -1 14 " is therefore " }{XPPEDIT 18 0 "-1 " 0 "" {MPLTEXT 1 0 79 "u := n -> (n+1)*x^n;\nLimit(abs(u(n+1)/u(n)),n=infinity);\nsimplify(%);\nv alue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"nG6\"6$%)oper atorG%&arrowGF(*&,&9$\"\"\"F/F/F/)%\"xGF.F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%$absG6#*&*&,&%\"nG\"\"\"\"\"#F-F-)%\"xG,&F ,F-F-F-F-F-*&F1F-)F0F,F-!\"\"/F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%$absG6#*&*&,&%\"nG\"\"\"\"\"#F-F-%\"xGF-F- ,&F,F-F-F-!\"\"/F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$ab sG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The series " }{XPPEDIT 18 0 "Sum((n+1)*x^n,n = 0 .. infinity)" "6#-%$SumG6$*&,&%\"nG\"\"\"F)F)F))%\"xGF(F)/F(;\"\"!%)infinityG" } {TEXT -1 23 " actually converges to " }{XPPEDIT 18 0 "1/(x-1)^2" "6#*& \"\"\"F$*$,&%\"xGF$F$!\"\"\"\"#F(" }{TEXT -1 7 " when " }{XPPEDIT 18 0 "abs(x)<1" "6#2-%$absG6#%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "Maple's procedure " }{TEXT 0 3 "sum" }{TEXT -1 61 " pr ovides this result without the necessity of assuming that " }{XPPEDIT 18 0 "abs(x)<1" "6#2-%$absG6#%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Sum(( n+1)*x^n,n=0..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%$SumG6$*&,&%\"nG\"\"\"F)F)F))%\"xGF(F)/F(;\"\"!%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$),&%\"xGF$!\"\"F$\"\"#F$F) " }}}{PARA 0 "" 0 "" {TEXT -1 45 "The following picture compares the g raphs of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x) = 1/((1-x)^2);" "6#/-%\"fG6#%\"xG*& \"\"\"F)*$,&F)F)F'!\"\"\"\"#F," }{TEXT -1 16 ", and the graph " } {XPPEDIT 18 0 "y = p(x);" "6#/%\"yG-%\"pG6#%\"xG" }{TEXT -1 36 " of th e degree 6 Taylor polynomial: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "p(x) = 1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+7*x^6;" "6#/-%\"pG 6#%\"xG,0\"\"\"F)*&\"\"#F)F'F)F)*&\"\"$F)*$F'F+F)F)*&\"\"%F)*$F'F-F)F) *&\"\"&F)*$F'F0F)F)*&\"\"'F)*$F'F3F)F)*&\"\"(F)*$F'F6F)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "f := x -> 1/(x-1)^2:\n'f(x)'=f(x);\np := x -> 1+2*x+ 3*x^2+4*x^3+5*x^4+6*x^5+7*x^6:\n'p(x)'=p(x);\nplot1 := plot([1/(1-x)^2 ,p(x)],x=-2..3,y=-1..8,color=[red,blue],numpoints=75,discont=true):\np lot2 := plot([[1,-1],[1,8]],color=black,linestyle=3):\nplots[display]( [plot1,plot2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&\" 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\"3+4\"3\"e)p3*\\F-$\"3y;;nfN!p%QF17$$\"3UBVKC$pzl&F-$\"3aB[?R!Hk!\\F1 7$$\"3)Q*=*=kgjN'F-$\"38()3+r]76kF17$$\"3R)['['[%**GqF-$\"3!4[]1-P8P)F 17$$\"33XKCV(for(F-$\"3)Qm@7wFh5\"Fi`l7$$\"3M\"Qy$G\"H!*R)F-$\"35MdL'* >&>Y\"Fi`l7$$\"3)o%f%f%>(e-*F-$\"3m!Rrn1k())=Fi`l7$$\"3PdS0aXIW(*F-$\" 3t1#>lR#)p_#Fi`l7$$\"3bCVKCzoQ5F1$\"3IkLZ-b1mKFi`l7$$\"3k8N^)y*>26F1$ \"35c\"**e:y9F%Fi`l7$$\"3!)f%fWQuF<\"F1$\"35LWw*G/2\\&Fi`l7$$\"3aYf%fa 'eX7F1$\"3/+0ClVH/sFi`l7$$\"3y-FqFlj38F1$\"3w%HUVJ[n0*Fi`l7$$\"3OYf%4# ))>!Q\"F1$\"3As@k8Twl6Fcbl7$$\"3!GVKC3?aW\"F1$\"3Yr?6v(GvX\"Fcbl7$$\"3 5ovc#*=w;:F1$\"3GOf?]h`Z=Fcbl7$$\"3mf%f%4yJy:F1$\"3)pfQwx$f`AFcbl7$$\" 3%R^8N\"GU[;F1$\"3pNn]JCf2GFcbl7$$\"3#o['['yShr\"F1$\"3'42!Rh,()\\MFcb l7$$\"3SYf%4_9Qy\"F1$\"3isTxW&oL@%Fcbl7$$\"3]Yf%fHR7&=F1$\"3_'=([D%zH6 &Fcbl7$$\"3t@;i'R8g\">F1$\"3g['e34%GEhFcbl7$$\"3)\\'['[\"f.')>F1$\"3;Y \"QAKb(3uFcbl7$$\"3SFq-_Cx_?F1$\"3=')y\\aT2O))Fcbl7$$\"36l['[^4I7#F1$ \"3a'4KvL7$e5Fccl7$$\"3;q-Fq4f'=#F1$\"3%4`%[bb\"3C\"Fccl7$$\"3+\"3\"3 \"QgoD#F1$\"3EM?U?Kvs9Fccl7$$\"3&ovc,$Qo;#>?Fccl7$$\"3ml['[OQ9Y#F1$\"3=$>DTP1iO#Fccl7$$ \"3zy$yLvIg_#F1$\"3r]PIOd*)GFFccl7$$\"3'[f%f%>#=#f#F1$\"3AdS6>goZJFccl 7$$\"37Yf%fC@_m#F1$\"3Y5mHw&R:n$Fccl7$$\"3SCVKC@NJFF1$\"3.qk-VE=2UFccl 7$$\"3!*f%fWL$)*)z#F1$\"3^[oc\")QL@[Fccl7$$\"3H%y$yLHPLGF1$\"3_!=@%*># Rh^Fccl7$$\"3o3\"3J`ix'GF1$\"3MrQQ2YM@bFccl7$$\"3M#*=*oEt$**GF1$\"3)3^ z$RAbqeFccl7$$\"3*fnv1+%)4$HF1$\"3)*o$*=jE4QiFccl7$$\"3+QyL+?\\lHF1$\" 39%e!3@<:hmFccl7$F[an$\"%3rF*-F`an6&FbanFfanFfanFcan-F$6%7$7$$\"\"\"F* $FcflF*7$Fbjo$\"\")F*-F`an6&FbanF*F*F*-%*LINESTYLEG6#F\\an-%+AXESLABEL SG6%Q\"x6\"Q\"yFa[p-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F[an;FdjoFfjo" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The convergence of the series is slow for values of " } {TEXT 291 1 "x" }{TEXT -1 12 " near 1 and " }{XPPEDIT 18 0 "-1" "6#,$ \"\"\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "xx := 0.95;\nSum((n+1)*xx^n,n = 0 . . 470);\nevalf(%);\nevalf(1/(xx-1)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#&*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&, &%\"nG\"\"\"F)F)F))$\"#&*!\"#F(F)/F(;\"\"!\"$q%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(*******R!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+++++S!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "xx := -0.95;\nSum((n+1)*xx^n,n = 0 .. 540);\neva lf(%);\nevalf(1/(xx-1)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$! #&*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&,&%\"nG\"\"\"F)F )F))$!#&*!\"#F(F)/F(;\"\"!\"$S&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +'y[)HE!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%y[)HE!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Convergence of the Maclaurin \+ series for " }{XPPEDIT 18 0 "sin*x;" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "In this section we shall investigate the convergence of the Maclau rin series for " }{XPPEDIT 18 0 "sin*x;" "6#*&%$sinG\"\"\"%\"xGF%" } {TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x-x ^3/3!+x^5/5!-x^7/7!+x^9/9!-x^11/11!+x^13/13!-` . . . `" "6#,2%\"xG\"\" \"*&F$\"\"$-%*factorialG6#F'!\"\"F+*&F$\"\"&-F)6#F-F+F%*&F$\"\"(-F)6#F 1F+F+*&F$\"\"*-F)6#F5F+F%*&F$\"#6-F)6#F9F+F+*&F$\"#8-F)6#F=F+F%%(~.~.~ .~GF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "Informal discussion of convergence from \+ \"first principles\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "taylor(sin(x),x,15);\nconvert(%,polynom); \nterms := [op(%)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"F% #!\"\"\"\"'\"\"$#F%\"$?\"\"\"&#F'\"%S]\"\"(#F%\"'!)GO\"\"*#F'\")+o\"*R \"#6#F%\"++3-Fi\"#8-%\"OG6#F%\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, 0%\"xG\"\"\"*$)F$\"\"$F%#!\"\"\"\"'*$)F$\"\"&F%#F%\"$?\"*$)F$\"\"(F%#F *\"%S]*$)F$\"\"*F%#F%\"'!)GO*$)F$\"#6F%#F*\")+o\"*R*$)F$\"#8F%#F%\"++3 -Fi" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&termsG7)%\"xG,$*$)F&\"\"$\" \"\"#!\"\"\"\"',$*$)F&\"\"&F+#F+\"$?\",$*$)F&\"\"(F+#F-\"%S],$*$)F&\" \"*F+#F+\"'!)GO,$*$)F&\"#6F+#F-\")+o\"*R,$*$)F&\"#8F+#F+\"++3-Fi" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Macla urin series for " }{XPPEDIT 18 0 "sin*x;" "6#*&%$sinG\"\"\"%\"xGF%" } {TEXT -1 151 " is not geometric, because the ratio of successive terms is not constant. In fact we can easily construct the ratios of the fi rst few successive terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ratios := [seq(terms[n+1]/terms[n], n=1..nops(terms)-1)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ratiosG7(, $*$)%\"xG\"\"#\"\"\"#!\"\"\"\"',$F'#F-\"#?,$F'#F-\"#U,$F'#F-\"#s,$F'#F -\"$5\",$F'#F-\"$c\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 53 "Looking at the first of these ratios, w e see that if " }{XPPEDIT 18 0 "abs(x^2) < 6" "6#2-%$absG6#*$%\"xG\"\" #\"\"'" }{TEXT -1 123 ", then the absolute value of the ratio of succe ssive terms is always less than 1. In this case the terms are approach ing 0 " }{TEXT 259 50 "faster than those of a convergent geometric ser ies" }{TEXT -1 30 ", so the Maclaurin series for " }{XPPEDIT 18 0 "sin *x;" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 11 " converges." }}{PARA 0 " " 0 "" {TEXT -1 44 "Looking at the second ratio, we see that if " } {XPPEDIT 18 0 "abs(x^2) < 20" "6#2-%$absG6#*$%\"xG\"\"#\"#?" }{TEXT -1 236 ", then the absolute value of the ratio of successive terms bey ond the first term is always less than 1. This means that the series o btained by omitting the first term of the Maclaurin series converges, \+ and so the whole series converges." }}{PARA 0 "" 0 "" {TEXT -1 85 "We \+ now try to generalise this argument so that it applies for an arbitrar y value for " }{TEXT 287 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 292 1 "n" } {TEXT -1 37 " th term of the Maclaurin series for " }{XPPEDIT 18 0 "si n*x" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "u[n] = (-1)^(n+1)*x^(2*n-1)/(2*n-1)!;" "6#/&%\"uG6#%\"nG*(),$\"\"\"!\"\",& F'F+F+F+F+)%\"xG,&*&\"\"#F+F'F+F+F+F,F+-%*factorialG6#,&*&F2F+F'F+F+F+ F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Replacing " } {TEXT 288 1 "n" }{TEXT -1 3 " by" }{XPPEDIT 18 0 " ``(n+1)" "6#-%!G6#, &%\"nG\"\"\"F(F(" }{TEXT -1 21 " in this formula, the" }{XPPEDIT 18 0 " ``(n+1)" "6#-%!G6#,&%\"nG\"\"\"F(F(" }{TEXT -1 13 "st term is " } {XPPEDIT 18 0 "u[n+1] = (-1)^(n+2)*x^(2*n+1)/(2*n+1)!;" "6#/&%\"uG6#,& %\"nG\"\"\"F)F)*(),$F)!\"\",&F(F)\"\"#F)F))%\"xG,&*&F/F)F(F)F)F)F)F)-% *factorialG6#,&*&F/F)F(F)F)F)F)F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The ratio of these two su ccessive terms is " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "u[n+1]/u[n] = ` `((-1)^(n+2)*x^(2*n+1)/(2*n+1)!)/``((-1)^(n+1)*x^(2*n-1)/(2*n-1)!);" " 6#/*&&%\"uG6#,&%\"nG\"\"\"F*F*F*&F&6#F)!\"\"*&-%!G6#*(),$F*F-,&F)F*\" \"#F*F*)%\"xG,&*&F6F*F)F*F*F*F*F*-%*factorialG6#,&*&F6F*F)F*F*F*F*F-F* -F06#*(),$F*F-,&F)F*F*F*F*)F8,&*&F6F*F)F*F*F*F-F*-F<6#,&*&F6F*F)F*F*F* F-F-F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "``((-1)^(n+2)*x^(2*n+1)/(2*n+1 )!)*``((2*n-1)!/((-1)^(n+1)*x^(2*n-1)));" "6#*&-%!G6#*(),$\"\"\"!\"\", &%\"nGF*\"\"#F*F*)%\"xG,&*&F.F*F-F*F*F*F*F*-%*factorialG6#,&*&F.F*F-F* F*F*F*F+F*-F%6#*&-F46#,&*&F.F*F-F*F*F*F+F**&),$F*F+,&F-F*F*F*F*)F0,&*& F.F*F-F*F*F*F+F*F+F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -x^2*(2*n-1)!/ (2*n+1)!;" "6#/%!G,$*(%\"xG\"\"#-%*factorialG6#,&*&F(\"\"\"%\"nGF.F.F. !\"\"F.-F*6#,&*&F(F.F/F.F.F.F.F0F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= - x^2/(2*n*(2*n+1))" "6#/%!G,$*&%\"xG\"\"#*(F(\"\"\"%\"nGF*,&*&F(F*F+F*F *F*F*F*!\"\"F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "This calculation can be performed using M aple, either by setting up \"" }{TEXT 293 1 "u" }{TEXT -1 34 "\" as an indexed variable (a Maple " }{TEXT 0 5 "table" }{TEXT -1 22 " data st ructure) . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "u := 'u':\nu[n] := (-1)^(n+1)*x^(2*n-1)/(2*n-1) !;\nu[n+1] := subs(n=n+1,u[n]);\nu[n+1]/u[n];\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#%\"nG*&*&)!\"\",&F'\"\"\"F-F-F-)%\" xG,&F'\"\"#F-F+F-F--%*factorialG6#F0F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#,&%\"nG\"\"\"F)F)*&*&)!\"\",&F(F)\"\"#F)F))%\"xG,&F(F/F )F)F)F)-%*factorialG6#F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*()!\" \",&%\"nG\"\"\"\"\"#F)F))%\"xG,&F(F*F)F)F)-%*factorialG6#,&F(F*F)F&F)F )*(-F/6#F-F))F&,&F(F)F)F)F))F,F1F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*&*$)%\"xG\"\"#\"\"\"F)*&%\"nGF),&F+F(F)F)F)!\"\"#F-F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " . . . or by se tting up \"" }{TEXT 293 1 "u" }{TEXT -1 22 "\" as a function . . . " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "u := 'u':\nu := n -> (-1)^(n+1)*x^(2*n-1)/(2*n-1)!;\nu(n+1);\nu( n+1)/u(n);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6# %\"nG6\"6$%)operatorG%&arrowGF(*&*&)!\"\",&9$\"\"\"F2F2F2)%\"xG,&F1\" \"#F2F/F2F2-%*factorialG6#F5F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*&*&)!\"\",&%\"nG\"\"\"\"\"#F)F))%\"xG,&F(F*F)F)F)F)-%*factorialG6#F -F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*()!\"\",&%\"nG\"\"\"\"\"#F)F ))%\"xG,&F(F*F)F)F)-%*factorialG6#,&F(F*F)F&F)F)*(-F/6#F-F))F&,&F(F)F) F)F))F,F1F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$)%\"xG\"\"#\"\" \"F)*&%\"nGF),&F+F(F)F)F)!\"\"#F-F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "n = 6" "6#/%\"nG\"\"'" }{TEXT -1 10 ", we get " }{XPPEDIT 18 0 "u[13]/u[12] = -x^2/156;" "6# /*&&%\"uG6#\"#8\"\"\"&F&6#\"#7!\"\",$*&%\"xG\"\"#\"$c\"F-F-" }{TEXT -1 55 ", which is the last member of the list of ratios above." }} {PARA 0 "" 0 "" {TEXT -1 17 "For any value of " }{TEXT 283 1 "x" } {TEXT -1 58 ", the magnitude of the ratio of successive terms, namely \+ " }{XPPEDIT 18 0 "abs(x^2)/(2*n*(2*n+1))" "6#*&-%$absG6#*$%\"xG\"\"# \"\"\"*(F)F*%\"nGF*,&*&F)F*F,F*F*F*F*F*!\"\"" }{TEXT -1 13 ", tends to 0." }}{PARA 0 "" 0 "" {TEXT -1 34 "This means that, for any value of \+ " }{TEXT 284 1 "x" }{TEXT -1 24 ", we can choose a value " }{TEXT 289 1 "N" }{TEXT -1 4 " of " }{TEXT 285 1 "n" }{TEXT -1 33 ", which is lar ge enough so that " }{XPPEDIT 18 0 "abs(x^2)/(2*N*(2*N+1)) < 1;" "6#2 *&-%$absG6#*$%\"xG\"\"#\"\"\"*(F*F+%\"NGF+,&*&F*F+F-F+F+F+F+F+!\"\"F+ " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 61 "Then the tail of the Maclaurin series consisting of the term " }{XPPEDIT 18 0 "u[N]=(-1)^( N+1)*x^(2*N-1)/(2*N-1)!" "6#/&%\"uG6#%\"NG*(),$\"\"\"!\"\",&F'F+F+F+F+ )%\"xG,&*&\"\"#F+F'F+F+F+F,F+-%*factorialG6#,&*&F2F+F'F+F+F+F,F," } {TEXT -1 176 ", and all subsequent terms, converges. This follows from the fact that the terms of this tail of the series tend to 0 faster t han those of a geometric series with common ratio " }{TEXT 286 1 "r" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "abs(x^2)/(2*N*(2*N+1)) < 1;" "6#2*&-% $absG6#*$%\"xG\"\"#\"\"\"*(F*F+%\"NGF+,&*&F*F+F-F+F+F+F+F+!\"\"F+" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Adding in the finite sum of the of the terms preceding " }{XPPEDIT 18 0 "u[N]" "6#&%\"uG6#%\"NG" }{TEXT -1 45 ", we see that th e Maclaurin series converges." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "x = 10 " "6#/%\"xG\"#5" }{TEXT -1 14 ", we can take " }{XPPEDIT 18 0 "N = 5" "6#/%\"NG\"\"&" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "abs(x^2)/(2* N*(2*N+1))" "6#*&-%$absG6#*$%\"xG\"\"#\"\"\"*(F)F*%\"NGF*,&*&F)F*F,F*F *F*F*F*!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "10/11 < 1;" "6#2*&\"#5 \"\"\"\"#6!\"\"F&" }{TEXT -1 30 ", and the Maclaurin series is:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "10-10^3/6+10^5/120-10^7/5040+10^9/362880-10^11/39916800 +10^13/6227020800-10^15/1307674368000+` . . . `;" "6#,4\"#5\"\"\"*&F$ \"\"$\"\"'!\"\"F)*&F$\"\"&\"$?\"F)F%*&F$\"\"(\"%S]F)F)*&F$\"\"*\"'!)GO F)F%*&F$\"#6\")+o\"*RF)F)*&F$\"#8\"++3-FiF)F%*&F$\"#:\".+!oVn28F)F)%(~ .~.~.~GF%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 33 "After the first 4 terms, the tail" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "10^9/362880-10^11/39916800+10^13/ 6227020800-10^15/1307674368000+` . . .`;" "6#,,*&\"#5\"\"*\"'!)GO!\"\" \"\"\"*&F%\"#6\")+o\"*RF(F(*&F%\"#8\"++3-FiF(F)*&F%\"#:\".+!oVn28F(F(% '~.~.~.GF)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 93 "has terms \+ which tend to zero more rapidly than those of a geometric series with \+ common ratio " }{XPPEDIT 18 0 "10/11;" "6#*&\"#5\"\"\"\"#6!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "(10^11/39916800)/(10^9/362880);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#5\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 41 "Absolute convergence using the ratio test" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "A formal way to de al with the argument in the previous subsection is to apply the " } {TEXT 259 10 "ratio test" }{TEXT -1 17 " for convergence." }}{PARA 0 " " 0 "" {TEXT -1 125 "This also involves using an expression for the ge neral term to construct an expression for the ratio of two successive \+ terms." }}{PARA 0 "" 0 "" {TEXT -1 114 "However, the comparison with g eometric series is taken care of automatically in the application of t he ratio test." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 94 "u := n -> (-1)^(n+1)*x^(2*n-1)/(2*n-1)!;\nLimi t(u(n+1)/u(n),n=infinity);\nsimplify(%);\nvalue(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"uGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&*&)!\"\",& 9$\"\"\"F2F2F2)%\"xG,&F1\"\"#F2F/F2F2-%*factorialG6#F5F/F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&*()!\"\",&%\"nG\"\"\"\"\" #F,F,)%\"xG,&F+F-F,F,F,-%*factorialG6#,&F+F-F,F)F,F,*(-F26#F0F,)F),&F+ F,F,F,F,)F/F4F,F)/F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%& LimitG6$,$*&*$)%\"xG\"\"#\"\"\"F,*&%\"nGF,,&F.F+F,F,F,!\"\"#F0F+/F.%)i nfinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Since, independently of \+ " }{TEXT 294 1 "x" }{TEXT -1 114 ", the value of the limit is 0, which is certainly less than 1, the ratio test shows that the Maclaurin ser ies for " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 29 " converges for all values of " }{TEXT 295 1 "x" }{TEXT -1 42 ". In fact the series converges absolutely." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "a := n -> (-1)^(n+1)*x^ (2*n-1)/(2*n-1)!;\nLimit(abs(a(n+1))/abs(a(n)),n=infinity);\nsimplify( %);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$ %)operatorG%&arrowGF(*&*&)!\"\",&9$\"\"\"F2F2F2)%\"xG,&F1\"\"#F/F2F2F2 -%*factorialG6#F5F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG 6$*&-%$absG6#*&)%\"xG,&%\"nG\"\"#\"\"\"F0F0-%*factorialG6#F-!\"\"F0-F( 6#*&)F,,&F.F/F4F0F0-F26#F9F4F4/F.%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$-%$absG6#*&*$)%\"xG\"\"#\"\"\"F/*&%\"nGF/, &F1F.F/F/F/!\"\"#F/F./F1%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 72 ": The discussi on in this section does not show the Maclaurin series for " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 23 " actually converg es to " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 119 "However, the very nature of the met hod of construction of the series would certainly seem to suggest that this is true." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 65 "Some \+ standard Maclaurin series and their intervals of convergence" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 113 "In each case the interval of convergence is given. In most cas es this can be verified by applying the ratio test." }}{PARA 0 "" 0 " " {TEXT -1 108 "It turns out that, in each case, the given Maclaurin s eries converges to the value of the original function." }}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x) = 1+x+x^ 2/2!+x^3/3!+x^4/4!+x^5/5!+` . . . `" "6#/-%$expG6#%\"xG,0\"\"\"F)F'F)* &F'\"\"#-%*factorialG6#F+!\"\"F)*&F'\"\"$-F-6#F1F/F)*&F'\"\"%-F-6#F5F/ F)*&F'\"\"&-F-6#F9F/F)%(~.~.~.~GF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ Sum(x^n/n!,n = 0 .. infinity);" "6#/%!G-%$SumG6$*&)%\"xG%\"nG\"\"\"-%* factorialG6#F+!\"\"/F+;\"\"!%)infinityG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "-infinity " 0 " " {MPLTEXT 1 0 28 "exp(x)=taylor(exp(x),x=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"xG+5F'\"\"\"\"\"!F)F)#F)\"\"#F,#F)\"\"'\" \"$#F)\"#C\"\"%#F)\"$?\"\"\"&#F)\"$?(F.#F)\"%S]\"\"(-%\"OG6#F)\"\")" } }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 262 36 "_________________ ___________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "Test for convergence" }}{PARA 0 "" 0 "" {TEXT -1 8 "We \+ find " }{XPPEDIT 18 0 "Limit(abs(u[n+1])/abs(u[n]),n = infinity) = Lim it(abs(u[n+1]/u[n]),n = infinity)" "6#/-%&LimitG6$*&-%$absG6#&%\"uG6#, &%\"nG\"\"\"F0F0F0-F)6#&F,6#F/!\"\"/F/%)infinityG-F%6$-F)6#*&&F,6#,&F/ F0F0F0F0&F,6#F/F5/F/F7" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u[n] = \+ x^n/n!;" "6#/&%\"uG6#%\"nG*&)%\"xGF'\"\"\"-%*factorialG6#F'!\"\"" } {TEXT -1 60 ", in order to apply the ratio test for absolute convergen ce." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "u := n -> x^n/n!;\nu(n+1)/u(n);\nsimplify(%);\nLimit( abs(%),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" uGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&)%\"xG9$\"\"\"-%*factorialG6#F/ !\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&)%\"xG,&%\"nG\"\"\" F)F)F)-%*factorialG6#F(F)F)*&-F+6#F'F))F&F(F)!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&%\"xG\"\"\",&%\"nGF%F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%$absG6#*&%\"xG\"\"\",&%\"nGF+F+F+!\"\"/F-% )infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "This means that the Mac laurin series for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 45 " converges absolutely for all real values of " }{TEXT 312 1 "x " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The interval of conve rgence is " }{XPPEDIT 18 0 "-infinity " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Gr aphical illustration" }}{PARA 0 "" 0 "" {TEXT -1 37 "The first picture shows the graph of " }{XPPEDIT 18 0 "y = exp(x)" "6#/%\"yG-%$expG6#% \"xG" }{TEXT -1 125 ", which is shown in red, along with the graphs of some polynomials obtained by truncating the corresponding Maclaurin s eries." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "m := 6:\nplot([exp(x),seq(convert(taylor(exp(x),x,i) ,polynom),i=2..m)],\nx=-3..3,y=-2..10,color=[red,seq(COLOR(HUE,i/(m+2) ),i=2..m)],\nthickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%'CURVESG6$7V7$$!\"$\"\"!$\"1$R'yOoqy\\!#<7$$!1+++vq@p G!#:$\"1,!*)o\\LVn&F-7$$!1++D^NUbFF1$\"1XlPp$3#ejF-7$$!1++]K3XFEF1$\"1 4#R.nViA(F-7$$!1++]F)H')\\#F1$\"1IY*=ja(>#)F-7$$!1++D'3@/P#F1$\"1/Bh6r 8W$*F-7$$!1++Dr^b^AF1$\"1_04\"QaB0\"!#;7$$!1++D,kZG@F1$\"1u^1A\\=!>\"F M7$$!1++Dh\")=,?F1$\"1tt(eyXFM7$$!1+++0)H% *\\\"F1$\"1/HHTUdKAFM7$$!1+++vl[p8F1$\"1I?;zXPUDFM7$$!1+++&>iUC\"F1$\" 1M)HwIM:)GFM7$$!1++DhkaI6F1$\"19X)[zn&GKFM7$$!1+++]XF`**FM$\"1!*[tlR-' p$FM7$$!1++++Az2))FM$\"1NLPs(=y()FM7$$!1b+++v`hH!#=$\"1oDEN%G/(**FM7$$\"1++](QIKH\"FM$ \"1***G:qd!Q6F17$$\"1****\\7:xWCFM$\"1i+5&[`pF\"F17$$\"1,++vuY)o$FM$\" 1%)oYtf1Y9F17$$\"1)******4FL(\\FM$\"1&)>f7&HVk\"F17$$\"1)****\\d6.B'FM $\"1rX*RHrX'=F17$$\"1++](o3lW(FM$\"1%Q9)Qhq0@F17$$\"1*****\\A))oz)FM$ \"1^rQ0'\\,T#F17$$\"1+++Ik-,5F1$\"113!pQt5s#F17$$\"1+++D-eI6F1$\"1z`u4$F17$$\"1++v=_(zC\"F1$\"1ynSEHG$[$F17$$\"1+++b*=jP\"F1$\"1??'Gt'Hg RF17$$\"1++v3/3(\\\"F1$\"1=db_MioWF17$$\"1++vB4JB;F1$\"1'=I/W[)p]F17$$ \"1+++DVsYw7#F1$\"1*e*)zGb[R)F17$$\"1++v)Q?Q D#F1$\"1JhP-?0C&*F17$$\"1+++5jypBF1$\"1@wDg1^p5!#97$$\"1++]Ujp-DF1$\"1 $=shkQ:A\"Fby7$$\"1+++gEd@EF1$\"1RVI?Utv8Fby7$$\"1+]PMh%\\o#F1$\"1-\\$ G=TdY\"Fby7$$\"1++v3'>$[FF1$\"1&GC:#ojh:Fby7$$\"1+++5h(*3GF1$\"1*3;$>? 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\"1+++++]P;Ffy-Fafl6$Fcfl#\"\"&F]amFf\\lFfflF\\gl7&F'-F(6%7U7$F,$!1*** **********\\'FQ7$F3$!1&*R'Qc'pj[FQ7$F9$!1`0W73?xOFQ7$F>$!13Tv`gQlDFQ7$ FC$!1zyO0.xW;FQ7$FH$!1I?CNf#*)*))F17$FM$!1M/1wN+cIF17$FS$\"1HN\\7#3D2# F17$FX$\"1Ik\"\\tPqi'F17$Fgn$\"157$G)p^i5FQ7$F\\o$\"10Li6OgP9FQ7$Fao$ \"1fxMo.;_FF57$F`v$\"102GV'3S4$F57$Fev$\"1\"[gveJpZ$F57$Fjv$\"16mL! eC'[RF57$F_w$\"1yb[U\\*)[WF57$Fdw$\"1#[nd(z3P]F57$Fiw$\"1#)Git.%Qo&F57 $F^x$\"1ykVP'R`W'F57$Fcx$\"1<9BY\\@psF57$Fhx$\"1&zZs%)RL@)F57$F]y$\"1. g2eddh#*F57$Fby$\"1Gl]b>FL5Ffy7$Fhy$\"12k,1!))*p6Ffy7$F]z$\"1#4=T'y118 Ffy7$Fgz$\"1Er#\\(Q&oY\"Ffy7$F\\[l$\"1*frqX^*\\:Ffy7$Fa[l$\"1dzFs9EP;F fy7$Ff[l$\"1y&)[eX)ft\"Ffy7$F[\\l$\"1++++++S=Ffy-Fafl6$Fcfl#F\\\\lF_fl Ff\\lFfflF\\gl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 43.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Numerical examples" }}{PARA 0 "" 0 "" {TEXT -1 98 "By tri al and error we can find the number of terms needed to obtain about 10 figure accuracy when " }{XPPEDIT 18 0 "x = 10" "6#/%\"xG\"#5" }{TEXT -1 10 " and when " }{XPPEDIT 18 0 "x = -10" "6#/%\"xG,$\"#5!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "xx := 10;\nSum(xx^i/i!,i=0..35);\nevalf(%);\neva lf(exp(xx));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)\"#5%\"iG\"\"\"-%*factorialG6#F) !\"\"/F);\"\"!\"#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zlk-A!\"&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zlk-A!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "In the case " }{XPPEDIT 18 0 "x = -10" "6#/%\"xG,$\"#5!\"\"" }{TEXT -1 129 ", it is necessary \+ to increase the precision for the calculation because of the occurrenc e of subtraction of nearly equal numbers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "xx := -10;\nSum(xx^i/ i!,i=0..49);\nevalf(evalf(%,20));\nevalf(exp(xx));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#xxG!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6 $*&)!#5%\"iG\"\"\"-%*factorialG6#F)!\"\"/F);\"\"!\"#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+wH**RX!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+wH**RX!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"% \"xGF%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x=x-x^3/3!+x^5/5!-x^7/7!+x^9/9!-x^11/11!+` . . . `" "6#/*&%$ sinG\"\"\"%\"xGF&,0F'F&*&F'\"\"$-%*factorialG6#F*!\"\"F.*&F'\"\"&-F,6# F0F.F&*&F'\"\"(-F,6#F4F.F.*&F'\"\"*-F,6#F8F.F&*&F'\"#6-F,6#F " 0 "" {MPLTEXT 1 0 29 "sin(x) =taylor(sin(x),x=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#% \"xG+/F'\"\"\"F)#!\"\"\"\"'\"\"$#F)\"$?\"\"\"&#F+\"%S]\"\"(#F)\"'!)GO \"\"*-%\"OG6#F)\"#5" }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 263 33 "_________________________________" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Test for convergence" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "We find " } {XPPEDIT 18 0 "Limit(abs(u[n+1])/abs(u[n]),n = infinity) = Limit(abs(u [n+1]/u[n]),n = infinity)" "6#/-%&LimitG6$*&-%$absG6#&%\"uG6#,&%\"nG\" \"\"F0F0F0-F)6#&F,6#F/!\"\"/F/%)infinityG-F%6$-F)6#*&&F,6#,&F/F0F0F0F0 &F,6#F/F5/F/F7" }{TEXT -1 9 ", where " }{XPPEDIT 18 0 "u[n] = (-1)^(n +1)*x^(2*n-1)/(2*n-1)!;" "6#/&%\"uG6#%\"nG*(),$\"\"\"!\"\",&F'F+F+F+F+ )%\"xG,&*&\"\"#F+F'F+F+F+F,F+-%*factorialG6#,&*&F2F+F'F+F+F+F,F," } {TEXT -1 60 ", in order to apply the ratio test for absolute convergen ce." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "u := n -> (-1)^(n+1)*x^(2*n-1)/(2*n-1)!;\nu(n+1)/u(n );\nsimplify(%);\nLimit(abs(%),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&*&)!\" \",&9$\"\"\"F2F2F2)%\"xG,&F1\"\"#F2F/F2F2-%*factorialG6#F5F/F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&*()!\"\",&%\"nG\"\"\"\"\"#F)F))%\"xG ,&F(F*F)F)F)-%*factorialG6#,&F(F*F)F&F)F)*(-F/6#F-F))F&,&F(F)F)F)F))F, F1F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$)%\"xG\"\"#\"\"\"F)*&% \"nGF),&F+F(F)F)F)!\"\"#F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&Lim itG6$,$-%$absG6#*&*$)%\"xG\"\"#\"\"\"F/*&%\"nGF/,&F1F.F/F/F/!\"\"#F/F. /F1%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "This means that th e Maclaurin series for " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\"xG F%" }{TEXT -1 45 " converges absolutely for all real values of " } {TEXT 296 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The int erval of convergence is " }{XPPEDIT 18 0 "-infinity " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 22 "Graphical illustration" }}{PARA 0 "" 0 "" {TEXT -1 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7$F^\\m$\"1wH)Hz@M.\"F17$Fh\\m$\"1@Q(QBdfz)F17$F^ao$\"1J(>mN(=c9F57$F] ]m$\"1<[/^)3T<#F57$Fb]m$\"1B5PJXV=IF57$Fg]m$\"1)[Ew4C[/%F57$$\"1+](o\" *[W!eF5$\"1%R2Bq%y#o%F57$F\\^m$\"1T'H\"yDY)Q&F57$$\"1,]i0j\"[$fF5$\"1- J$=a@\"ohF57$Fa^m$\"1D9dG9dGqF5-Febm6$Fgbm#\"\"&Fb^mF\\_mFjbmF`cm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF %" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " cos*x=1-x^2/2!+x^4/4!-x^6/6!+x^8/8!-x^10/10!+` . . . `" "6#/*&%$cosG\" \"\"%\"xGF&,0F&F&*&F'\"\"#-%*factorialG6#F*!\"\"F.*&F'\"\"%-F,6#F0F.F& *&F'\"\"'-F,6#F4F.F.*&F'\"\")-F,6#F8F.F&*&F'\"#5-F,6#F " 0 "" {MPLTEXT 1 0 29 "cos(x)= taylor(cos(x),x=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#% \"xG+/F'\"\"\"\"\"!#!\"\"\"\"#F-#F)\"#C\"\"%#F,\"$?(\"\"'#F)\"&?.%\"\" )-%\"OG6#F)\"#5" }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 264 31 "_______________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Test for convergence" }}{PARA 0 "" 0 " " {TEXT -1 9 "We find " }{XPPEDIT 18 0 "Limit(abs(u[n+1])/abs(u[n]),n = infinity) = Limit(abs(u[n+1]/u[n]),n = infinity)" "6#/-%&LimitG6$*& -%$absG6#&%\"uG6#,&%\"nG\"\"\"F0F0F0-F)6#&F,6#F/!\"\"/F/%)infinityG-F% 6$-F)6#*&&F,6#,&F/F0F0F0F0&F,6#F/F5/F/F7" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "u[n] = (-1)^n*x^(2*n)/(2*n)!;" "6#/&%\"uG6#%\"nG*(),$\" \"\"!\"\"F'F+)%\"xG*&\"\"#F+F'F+F+-%*factorialG6#*&F0F+F'F+F," }{TEXT -1 60 ", in order to apply the ratio test for absolute convergence." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "u := n -> (-1)^n*x^(2*n)/(2*n)!;\nu(n+1)/u(n);\nsimplify(%);\nLi mit(abs(%),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"uGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&*&)!\"\"9$\"\"\")%\"xG,$F0 \"\"#F1F1-%*factorialG6#F4F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&*()!\"\",&%\"nG\"\"\"F)F)F))%\"xG,&F(\"\"#F-F)F)-%*factorialG6#,$F(F -F)F)*(-F/6#F,F))F&F(F))F+F1F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&*$)%\"xG\"\"#\"\"\"F)*&,&%\"nGF(F)F)F),&F,F)F)F)F)!\"\"#F.F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$-%$absG6#*&*$)%\"xG\"\"# \"\"\"F/*&,&%\"nGF.F/F/F/,&F2F/F/F/F/!\"\"#F/F./F2%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "This means that the Maclaurin series \+ for " }{XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 45 " \+ converges absolutely for all real values of " }{TEXT 297 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The interval of convergence is " }{XPPEDIT 18 0 "-infinity \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "sinh*x" "6#*&%%sinhG\" \"\"%\"xGF%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sinh*x = x+x^3/3!+x^5/5!+x^7/7!+x^9/9!+x^11/11!+` . . . `;" "6#/*&%%sinhG\"\"\"%\"xGF&,0F'F&*&F'\"\"$-%*factorialG6#F*!\"\"F& *&F'\"\"&-F,6#F0F.F&*&F'\"\"(-F,6#F4F.F&*&F'\"\"*-F,6#F8F.F&*&F'\"#6-F ,6#F " 0 "" {MPLTEXT 1 0 31 "sinh(x)=taylor(sinh(x),x=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6#%\"xG+/F'\"\"\"F)#F)\"\"'\"\"$#F)\"$?\"\"\"& #F)\"%S]\"\"(#F)\"'!)GO\"\"*-%\"OG6#F)\"#5" }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 265 35 "__________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Test for convergence" }}{PARA 0 "" 0 "" {TEXT -1 9 "We find " }{XPPEDIT 18 0 "Limit(abs(u[n +1])/abs(u[n]),n = infinity) = Limit(abs(u[n+1]/u[n]),n = infinity)" " 6#/-%&LimitG6$*&-%$absG6#&%\"uG6#,&%\"nG\"\"\"F0F0F0-F)6#&F,6#F/!\"\"/ F/%)infinityG-F%6$-F)6#*&&F,6#,&F/F0F0F0F0&F,6#F/F5/F/F7" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u[n] = x^(2*n-1)/(2*n-1)!;" "6#/&%\"uG6#%\" nG*&)%\"xG,&*&\"\"#\"\"\"F'F.F.F.!\"\"F.-%*factorialG6#,&*&F-F.F'F.F.F .F/F/" }{TEXT -1 60 ", in order to apply the ratio test for absolute c onvergence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 91 "u := n -> x^(2*n-1)/(2*n-1)!;\nu(n+1)/u(n);\nsimpli fy(%);\nLimit(abs(%),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&)%\"xG,&9$\" \"#\"\"\"!\"\"F2-%*factorialG6#F/F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&)%\"xG,&%\"nG\"\"#\"\"\"F*F*-%*factorialG6#,&F(F)F* !\"\"F*F**&-F,6#F'F*)F&F.F*F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&* $)%\"xG\"\"#\"\"\"F)*&%\"nGF),&F+F(F)F)F)!\"\"#F)F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%&LimitG6$,$-%$absG6#*&*$)%\"xG\"\"#\"\"\"F/*&%\"nGF /,&F1F.F/F/F/!\"\"#F/F./F1%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "This means that the Maclaurin series for " }{XPPEDIT 18 0 "sinh*x; " "6#*&%%sinhG\"\"\"%\"xGF%" }{TEXT -1 45 " converges absolutely for a ll real values of " }{TEXT 313 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 31 "The interval of convergence is " }{XPPEDIT 18 0 "-infin ity " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Graphical illustration" }}{PARA 0 "" 0 "" {TEXT -1 37 "The first picture shows the graph of " } {XPPEDIT 18 0 "y = sinh*x;" "6#/%\"yG*&%%sinhG\"\"\"%\"xGF'" }{TEXT -1 125 ", which is shown in red, along with the graphs of some polynom ials obtained by truncating the corresponding Maclaurin series." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "m := 4:\nplot([sinh(x),seq(convert(taylor(sinh(x),x,2*i),polynom) ,\ni=1..m)],x=-5..5,y=-40..40,\ncolor=[red,seq(COLOR(HUE,2*i/(2*m+2)), i=1..m)],\nthickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 345 301 301 {PLOTDATA 2 "6*-%'CURVESG6$7W7$$!\"&\"\"!$!1v)yx0@.U(!#97$$!1mm;HU,\"* 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F17$Fav$\"1()e24\"=sb#F17$Ffv$\"1h)Q$oQ09KF17$F[w$\"1())f(HC^PRF17$F`w $\"14Ho\"Rl5!\\F17$Few$\"1t*pc8)\\4gF17$Fjw$\"1r=$4*)=ZU(F17$F_x$\"1j< %)G=_<\"*F17$Fdx$\"1*H'G1\"[*G6F-7$Fix$\"1$\\yk/-^Q\"F-7$F^y$\"1#ev.ao [q\"F-7$Fcy$\"12\"3U/e;4#F-7$Fhy$\"1?NCY&)H?DF-7$F]z$\"19/%o$pu9JF-7$F bz$\"1P[c<*=tv$F-7$Fgz$\"1\\Bw)R!ezXF-7$Fa[l$\"1G2MZ?%G_&F-7$F[\\l$\"1 1#\\j?*fPnF--F[`l6$F]`l#\"\"%F\\\\l-%+AXESLABELSG6$Q\"x6\"Q\"yF\\]n-%* THICKNESSG6#Fiil-%%VIEWG6$;F(F[\\l;$!#SF*$\"#SF*" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "The same graphs can be used to construct an animation in which single polynomials of progressively higher degree are shown alo ng with the graph of " }{XPPEDIT 18 0 "y = sinh*x;" "6#/%\"yG*&%%sinhG \"\"\"%\"xGF'" }{TEXT -1 24 ", which is shown in red." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "m := 4; \nfrms:=seq(plot([sinh(x),\nconvert(taylor(sinh(x),x,2*i),polynom)],\n x=-5..5,y=-40..40,color=[red,COLOR(HUE,2*i/(2*m+2))],\nthickness=2),i= 1..m):\nplots[display](frms,insequence=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"%" }}{PARA 13 "" 1 "" {GLPLOT2D 364 320 320 {PLOTDATA 2 "6#-%(ANIMATEG6&7&-%'CURVESG6%7W7$$!\"&\"\"!$!1v)yx0@.U(!# 97$$!1mm;HU,\"*[!#:$!1n&fDq[Sl'F17$$!1LLLe%G?y%F5$!1'4JOs!*o'fF17$$!1+ ](=_+so%F5$!1QF'42$)pU&F17$$!1mmT&esBf%F5$!1zNgiK\"f$\\F17$$!1LL$3s%3z VF5$!1mj)fzAw)RF17$$!1ML$e/$QkTF5$!1$)*fJSro@$F17$$!1nmT5=q]RF5$!1x\") zg$Hwf#F17$$!1LL3_>f_PF5$!1*H!yM!*RI@F17$$!1++vo1YZNF5$!1i)z2Z6[t\"F17 $$!1LL3-OJNLF5$!1a%[7LzDS\"F17$$!1++v$*o%Q7$F5$!1$H^2!R[M6F17$$!1mmm\" RFj!HF5$!1:ogeVT<\"*F57$$!1LL$e4OZr#F5$!1'\\61+,s^(F57$$!1+++v'\\!*\\# F5$!1Zh=W&zV/'F57$$!1+++DwZ#G#F5$!1JImX6X\\[F57$$!1+++D.xt?F5$!1-&>#zQ _9RF57$$!1LL3-TC%)=F5$!1ftv?pr9KF57$$!1mmm\"4z)e;F5$!1Y\\%pNL:`#F57$$! 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1X^@DONIYF17$FH$!1!y*eqj9#z$F17$FM$!1&Q>'y%GZ4$F17$FR$!1S$ocJhG_#F17$F W$!1B`NT_-%3#F17$Ffn$!1YHGptC2UY'Q@$F57$Fcq$!1onH>XEJDF57$ Fhq$!1vt1m;\"\\0#F57$F]r$!1D6l/f9%f\"F57$Fbr$!1eC\\gqu_7F57$Fgr$!1>*pB 72gL*Fir7$F]s$!1)3V;PSTq'Fir7$Fbs$!1*)y.$o'**eUFir7$Fgs$!1&f\"oeY.*=#F irF[t7$Fbt$\"1c!oY\\6@<#Fir7$Fgt$\"1)ojX42$)=%Fir7$F\\u$\"1#f)4Hd-UlFi r7$Fau$\"1[(eYbq6F*Fir7$Ffu$\"1J(\\KG4`B\"F57$F[v$\"10/\"QGF^e\"F57$F` v$\"1T+.MFw]?F57$Fev$\"1()e24\"=sb#F57$Fjv$\"1h)Q$oQ09KF57$F_w$\"1())f (HC^PRF57$Fdw$\"14Ho\"Rl5!\\F57$Fiw$\"1t*pc8)\\4gF57$F^x$\"1r=$4*)=ZU( F57$Fcx$\"1j<%)G=_<\"*F57$Fhx$\"1*H'G1\"[*G6F17$F]y$\"1$\\yk/-^Q\"F17$ Fby$\"1#ev.ao[q\"F17$Fgy$\"12\"3U/e;4#F17$F\\z$\"1?NCY&)H?DF17$Faz$\"1 9/%o$pu9JF17$Ffz$\"1P[c<*=tv$F17$F[[l$\"1\\Bw)R!ezXF17$Fe[l$\"1G2MZ?%G _&F17$F_\\l$\"11#\\j?*fPnF1-Fc`l6$Fe`l#\"\"%F`\\lFj\\lFh`lF^al" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "cosh*x" "6#*&%%coshG\"\"\"%\"xGF%" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh *x = 1+x^2/2!+x^4/4!+x^6/6!+x^8/8!+x^10/10!+` . . . `;" "6#/*&%%coshG \"\"\"%\"xGF&,0F&F&*&F'\"\"#-%*factorialG6#F*!\"\"F&*&F'\"\"%-F,6#F0F. F&*&F'\"\"'-F,6#F4F.F&*&F'\"\")-F,6#F8F.F&*&F'\"#5-F,6#F " 0 "" {MPLTEXT 1 0 31 "cosh(x)=taylor(cosh(x), x=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%coshG6#%\"xG+/F'\"\"\" \"\"!#F)\"\"#F,#F)\"#C\"\"%#F)\"$?(\"\"'#F)\"&?.%\"\")-%\"OG6#F)\"#5" }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 266 32 "________________ ________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "Test for convergence" }}{PARA 0 "" 0 "" {TEXT -1 9 "We \+ find " }{XPPEDIT 18 0 "Limit(abs(u[n+1])/abs(u[n]),n = infinity) = Li mit(abs(u[n+1]/u[n]),n = infinity)" "6#/-%&LimitG6$*&-%$absG6#&%\"uG6# ,&%\"nG\"\"\"F0F0F0-F)6#&F,6#F/!\"\"/F/%)infinityG-F%6$-F)6#*&&F,6#,&F /F0F0F0F0&F,6#F/F5/F/F7" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u[n] = x^(2*n)/(2*n)!;" "6#/&%\"uG6#%\"nG*&)%\"xG*&\"\"#\"\"\"F'F-F--%*facto rialG6#*&F,F-F'F-!\"\"" }{TEXT -1 60 ", in order to apply the ratio te st for absolute convergence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "u := n -> x^(2*n)/(2*n)!;\nu (n+1)/u(n);\nsimplify(%);\nLimit(abs(%),n=infinity);\nvalue(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"nG6\"6$%)operatorG%&arrow GF(*&)%\"xG,$9$\"\"#\"\"\"-%*factorialG6#F/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&)%\"xG,&%\"nG\"\"#F)\"\"\"F*-%*factorialG6#,$F (F)F*F**&-F,6#F'F*)F&F.F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*& *$)%\"xG\"\"#\"\"\"F)*&,&%\"nGF(F)F)F),&F,F)F)F)F)!\"\"#F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$-%$absG6#*&*$)%\"xG\"\"#\"\"\"F /*&,&%\"nGF.F/F/F/,&F2F/F/F/F/!\"\"#F/F./F2%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "This means that the Maclaurin series for " } {XPPEDIT 18 0 "cosh*x" "6#*&%%coshG\"\"\"%\"xGF%" }{TEXT -1 45 " conve rges absolutely for all real values of " }{TEXT 298 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The interval of convergence is " } {XPPEDIT 18 0 "-infinity " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "(1+x)^m" "6#),&\"\"\"F%%\"xGF% %\"mG" }{TEXT -1 24 " .. the binomial series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1+x) ^m=1+m*x+m*(m-1)/2!" "6#/),&\"\"\"F&%\"xGF&%\"mG,(F&F&*&F(F&F'F&F&*(F( F&,&F(F&F&!\"\"F&-%*factorialG6#\"\"#F-F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+m*(m-1)*(m-2)/3!" "6#,&*$%\"xG\"\"#\"\"\"**%\"mGF',&F)F'F'! \"\"F',&F)F'F&F+F'-%*factorialG6#\"\"$F+F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^3+m*(m-1)*(m-2)*(m-3)/4!" "6#,&*$%\"xG\"\"$\"\"\"*,%\"mGF',&F) F'F'!\"\"F',&F)F'\"\"#F+F',&F)F'F&F+F'-%*factorialG6#\"\"%F+F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x^4+` . . . `" "6#,&*$%\"xG\"\"%\"\"\"% (~.~.~.~GF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "`` = Sum(m!*x^n/(n!*(m-n)!),n = 0 .. infinity);" "6#/%! G-%$SumG6$*(-%*factorialG6#%\"mG\"\"\")%\"xG%\"nGF-*&-F*6#F0F--F*6#,&F ,F-F0!\"\"F-F7/F0;\"\"!%)infinityG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "- 1 < x;" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\" " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "(1+x)^m=taylor((1+x)^m,x=0,7);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/),&\"\"\"F&%\"xGF&%\"mG+3F'F&\"\"!F(F&,$*&F(F&, &F(F&!\"\"F&F&#F&\"\"#F0,$*(F(F&F-F&,&F(F&!\"#F&F&#F&\"\"'\"\"$,$**F(F &F-F&F3F&,&F(F&!\"$F&F&#F&\"#C\"\"%,$*,F(F&F-F&F3F&F:F&,&F(F&!\"%F&F&# F&\"$?\"\"\"&,$*.F(F&F-F&F3F&F:F&FAF&,&F(F&!\"&F&F&#F&\"$?(F6-%\"OG6#F &\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 27 "This series c onverges when " }{XPPEDIT 18 0 "abs(x)<1" "6#2-%$absG6#%\"xG\"\"\"" } {TEXT -1 20 ", and diverges when " }{XPPEDIT 18 0 "1 <= abs(x);" "6#1 \"\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 5 "W hen " }{TEXT 299 1 "m" }{TEXT -1 44 " is a positive integer the series is finite." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "(1+x)^4=taylor((1+x)^4,x=0,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&\"\"\"F'%\"xGF'\"\"%F'+-F(F'\"\"!F)F'\"\"'\"\"#F) \"\"$F'F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "m = -1" "6#/%\"mG,$\"\"\"!\"\"" }{TEXT -1 25 " the series is geometric." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "(1+x)^(-1)=taylor((1+x)^(-1) ,x=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\"\"F%,&F%F%%\"xGF%! \"\"+5F'F%\"\"!F(F%F%\"\"#F(\"\"$F%\"\"%F(\"\"&F%\"\"'F(\"\"(-%\"OG6#F %\"\")" }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 267 29 "________ _____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Test for convergence" }}{PARA 0 "" 0 "" {TEXT -1 9 " We find " }{XPPEDIT 18 0 "Limit(abs(u[n+1])/abs(u[n]),n = infinity) = Limit(abs(u[n+1]/u[n]),n = infinity)" "6#/-%&LimitG6$*&-%$absG6#&%\"u G6#,&%\"nG\"\"\"F0F0F0-F)6#&F,6#F/!\"\"/F/%)infinityG-F%6$-F)6#*&&F,6# ,&F/F0F0F0F0&F,6#F/F5/F/F7" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u[n ] = m!*x^n/(n!*(m-n)!);" "6#/&%\"uG6#%\"nG*(-%*factorialG6#%\"mG\"\"\" )%\"xGF'F-*&-F*6#F'F--F*6#,&F,F-F'!\"\"F-F6" }{TEXT -1 60 ", in order \+ to apply the ratio test for absolute convergence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "u := n -> m! *x^n/(n!*(m-n)!);\nu(n+1)/u(n);\nsimplify(%);\nLimit(abs(%),n=infinity );\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"nG6\"6$% )operatorG%&arrowGF(*&*&-%*factorialG6#%\"mG\"\"\")%\"xG9$F2F2*&-F/6#F 5F2-F/6#,&F1F2F5!\"\"F2F " 0 "" {MPLTEXT 1 0 64 "x*signum(x);\nassume(x_,real);\nsubs(x_=x,simplify(subs(x=x_,%))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"-%'signumG6#F$F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$absG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The ratio test shows that the binomial series converges absolutely when " }{XPPEDIT 18 0 "abs(x )<1" "6#2-%$absG6#%\"xG\"\"\"" }{TEXT -1 19 " and diverges when " } {XPPEDIT 18 0 "abs(x)>1" "6#2\"\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 38 "The series clearly also diverges when " } {XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 10 " and when " } {XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 31 "The interval of convergence is " } {XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1 ;" "6#2%!G\"\"\"" }{TEXT -1 35 ", and the radius of convergence is " } {XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "ln(1+x)" "6#-%#lnG6#,&\"\"\"F'%\"xGF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "ln(x+1)=x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/6+x^7/7+` . . . ` " "6#/-%#lnG6#,&%\"xG\"\"\"F)F),2F(F)*&F(\"\"#F,!\"\"F-*&F(\"\"$F/F-F) *&F(\"\"%F1F-F-*&F(\"\"&F3F-F)*&F(\"\"'F5F-F-*&F(\"\"(F7F-F)%(~.~.~.~G F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = Sum((-1)^n*x^(n+1)/(n+1),n = \+ 0 .. infinity);" "6#/%!G-%$SumG6$*(),$\"\"\"!\"\"%\"nGF+)%\"xG,&F-F+F+ F+F+,&F-F+F+F+F,/F-;\"\"!%)infinityG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G\" \"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ln(x+1)=t aylor(ln(x+1),x=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#,&\" \"\"F(%\"xGF(+3F)F(F(#!\"\"\"\"#F-#F(\"\"$F/#F,\"\"%F1#F(\"\"&F3#F,\" \"'F5#F(\"\"(F7-%\"OG6#F(\"\")" }}}{PARA 256 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 268 34 "__________________________________" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Test for convergence" }}{PARA 0 "" 0 "" {TEXT -1 9 "We find " }{XPPEDIT 18 0 "Limit(abs(u[n +1])/abs(u[n]),n = infinity) = Limit(abs(u[n+1]/u[n]),n = infinity)" " 6#/-%&LimitG6$*&-%$absG6#&%\"uG6#,&%\"nG\"\"\"F0F0F0-F)6#&F,6#F/!\"\"/ F/%)infinityG-F%6$-F)6#*&&F,6#,&F/F0F0F0F0&F,6#F/F5/F/F7" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u[n] = (-1)^n*x^(n+1)/(n+1);" "6#/&%\"uG6#% \"nG*(),$\"\"\"!\"\"F'F+)%\"xG,&F'F+F+F+F+,&F'F+F+F+F," }{TEXT -1 60 " , in order to apply the ratio test for absolute convergence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "u \+ := n -> (-1)^n*x^(n+1)/(n+1);\nu(n+1)/u(n);\nsimplify(%);\nLimit(abs(% ),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6 #%\"nG6\"6$%)operatorG%&arrowGF(*&*&)!\"\"9$\"\"\")%\"xG,&F0F1F1F1F1F1 F4F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*()!\"\",&%\"nG\"\"\"F )F)F))%\"xG,&F(F)\"\"#F)F)F'F)F)*(F,F))F&F(F))F+F'F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"xG\"\"\",&%\"nGF'F'F'F'F',&F)F'\"\"#F'!\" \"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%$absG6#*&*&%\"xG \"\"\",&%\"nGF,F,F,F,F,,&F.F,\"\"#F,!\"\"/F.%)infinityG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$absG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The ratio test shows that the Maclaurin series for " } {XPPEDIT 18 0 "ln(1+x)" "6#-%#lnG6#,&\"\"\"F'%\"xGF'" }{TEXT -1 27 " c onverges absolutely when " }{XPPEDIT 18 0 "abs(x)<1" "6#2-%$absG6#%\"x G\"\"\"" }{TEXT -1 20 ", and diverges when " }{XPPEDIT 18 0 "abs(x)>1 " "6#2\"\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "The series also converges to " }{XPPEDIT 18 0 "ln(2)" "6#-%#lnG 6#\"\"#" }{TEXT -1 6 ",when " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 61 ", by the alternating series test, but the series diverge s to " }{XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When \+ " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 20 ", the series b ecomes" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^(n -1)/n,n = 1 .. infinity) = 1-1/2+1/3-1/4+` . . . `;" "6#/-%$SumG6$*&), $\"\"\"!\"\",&%\"nGF*F*F+F*F-F+/F-;F*%)infinityG,,F*F**&F*F*\"\"#F+F+* &F*F*\"\"$F+F**&F*F*\"\"%F+F+%(~.~.~.~GF*" }{TEXT -1 1 "," }}{PARA 0 " " 0 "" {TEXT -1 47 "which converges by the alternating series test." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 " x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 20 ", the se ries becomes" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum( (-1)/n,n = 1 .. infinity) = -Sum(1/n,n = 1 .. infinity);" "6#/-%$SumG6 $*&,$\"\"\"!\"\"F)%\"nGF*/F+;F)%)infinityG,$-F%6$*&F)F)F+F*/F+;F)F.F* " }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -``(1+1/2+1/3+1/4+` . . . `);" "6# /%!G,$-F$6#,,\"\"\"F)*&F)F)\"\"#!\"\"F)*&F)F)\"\"$F,F)*&F)F)\"\"%F,F)% (~.~.~.~GF)F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 68 "which is the negative of the harmonic series, and so it diverges to " } {XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The inter val of convergence of the series " }{XPPEDIT 18 0 "Sum((-1)^n*x^(n+1)/ (n+1),n = 0 .. infinity)" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"nGF))%\"xG,&F +F)F)F)F),&F+F)F)F)F*/F+;\"\"!%)infinityG" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= \+ 1;" "6#1%!G\"\"\"" }{TEXT -1 37 ", and the radius of convergence is 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Sum((-1)^n/(n+1),n=0..infinity);\nvalue(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$SumG6$*&)!\"\"%\"nG\"\"\",&F)F*F*F*F(/F);\"\"!%)in finityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#\"\"#" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Sum(- 1/(n+1),n=0..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$SumG6$,$*&\"\"\"F(,&%\"nGF(F(F(!\"\"F+/F*;\"\"!%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 4 "Note" }{TEXT -1 61 ": This series can be obtained by integrating the series for " }{XPPEDIT 18 0 "1/(1+x)" "6#*&\"\"\"F$,&F$F$%\"xGF$!\"\"" }{TEXT -1 15 " term by t erm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(taylor(1/(1+x),x =0,7),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$+3%\" xG\"\"\"\"\"!!\"\"F(F(\"\"#F*\"\"$F(\"\"%F*\"\"&F(\"\"'-%\"OG6#F(\"\"( F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"F%#!\"\"\"\"#F(#F% \"\"$F*#F'\"\"%F,#F%\"\"&F.#F'\"\"'F0#F%\"\"(F2-%\"OG6#F%\"\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "arctan*x" "6#*&%'arct anG\"\"\"%\"xGF%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "arctan*x = x-x^3/3+x^5/5-x^7/7+x^9/9-x^11/11+` . . . ` " "6#/*&%'arctanG\"\"\"%\"xGF&,0F'F&*&F'\"\"$F*!\"\"F+*&F'\"\"&F-F+F&* &F'\"\"(F/F+F+*&F'\"\"*F1F+F&*&F'\"#6F3F+F+%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "`` = Sum((-1)^(n+1)*x^(2*n-1)/(2*n-1),n = 1 .. infin ity);" "6#/%!G-%$SumG6$*(),$\"\"\"!\"\",&%\"nGF+F+F+F+)%\"xG,&*&\"\"#F +F.F+F+F+F,F+,&*&F3F+F.F+F+F+F,F,/F.;F+%)infinityG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= \+ 1;" "6#1%!G\"\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "arctan(x)=taylor(arctan(x),x =0,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#%\"xG+1F'\"\" \"F)#!\"\"\"\"$F,#F)\"\"&F.#F+\"\"(F0#F)\"\"*F2#F+\"#6F4-%\"OG6#F)\"#7 " }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 269 33 "______________ ___________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "Test for convergence" }}{PARA 0 "" 0 "" {TEXT -1 8 "We \+ find " }{XPPEDIT 18 0 "Limit(abs(u[n+1])/abs(u[n]),n = infinity) = Lim it(abs(u[n+1]/u[n]),n = infinity);" "6#/-%&LimitG6$*&-%$absG6#&%\"uG6# ,&%\"nG\"\"\"F0F0F0-F)6#&F,6#F/!\"\"/F/%)infinityG-F%6$-F)6#*&&F,6#,&F /F0F0F0F0&F,6#F/F5/F/F7" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u[n] = (-1)^(n+1)*x^(2*n-1)/(2*n-1);" "6#/&%\"uG6#%\"nG*(),$\"\"\"!\"\",&F'F +F+F+F+)%\"xG,&*&\"\"#F+F'F+F+F+F,F+,&*&F2F+F'F+F+F+F,F," }{TEXT -1 60 ", in order to apply the ratio test for absolute convergence." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "u := n -> (-1)^(n+1)*x^(2*n-1)/(2*n-1);\nu(n+1)/u(n);\nsimplify(% );\nLimit(abs(%),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&*&)!\"\",&9$\"\"\"F2F 2F2)%\"xG,&F1\"\"#F2F/F2F2F5F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*&*()!\"\",&%\"nG\"\"\"\"\"#F)F))%\"xG,&F(F*F)F)F),&F(F*F)F&F)F)*(F- F))F&,&F(F)F)F)F))F,F.F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)% \"xG\"\"#\"\"\",&%\"nGF(F)!\"\"F)F),&F+F(F)F)F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%$absG6#*&*&)%\"xG\"\"#\"\"\",&%\"nGF-F.!\" \"F.F.,&F0F-F.F.F1/F0%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$ )-%$absG6#%\"xG\"\"#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 301 1 "x" }{TEXT -1 19 " is a real n umber, " }{XPPEDIT 18 0 "x^2*signum(x)^2 = x^2;" "6#/*&%\"xG\"\"#-%'si gnumG6#F%F&*$F%F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "x^2*signum(x)^2;\nassume(x_, real);\nsubs(x_=x,simplify(subs(x=x_,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG\"\"#\"\"\")-%'signumG6#F%F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"xG\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The ratio test shows that the Maclaurin series for " } {XPPEDIT 18 0 "arctan*x;" "6#*&%'arctanG\"\"\"%\"xGF%" }{TEXT -1 27 " \+ converges absolutely when " }{XPPEDIT 18 0 "abs(x)<1" "6#2-%$absG6#%\" xG\"\"\"" }{TEXT -1 20 ", and diverges when " }{XPPEDIT 18 0 "abs(x)>1 " "6#2\"\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x = 1" "6 #/%\"xG\"\"\"" }{TEXT -1 20 ", the series becomes" }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum((-1)^(n+1)/(2*n-1),n = 1 .. infi nity) = 1-1/3+1/5-1/7+` . . . `;" "6#/-%$SumG6$*&),$\"\"\"!\"\",&%\"nG F*F*F*F*,&*&\"\"#F*F-F*F*F*F+F+/F-;F*%)infinityG,,F*F**&F*F*\"\"$F+F+* &F*F*\"\"&F+F**&F*F*\"\"(F+F+%(~.~.~.~GF*" }{TEXT -1 1 "," }}{PARA 0 " " 0 "" {TEXT -1 47 "which converges by the alternating series test." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 " x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 20 ", the se ries becomes" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum( (-1)/(2*n-1),n = 1 .. infinity) = -Sum(1/(2*n-1),n = 1 .. infinity);" "6#/-%$SumG6$*&,$\"\"\"!\"\"F),&*&\"\"#F)%\"nGF)F)F)F*F*/F.;F)%)infini tyG,$-F%6$*&F)F),&*&F-F)F.F)F)F)F*F*/F.;F)F1F*" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = -``(1+1/3+1/5+1/7+` . . . `);" "6#/%!G,$-F$6#,,\" \"\"F)*&F)F)\"\"$!\"\"F)*&F)F)\"\"&F,F)*&F)F)\"\"(F,F)%(~.~.~.~GF)F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "which diverges to " } {XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 32 "This follows because the series " } {XPPEDIT 18 0 "Sum(2/(2*n-1),n = 1 .. infinity)" "6#-%$SumG6$*&\"\"#\" \"\",&*&F'F(%\"nGF(F(F(!\"\"F,/F+;F(%)infinityG" }{TEXT -1 52 " diverg es to infinity by comparison with the series " }{XPPEDIT 18 0 "Sum(1/n ,n = 1 .. infinity)" "6#-%$SumG6$*&\"\"\"F'%\"nG!\"\"/F(;F'%)infinityG " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "Sum(1/(2*n-1),n=1..infinity);\nvalue(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F',&%\"nG\"\"#!\"\"F' F+/F);F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The inte rval of convergence of the series " }{XPPEDIT 18 0 "Sum((-1)^(n+1)*x^( 2*n-1)/(2*n-1),n = 1 .. infinity)" "6#-%$SumG6$*(),$\"\"\"!\"\",&%\"nG F)F)F)F))%\"xG,&*&\"\"#F)F,F)F)F)F*F),&*&F1F)F,F)F)F)F*F*/F,;F)%)infin ityG" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"% \"xG" }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 37 ", and the radius of convergence is 1." }}{PARA 0 "" 0 "" {TEXT -1 24 "The serie s converges to " }{XPPEDIT 18 0 "arctan(1)=Pi/4" "6#/-%'arctanG6#\"\" \"*&%#PiGF'\"\"%!\"\"" }{TEXT -1 8 " , when " }{XPPEDIT 18 0 "x = 1" " 6#/%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Sum((-1)^(n+1)/(2*n-1),n=1.. infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)! \"\",&%\"nG\"\"\"F+F+F+,&F*\"\"#F(F+F(/F*;F+%)infinityG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 61 " : This series can be obtained by integrating the series for " } {XPPEDIT 18 0 "1/(1+x^2);" "6#*&\"\"\"F$,&F$F$*$%\"xG\"\"#F$!\"\"" } {TEXT -1 15 " term by term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(taylor(1/(1+x^2),x=0,11),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$+1%\"xG\"\"\"\"\"!!\"\"\"\"#F(\"\"%F*\"\"'F(\" \")F*\"#5-%\"OG6#F(\"#7F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\" \"\"F%#!\"\"\"\"$F(#F%\"\"&F*#F'\"\"(F,#F%\"\"*F.#F'\"#6F0-%\"OG6#F%\" #8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Graphical i llustration" }}{PARA 0 "" 0 "" {TEXT -1 37 "The first picture shows th e graph of " }{XPPEDIT 18 0 "y = arctan*x;" "6#/%\"yG*&%'arctanG\"\"\" %\"xGF'" }{TEXT -1 125 ", which is shown in red, along with the graphs of some polynomials obtained by truncating the corresponding Maclauri n series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "m := 8:\nplot([arctan(x),seq(convert(taylor(arctan(x ),x,2*i),polynom),\ni=1..m)],x=-2..2,y=-1.5..1.5,\ncolor=[red,seq(COLO R(HUE,2*i/(2*m+2)),i=1..m)],\nthickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 416 200 200 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" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "arctanh(x)=taylor(arctanh(x),x=0,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(arctanhG6#%\"xG+1F'\"\"\"F)#F)\"\"$F+#F)\"\"&F -#F)\"\"(F/#F)\"\"*F1#F)\"#6F3-%\"OG6#F)\"#7" }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 270 33 "_________________________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Test f or convergence" }}{PARA 0 "" 0 "" {TEXT -1 9 "We find " }{XPPEDIT 18 0 "Limit(abs(u[n+1])/abs(u[n]),n = infinity) = Limit(abs(u[n+1]/u[n]), n = infinity)" "6#/-%&LimitG6$*&-%$absG6#&%\"uG6#,&%\"nG\"\"\"F0F0F0-F )6#&F,6#F/!\"\"/F/%)infinityG-F%6$-F)6#*&&F,6#,&F/F0F0F0F0&F,6#F/F5/F/ F7" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u[n] = x^(2*n-1)/(2*n-1);" "6#/&%\"uG6#%\"nG*&)%\"xG,&*&\"\"#\"\"\"F'F.F.F.!\"\"F.,&*&F-F.F'F.F.F .F/F/" }{TEXT -1 60 ", in order to apply the ratio test for absolute c onvergence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 90 "u := n -> x^(2*n-1)/(2*n-1);\nu(n+1)/u(n);\nsimplif y(%);\nLimit(abs(%),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&)%\"xG,&9$\" \"#\"\"\"!\"\"F2F/F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&)%\" xG,&%\"nG\"\"#\"\"\"F*F*,&F(F)F*!\"\"F*F**&F'F*)F&F+F*F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&*&)%\"xG\"\"#\"\"\",&%\"nGF'F(!\"\"F(F(,&F*F'F (F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%$absG6#*&*&)%\"x G\"\"#\"\"\",&%\"nGF-F.!\"\"F.F.,&F0F-F.F.F1/F0%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$absG6#%\"xG\"\"#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 316 1 "x " }{TEXT -1 19 " is a real number, " }{XPPEDIT 18 0 "x^2*signum(x)^2 = x^2;" "6#/*&%\"xG\"\"#-%'signumG6#F%F&*$F%F&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "x^2*signum(x)^2;\nassume(x_,real);\nsubs(x_=x,simplify(subs(x=x_,% )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG\"\"#\"\"\")-%'signumG 6#F%F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"xG\"\"#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The ratio test shows that the M aclaurin series for " }{XPPEDIT 18 0 "arctanh*x;" "6#*&%(arctanhG\"\" \"%\"xGF%" }{TEXT -1 27 " converges absolutely when " }{XPPEDIT 18 0 " abs(x)<1" "6#2-%$absG6#%\"xG\"\"\"" }{TEXT -1 19 " and diverges when \+ " }{XPPEDIT 18 0 "abs(x)>1" "6#2\"\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "The series also diverges when " } {XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 10 " and when " } {XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Sum(1/(2*n-1),n=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F',&%\"nG\"\"#!\"\"F'F+/F);F'%)infinit yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 61 ": This series can be obtain ed by integrating the series for " }{XPPEDIT 18 0 "1/(1-x^2);" "6#*& \"\"\"F$,&F$F$*$%\"xG\"\"#!\"\"F)" }{TEXT -1 15 " term by term." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(taylor(1/(1-x^2),x=0,11),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$+1%\"xG\"\"\"\"\"!F(\"\"#F(\"\"%F(\"\"'F(\"\") F(\"#5-%\"OG6#F(\"#7F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\" \"F%#F%\"\"$F'#F%\"\"&F)#F%\"\"(F+#F%\"\"*F-#F%\"#6F/-%\"OG6#F%\"#8" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "arctanh*x = 1/2;" "6#/*&%(arctanhG\"\"\"%\"xGF&*&F&F& \"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ln(1+x)-ln(1-x)];" "6#7#, &-%#lnG6#,&\"\"\"F)%\"xGF)F)-F&6#,&F)F)F*!\"\"F." }{TEXT -1 63 ", we c an also obtain this series from the series expansions of " }{XPPEDIT 18 0 "ln(1+x)" "6#-%#lnG6#,&\"\"\"F'%\"xGF'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "ln(1-x)" "6#-%#lnG6#,&\"\"\"F'%\"xG!\"\"" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert(arctanh(x),ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#,&\"\"\"F(%\"xGF(#F(\"\"#-F%6#,&F(F(F)!\"\"#F/F+" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "1/2*(taylor(ln(1+x),x,13)-taylor(ln(1-x),x,13));\nconvert(%,poly nom);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&+=%\"xG\"\"\"F&#!\"\"\"\"#F )#F&\"\"$F+#F(\"\"%F-#F&\"\"&F/#F(\"\"'F1#F&\"\"(F3#F(\"\")F5#F&\"\"*F 7#F(\"#5F9#F&\"#6F;#F(\"#7F=-%\"OG6#F&\"#8#F&F)+=F%F(F&F'F)#F(F+F+F,F- #F(F/F/F0F1#F(F3F3F4F5#F(F7F7F8F9#F(F;F;FFAF'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,.%\"xG\"\"\"*$)F$\"\"$F%#F%F(*$)F$\"\"&F%#F%F,*$)F$ \"\"(F%#F%F0*$)F$\"\"*F%#F%F4*$)F$\"#6F%#F%F8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Graphical illustration" }}{PARA 0 "" 0 "" {TEXT -1 38 "The first picture shows the graph of " }{XPPEDIT 18 0 "y = arctanh*x;" "6#/%\"yG*&%(arctanhG\"\"\"%\"xGF'" }{TEXT -1 125 " , which is shown in red, along with the graphs of some polynomials obt ained by truncating the corresponding Maclaurin series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "m := 5 :\nplot([arctanh(x),seq(convert(taylor(arctanh(x),x,2*i),polynom),\ni= 1..m)],x=-1.2..1.2,y=-2..2,\ncolor=[red,seq(COLOR(HUE,2*i/(2*m+2)),i=1 ..m)],\nthickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 384 313 313 {PLOTDATA 2 "6+-%'CURVESG6$7T7$$!1+++5$>X***!#;$!1!o>vB\"*45%!#:7$$!1+ ++=K\\y**F*$!1>dz`;-QJF-7$$!1+++L5WY**F*$ !1,h5*Q6+'HF-7$$!1+++S\\TI**F*$!1/n'=1A!)*F*$!1jM&yxhJI#F-7$$! 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F.7$F_`l$\"1rZ+l#Hda#F.7$Fb`l$\"1a(p!)H)prIF.7$F^w$\"1`)[Zs;en$F.7$Ff` l$\"1'Q/(yLLTUF.7$Fi`l$\"1>,2#p[[([F.7$F\\al$\"1fG-l\\r\"[&F.7$Fbx$\"1 eQ+ZTl*='F.7$F`al$\"1vtuO**e3pF.7$Fcal$\"192Bzg)*GxF.7$Ffal$\"1&oeY&=. <')F.7$Fial$\"1ja)fpxHm*F.7$F[z$\"1h\"fo8O93\"F17$F`z$\"1i*ycY=!=7F17$ F^bl$\"1G?zEbsx8F17$Fjz$\"1K#)=Q*)y^:F17$Fbbl$\"1s#fc#fq#z\"F17$Febl$ \"1*3R!\\!Gh0#F17$Fhbl$\"12.AWs0,CF17$F[cl$\"1zv*zjG*4GF17$F^cl$\"1r&G %**e&)eLF1-Facl6$Fccl#\"\"&FfclFh[lFgclF]dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {XPPEDIT 18 0 "arcsin*x" "6#*&%'arcsinG\"\"\"%\"xGF%" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin*x = x+1/6; " "6#/*&%'arcsinG\"\"\"%\"xGF&,&F'F&*&F&F&\"\"'!\"\"F&" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x^3+3/40;" "6#,&*$%\"xG\"\"$\"\"\"*&F&F'\"#S!\"\"F' " }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^5+5/112" "6#,&*$%\"xG\"\"&\"\"\"*& F&F'\"$7\"!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^7+35/1152" "6#,&* $%\"xG\"\"(\"\"\"*&\"#NF'\"%_6!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^9+63/2816" "6#,&*$%\"xG\"\"*\"\"\"*&\"#jF'\"%;G!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^11+` . . . `" "6#,&*$%\"xG\"#6\"\"\"%(~.~.~.~G F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = x+Sum((2*n-1)!*x^(2*n+1)/(2^(2 *n-1)*n!*(n-1)!*(2*n+1)),n = 1 .. infinity);" "6#/%!G,&%\"xG\"\"\"-%$S umG6$*(-%*factorialG6#,&*&\"\"#F'%\"nGF'F'F'!\"\"F')F&,&*&F1F'F2F'F'F' F'F'**)F1,&*&F1F'F2F'F'F'F3F'-F-6#F2F'-F-6#,&F2F'F'F3F',&*&F1F'F2F'F'F 'F'F'F3/F2;F'%)infinityGF'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1 <= x; " "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "arcsin(x)=taylor(arcsin(x),x =0,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arcsinG6#%\"xG+1F'\"\" \"F)#F)\"\"'\"\"$#F,\"#S\"\"&#F/\"$7\"\"\"(#\"#N\"%_6\"\"*#\"#j\"%;G\" #6-%\"OG6#F)\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The expression for the general term given above can be ch ecked empirically as follows." }}{PARA 0 "" 0 "" {TEXT -1 3 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "(2*n-1)!*x^(2*n+1)/(2^(2*n-1 )*n!*(n-1)!*(2*n+1));\nx+add(%,n=1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&-%*factorialG6#,&%\"nG\"\"#!\"\"\"\"\"F,)%\"xG,&F)F*F,F,F,F, **)F*F(F,-F&6#F)F,-F&6#,&F)F,F+F,F,F/F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.%\"xG\"\"\"*$)F$\"\"$F%#F%\"\"'*$)F$\"\"&F%#F(\"#S*$) F$\"\"(F%#F-\"$7\"*$)F$\"\"*F%#\"#N\"%_6*$)F$\"#6F%#\"#j\"%;G" }}} {PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 314 27 "___________________ ________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Test for convergence" }}{PARA 0 "" 0 "" {TEXT -1 9 "We find " }{XPPEDIT 18 0 "Limit(abs(u[n+1])/abs(u[n]),n = infinity) = Limit(abs( u[n+1]/u[n]),n = infinity)" "6#/-%&LimitG6$*&-%$absG6#&%\"uG6#,&%\"nG \"\"\"F0F0F0-F)6#&F,6#F/!\"\"/F/%)infinityG-F%6$-F)6#*&&F,6#,&F/F0F0F0 F0&F,6#F/F5/F/F7" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u[n] = (2*n-1 )!*x^(2*n+1)/(2^(2*n-1)*n!*(n-1)!*(2*n+1));" "6#/&%\"uG6#%\"nG*(-%*fac torialG6#,&*&\"\"#\"\"\"F'F/F/F/!\"\"F/)%\"xG,&*&F.F/F'F/F/F/F/F/**)F. ,&*&F.F/F'F/F/F/F0F/-F*6#F'F/-F*6#,&F'F/F/F0F/,&*&F.F/F'F/F/F/F/F/F0" }{TEXT -1 84 ", in order to apply the ratio test for absolute converge nce (omitting the 1st term)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "u := n -> (2*n-1)!*x^(2*n+1 )/(2^(2*n-1)*n!*(n-1)!*(2*n+1));\nu(n+1)/u(n);\nsimplify(%);\nLimit(ab s(%),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG f*6#%\"nG6\"6$%)operatorG%&arrowGF(*&*&-%*factorialG6#,&9$\"\"#\"\"\"! \"\"F4)%\"xG,&F2F3F4F4F4F4**)F3F1F4-F/6#F2F4-F/6#,&F2F4F4F5F4F8F4F5F(F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*,-%*factorialG6#,&%\"nG\"\"# \"\"\"F+F+)%\"xG,&F)F*\"\"$F+F+)F*,&F)F*F+!\"\"F+-F&6#,&F)F+F+F2F+F(F+ F+*,)F*F(F+-F&6#,&F)F+F+F+F+F.F+-F&6#F1F+)F-F(F+F2" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&*&),&%\"nG\"\"#\"\"\"F*F)F*)%\"xGF)F*F**&,&F(F*F* F*F*,&F(F)\"\"$F*F*!\"\"#F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&Li mitG6$,$-%$absG6#*&*&),&%\"nG\"\"#\"\"\"F0F/F0)%\"xGF/F0F0*&,&F.F0F0F0 F0,&F.F/\"\"$F0F0!\"\"#F0F//F.%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$absG6#%\"xG\"\"#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The ratio test shows that the M aclaurin series for " }{XPPEDIT 18 0 "arcsin*x;" "6#*&%'arcsinG\"\"\"% \"xGF%" }{TEXT -1 27 " converges absolutely when " }{XPPEDIT 18 0 "abs (x)<1" "6#2-%$absG6#%\"xG\"\"\"" }{TEXT -1 19 " and diverges when " } {XPPEDIT 18 0 "abs(x)>1" "6#2\"\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 29 "The series also converges to " }{XPPEDIT 18 0 "arcsin(1) = Pi/2;" "6#/-%'arcsinG6#\"\"\"*&%#PiGF'\"\"#!\"\"" } {TEXT -1 8 " , when " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 18 ", by Raabe's test." }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 26 ", the Maclaurin s eries is " }{XPPEDIT 18 0 "1+Sum((2*n-1)!/(2^(2*n-1)*n!*(n-1)!*(2*n+1) ),n = 1 .. infinity) = 1+1/6+3/40+5/112+35/1152+63/2816+` . . . `;" "6 #/,&\"\"\"F%-%$SumG6$*&-%*factorialG6#,&*&\"\"#F%%\"nGF%F%F%!\"\"F%**) F/,&*&F/F%F0F%F%F%F1F%-F+6#F0F%-F+6#,&F0F%F%F1F%,&*&F/F%F0F%F%F%F%F%F1 /F0;F%%)infinityGF%,0F%F%*&F%F%\"\"'F1F%*&\"\"$F%\"#SF1F%*&\"\"&F%\"$7 \"F1F%*&\"#NF%\"%_6F1F%*&\"#jF%\"%;GF1F%%(~.~.~.~GF%" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "c := n -> (2*n-1)!/(2^(2*n-1)*n!*(n-1)!*(2*n+1));\nn*(c(n)/c( n+1)-1);\nsimplify(%);\nLimit(%,n=infinity);\nvalue(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"cGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&-%*fa ctorialG6#,&9$\"\"#!\"\"\"\"\"F4**)F2F0F4-F.6#F1F4-F.6#,&F1F4F3F4F4,&F 1F2F4F4F4F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"nG\"\"\",&*& **-%*factorialG6#,&F$\"\"#!\"\"F%F%)F-,&F$F-F%F%F%-F*6#,&F$F%F%F%F%,&F $F-\"\"$F%F%F%**)F-F,F%-F*6#,&F$F%F.F%F%F0F%-F*6#F0F%F.F%F.F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&%\"nG\"\"\",&F%\"\"'\"\"&F&F&F&*$) ,&F%\"\"#F&F&F-F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$* &*&%\"nG\"\"\",&F(\"\"'\"\"&F)F)F)*$),&F(\"\"#F)F)F0F)!\"\"/F(%)infini tyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"#" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 " Limit(n*(c(n)/c(n+1)-1),n=infinity)>1" "6#2\"\"\"-%&LimitG6$*&%\"nGF$, &*&-%\"cG6#F)F$-F-6#,&F)F$F$F$!\"\"F$F$F2F$/F)%)infinityG" }{TEXT -1 38 ", the Maclaurin series converges when " }{XPPEDIT 18 0 "x = 1" "6# /%\"xG\"\"\"" }{TEXT -1 17 " by Raabe's test." }}{PARA 0 "" 0 "" {TEXT -1 51 "The following calculation suggests that the sum is " } {XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "1+Sum((2*n-1)!/(2^(2*n-1)*n!*(n-1)!*(2*n+1)),n=1..infinity);\nvalu e(%);\nevalf(%);\nevalf(Pi/2);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&\"\"\"F$-%$SumG6$*&-%*factorialG6#,&%\"nG\"\"#!\"\"F$F$**)F.F,F$-F*6 #F-F$-F*6#,&F-F$F/F$F$,&F-F.F$F$F$F//F-;F$%)infinityGF$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%*hypergeomG6%7%F$#\"\"$\"\"#F)7$F+# \"\"&F+F$#F$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Fjzq:!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Fjzq:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 98 ", changes all the signs o f the terms of the previous series, so the Maclaurin series converges \+ to " }{XPPEDIT 18 0 "-Pi/2" "6#,$*&%#PiG\"\"\"\"\"#!\"\"F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 4 "Note" }{TEXT -1 61 ": This series can be obtained by in tegrating the series for " }{XPPEDIT 18 0 "1/sqrt(1-x^2);" "6#*&\"\" \"F$-%%sqrtG6#,&F$F$*$%\"xG\"\"#!\"\"F," }{TEXT -1 15 " term by term. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Int(1/sqrt(1-x^2),x);\nInt(taylor(1/sqrt(1-x^2),x=0,13),x);\nv alue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$,&F'F '*$)%\"xG\"\"#F'!\"\"#F'F-F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I ntG6$+3%\"xG\"\"\"\"\"!#F(\"\"#F+#\"\"$\"\")\"\"%#\"\"&\"#;\"\"'#\"#N \"$G\"F.#\"#j\"$c#\"#5#\"$J#\"%C5\"#7-%\"OG6#F(\"#9F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"F%#F%\"\"'\"\"$#F(\"#S\"\"&#F+\"$7\" \"\"(#\"#N\"%_6\"\"*#\"#j\"%;G\"#6#\"$J#\"&7L\"\"#8-%\"OG6#F%\"#:" }}} {PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 315 36 "___________________ _________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 22 "Graphical illustration" }}{PARA 0 "" 0 "" {TEXT -1 37 " The first picture shows the graph of " }{XPPEDIT 18 0 "y = arcsin*x;" "6#/%\"yG*&%'arcsinG\"\"\"%\"xGF'" }{TEXT -1 125 ", which is shown in \+ red, along with the graphs of some polynomials obtained by truncating \+ the corresponding Maclaurin series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "m := 5:\nplot([arcsin(x),s eq(convert(taylor(arcsin(x),x,2*i),polynom),\ni=1..m)],x=-1.2..1.2,y=- 2..2,\ncolor=[red,seq(COLOR(HUE,2*i/(2*m+2)),i=1..m)],\nthickness=2); " }}{PARA 13 "" 1 "" {GLPLOT2D 394 305 305 {PLOTDATA 2 "6+-%'CURVESG6$ 7L7$$!1+++5$>X***!#;$!1hqhRooP:!#:7$$!1+++G=5Q(*F*$!1#=,!*)yUT8F-7$$!1 +++XVo\"[*F*$!1mg9(e@uC\"F-7$$!1+++&o?i+*F*$!1l$Gtu)>@6F-7$$!1*****\\g 0R^)F*$!1'*G+m1j=5F-7$$!1+++XEv/!)F*$!1v_;Vu(3G*F*7$$!1+++XDB(\\(F*$!1 ,y-$yPkZ)F*7$$!1******Rd=vpF*$!1QJ&Q&pG>xF*7$$!1+++ImO:lF*$!1@&*z)*G3' 4(F*7$$!1+++?#>x*fF*$!1w7j \"eOIF*7$$!1+++![X$=DF*$!1E$**)HVvXDF*7$$!1+++lNs+?F*$!1^e!*)pF*$\"1Gp%)\\7p-?F*7$$\"1+++IY7#\\#F* $\"1gnDd(p'=DF*7$$\"1+++![.'yHF*$\"1Mq.zZ]CIF*7$$\"1+++!Hb(=NF*$\"1*>? 2@Sdf$F*7$$\"1+++?d5/SF*$\"18og1'['>TF*7$$\"1++++4KAXF*$\"1y+6'yjEp%F* 7$$\"1+++!)3!>*\\F*$\"1*3;)>#QmA&F*7$$\"1+++?eF0bF*$\"1R!)G&zg*HeF*7$$ \"1,++S;K))fF*$\"1U!z_6@/U'F*7$$\"1++++PC$\\'F*$\"1hF)e5dp1(F*7$$\"1++ ++t*o)pF*$\"1H7^uRkNxF*7$$\"1+++!)oq.vF*$\"1I[oZoA'[)F*7$$\"1+++?fX,!) 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "arcsinh (x)=taylor(arcsinh(x),x=0,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%( arcsinhG6#%\"xG+1F'\"\"\"F)#!\"\"\"\"'\"\"$#F-\"#S\"\"&#!\"&\"$7\"\"\" (#\"#N\"%_6\"\"*#!#j\"%;G\"#6-%\"OG6#F)\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The expression for the general \+ term given above can be checked empirically as follows." }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "(-1) ^n*(2*n-1)!*x^(2*n+1)/(2^(2*n-1)*n!*(n-1)!*(2*n+1));\nx+add(%,n=1..5); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*0)!\"\"%\"nG\"\"\"-%*factorialG6# ,&*&\"\"#F'F&F'F'F'F%F')%\"xG,&*&F-F'F&F'F'F'F'F')F-F+F%-F)6#F&F%-F)6# ,&F&F'F'F%F%F0F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.%\"xG\"\"\"*&#F% \"\"'F%*$)F$\"\"$F%F%!\"\"*&#F+\"#SF%*$)F$\"\"&F%F%F%*&#F2\"$7\"F%*$)F $\"\"(F%F%F,*&#\"#N\"%_6F%*$)F$\"\"*F%F%F%*&#\"#j\"%;GF%*$)F$\"#6F%F%F ," }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 271 27 "_____________ ______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "Test for convergence" }}{PARA 0 "" 0 "" {TEXT -1 9 "We \+ find " }{XPPEDIT 18 0 "Limit(abs(u[n+1])/abs(u[n]),n = infinity) = Li mit(abs(u[n+1]/u[n]),n = infinity)" "6#/-%&LimitG6$*&-%$absG6#&%\"uG6# ,&%\"nG\"\"\"F0F0F0-F)6#&F,6#F/!\"\"/F/%)infinityG-F%6$-F)6#*&&F,6#,&F /F0F0F0F0&F,6#F/F5/F/F7" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u[n] = (-1)^n*(2*n-1)!*x^(2*n+1)/(2^(2*n-1)*n!*(n-1)!*(2*n+1));" "6#/&%\"uG6 #%\"nG**),$\"\"\"!\"\"F'F+-%*factorialG6#,&*&\"\"#F+F'F+F+F+F,F+)%\"xG ,&*&F2F+F'F+F+F+F+F+**)F2,&*&F2F+F'F+F+F+F,F+-F.6#F'F+-F.6#,&F'F+F+F,F +,&*&F2F+F'F+F+F+F+F+F," }{TEXT -1 84 ", in order to apply the ratio t est for absolute convergence (omitting the 1st term)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "u := n - > (-1)^n*(2*n-1)!*x^(2*n+1)/(2^(2*n-1)*n!*(n-1)!*(2*n+1));\nu(n+1)/u(n );\nsimplify(%);\nLimit(abs(%),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*0)!\"\"9 $\"\"\"-%*factorialG6#,&*&\"\"#F0F/F0F0F0F.F0)%\"xG,&*&F6F0F/F0F0F0F0F 0)F6F4F.-F26#F/F.-F26#,&F/F0F0F.F.F9F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*:)!\"\",&%\"nG\"\"\"F(F(F(-%*factorialG6#,&*&\"\"#F(F' F(F(F(F(F()%\"xG,&*&F.F(F'F(F(\"\"$F(F()F.F,F%-F*6#F&F%F1F%)F%F'F%-F*6 #,&*&F.F(F'F(F(F(F%F%)F0F,F%)F.F:F(-F*6#,&F'F(F(F%F(F,F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*,\"\"#!\"\",&*&F%\"\"\"%\"nGF)F)F)F)F%%\"xGF% ,&F*F)F)F)F&,&*&F%F)F*F)F)\"\"$F)F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&#\"\"\"\"\"#F)-%$absG6#**,&*&F*F)%\"nGF)F)F)F)F*%\"x GF*,&F1F)F)F)!\"\",&*&F*F)F1F)F)\"\"$F)F4F)F)/F1%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$absG6#%\"xG\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The ratio test shows that the Maclaurin series for " }{XPPEDIT 18 0 "arcsinh*x;" "6#*&%(arcsinhG\"\"\"%\"xGF% " }{TEXT -1 27 " converges absolutely when " }{XPPEDIT 18 0 "abs(x)<1 " "6#2-%$absG6#%\"xG\"\"\"" }{TEXT -1 19 " and diverges when " } {XPPEDIT 18 0 "abs(x)>1" "6#2\"\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\" \"\"" }{TEXT -1 26 ", the Maclaurin series is " }{XPPEDIT 18 0 "1+Sum( (-1)^n*(2*n-1)!/(2^(2*n-1)*n!*(n-1)!*(2*n+1)),n = 1 .. infinity) = 1-1 /6+3/40-5/112+35/1152-63/2816+` . . . `;" "6#/,&\"\"\"F%-%$SumG6$*(),$ F%!\"\"%\"nGF%-%*factorialG6#,&*&\"\"#F%F-F%F%F%F,F%**)F3,&*&F3F%F-F%F %F%F,F%-F/6#F-F%-F/6#,&F-F%F%F,F%,&*&F3F%F-F%F%F%F%F%F,/F-;F%%)infinit yGF%,0F%F%*&F%F%\"\"'F,F,*&\"\"$F%\"#SF,F%*&\"\"&F%\"$7\"F,F,*&\"#NF% \"%_6F,F%*&\"#jF%\"%;GF,F,%(~.~.~.~GF%" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 48 "which converges by the alternating series test. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The serie s converges to " }{XPPEDIT 18 0 "arcsinh(1)=ln(1+sqrt(2))" "6#/-%(arcs inhG6#\"\"\"-%#lnG6#,&F'F'-%%sqrtG6#\"\"#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "1 +Sum((-1)^n*(2*n-1)!/(2^(2*n-1)*n!*(n-1)!*(2*n+1)),n=1..infinity);\nva lue(%);\nevalf(%);\nevalf(ln(1+sqrt(2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%$SumG6$*.)!\"\"%\"nGF$-%*factorialG6#,&*&\" \"#F$F+F$F$F$F*F$)F1F/F*-F-6#F+F*-F-6#,&F+F$F$F*F*,&*&F1F$F+F$F$F$F$F* /F+;F$%)infinityGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&#F$ \"\"'F$-%*hypergeomG6%7%F$#\"\"$\"\"#F,7$F.#\"\"&F.!\"\"F$F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+qet8))!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+pet8))!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 98 ", change s all the signs of the terms of the previous series, so the Maclaurin \+ series converges to " }{XPPEDIT 18 0 "-arcsinh(1) = -ln(1+sqrt(2))" "6 #/,$-%(arcsinhG6#\"\"\"!\"\",$-%#lnG6#,&F(F(-%%sqrtG6#\"\"#F(F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 259 4 "Note" }{TEXT -1 61 ": This series can be obtained \+ by integrating the series for " }{XPPEDIT 18 0 "1/sqrt(1+x^2);" "6#*& \"\"\"F$-%%sqrtG6#,&F$F$*$%\"xG\"\"#F$!\"\"" }{TEXT -1 15 " term by t erm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Int(1/sqrt(1+x^2),x);\nInt(taylor(1/sqrt(1+x^2),x=0,1 3),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\" F'*$,&F'F'*$)%\"xG\"\"#F'F'#F'F-!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$+3%\"xG\"\"\"\"\"!#!\"\"\"\"#F,#\"\"$\"\")\"\"%#!\"&\"# ;\"\"'#\"#N\"$G\"F/#!#j\"$c#\"#5#\"$J#\"%C5\"#7-%\"OG6#F(\"#9F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"F%#!\"\"\"\"'\"\"$#F)\"#S \"\"&#!\"&\"$7\"\"\"(#\"#N\"%_6\"\"*#!#j\"%;G\"#6#\"$J#\"&7L\"\"#8-%\" OG6#F%\"#:" }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 272 36 "____ ________________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Graphical illustration" }}{PARA 0 "" 0 "" {TEXT -1 38 "The first picture shows the graph of " }{XPPEDIT 18 0 "y = arcsinh*x;" "6#/%\"yG*&%(arcsinhG\"\"\"%\"xGF'" }{TEXT -1 125 " , which is shown in red, along with the graphs of some polynomials obt ained by truncating the corresponding Maclaurin series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "m := 5 :\nplot([arcsinh(x),seq(convert(taylor(arcsinh(x),x,2*i),polynom),\n \+ i=1..m)],x=-3..3,y=-3..3,\n color=[red,seq(COLOR(HUE,2*i/(2*m+2)) ,i=1..m)],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 394 305 305 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!\"$\"\"!$!3$o1K#fkW==!#<7$$!3!****** \\2<#pGF-$!3gi7aotDw\"F-7$$!3/+++vl[p8F-$!3[Wc*yd<,7\"F-7$$!3\"******\\>iUC\"F-$!3j/cq 6O+W5F-7$$!3-++DhkaI6F-$!3+g$R&4VR2(*!#=7$$!3s******\\XF`**Fcp$!3K5(eD Od1y)Fcp7$$!3u*******>#z2))Fcp$!377Uev!=^%zFcp7$$!3S++]7RKvuFcp$!3Ga*G NC><\"pFcp7$$!3s,+++P'eH'Fcp$!38*\\f#)4*GSfFcp7$$!3q)***\\7*3=+&Fcp$!3 f$4&eKht8[Fcp7$$!3[)***\\PFcpPFcp$!3E`RN2tb&o$Fcp7$$!3;)****\\7VQ[#Fcp $!3g(**Q-<()*eCFcp7$$!32)***\\i6:.8Fcp$!3u)pn$32\\*H\"Fcp7$$!3Wb+++v`h H!#?$!3EH)))3*4#))Fcp7$$\"36+++D-eI 6F-$\"3(4Nc]+=wq*Fcp7$$\"3u***\\(=_(zC\"F-$\"3#***\\*o^Fj/\"F-7$$\"3M+ ++b*=jP\"F-$\"35'[o\\?ST7\"F-7$$\"3g***\\(3/3(\\\"F-$\"3E$\\XviUJ>\"F- 7$$\"33++vB4JB;F-$\"3S7TtNPFh7F-7$$\"3u*****\\KCnu\"F-$\"3$3)3YH9FC8F- 7$$\"3s***\\(=n#f(=F-$\"3QPSw7#F-$\"3U!4%)Q%RH*\\\"F-7$$\"3O++v)Q?QD#F-$\"3a*yQYY3< b\"F-7$$\"3G+++5jypBF-$\"3Qcs#o_axf\"F-7$$\"3<++]Ujp-DF-$\"3ST`cx?B[;F -7$$\"3++++gEd@EF-$\"3q0R#[Tf9p\"F-7$$\"39++v3'>$[FF-$\"3:scRpJqNF>7$FCF C7$FHFH7$FMFM7$FRFR7$FWFW7$FfnFfn7$F[oF[o7$F`oF`o7$FeoFeo7$FjoFjo7$F_p F_p7$FepFep7$FjpFjp7$F_qF_q7$FdqFdq7$FiqFiq7$F^rF^r7$FcrFcr7$FhrFhr7$F ]sF]s7$FcsFcs7$FhsFhs7$F]tF]t7$FbtFbt7$FgtFgt7$F\\uF\\u7$FauFau7$FfuFf u7$F[vF[v7$F`vF`v7$FevFev7$FjvFjv7$F_wF_w7$FdwFdw7$FiwFiw7$F^xF^x7$Fcx Fcx7$FhxFhx7$F]yF]y7$FbyFby7$FgyFgy7$F\\zF\\z7$FazFaz7$FfzFfz-%&COLORG 6$%$HUEG#\"\"\"\"\"'-F$6$7\\o7$F($\"3++++++++:F-7$$!3&*****\\P&3Y$HF-$ \"39wN^Vt\\x7F-7$F/$\"39!G,/?Vv1\"F-7$$!3')**\\78.K7GF-$\"3'f))p$))za[ *)Fcp7$F4$\"3?S*\\Ld0FJ(Fcp7$$!3'***\\(=>P9p#F-$\"3!3)[T@D]zbFcp7$F9$ \"3s>NP\\P\\cRFcp7$$!3w******H./jDF-$\"3))H$4^En8V#Fcp7$F>$\"3u_H\"R'R d75Fcp7$$!3))**\\(oXDXV#F-$!3Q)GQpL([lH!#>7$FC$!3hA4Ukxi0:Fcp7$FH$!31) R#>DUy\"\\$Fcp7$FM$!3],!G=P1L@&Fcp7$FR$!3g2&*4M4xamFcp7$FW$!3S.,&zT5*o xFcp7$Ffn$!38[5QL%G.g)Fcp7$F[o$!3OhDaw>)e3*Fcp7$$!3%****\\7tNTc\"F-$!3 3CIU.-aj#*Fcp7$F`o$!3/P9k0.rv$*Fcp7$$!3'********=eWV\"F-$!3([Z0eay^U*F cp7$Feo$!3[1*3\"[Q49%*Fcp7$$!3(******\\QuoI\"F-$!3CVS/<4o[$*Fcp7$Fjo$! 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 36 "Suppose that we have a power series " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(a[n]*x^n,n = 0 .. infinity) = a[ 0]+a[1]*x+a[2]*x^2+a[3]*x^3+`. . . `" "6#/-%$SumG6$*&&%\"aG6#%\"nG\"\" \")%\"xGF+F,/F+;\"\"!%)infinityG,,&F)6#F1F,*&&F)6#F,F,F.F,F,*&&F)6#\" \"#F,*$F.F0" "6#f*6#-%$absG6#*&&%\"aG6# %\"nG\"\"\")&%\"xG6#\"\"!F,F-7\"6$%)operatorG%&arrowG6\"F2F7F7F7" } {TEXT -1 4 " as " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)op eratorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 98 "This means, in particular, that it must be possible to \+ find a sufficiently large positive integer " }{TEXT 303 1 "N" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "abs(a[n]*x[0]^n)<1/2" "6#2-%$absG6 #*&&%\"aG6#%\"nG\"\"\")&%\"xG6#\"\"!F+F,*&F,F,\"\"#!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "n > N" "6#2%\"NG%\"nG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 11 "Then, when " }{XPPEDIT 18 0 "N \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Exam ple to illustrate the argument above" }}{PARA 0 "" 0 "" {TEXT -1 35 "C onsider the Maclaurin series for l" }{XPPEDIT 18 0 "n(1+x)" "6#-%\"nG6 #,&\"\"\"F'%\"xGF'" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "Sum((-1)^(n+1)*x^n/n,n = 1 .. infinity) = x-x^2/2+x^ 3/3-x^4/4+x^5/5-x^6/6+`. . . `;" "6#/-%$SumG6$*(),$\"\"\"!\"\",&%\"nGF *F*F*F*)%\"xGF-F*F-F+/F-;F*%)infinityG,0F/F**&F/\"\"#F5F+F+*&F/\"\"$F7 F+F**&F/\"\"%F9F+F+*&F/\"\"&F;F+F**&F/\"\"'F=F+F+%'.~.~.~GF*" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 23 " the series (i) becomes" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^(n+1)/n,n = 1 .. infinity) = 1-1/2+1/3-1/4+1/5-1/6+`. . . `" "6#/-%$SumG6$*&),$\" \"\"!\"\",&%\"nGF*F*F*F*F-F+/F-;F*%)infinityG,0F*F**&F*F*\"\"#F+F+*&F* F*\"\"$F+F**&F*F*\"\"%F+F+*&F*F*\"\"&F+F**&F*F*\"\"'F+F+%'.~.~.~GF*" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "which converges by the \+ " }{TEXT 259 23 "alternating series test" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 70 "Hence the argument above shows that (i) converges \+ for any real number " }{TEXT 307 1 "x" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "abs(x)<1" "6#2-%$absG6#%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "x = -9/10;" "6 #/%\"xG,$*&\"\"*\"\"\"\"#5!\"\"F*" }{TEXT -1 25 " the series (i) becom es: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(-``(1/n)* (9/10)^n,n = 1 .. infinity) = -9/10-``(1/2)*(9/10)^2-``(1/3)*(9/10)^3- ``(1/4)*(9/10)^4-``(1/5)*(9/10)^5-`. . . `;" "6#/-%$SumG6$,$*&-%!G6#*& \"\"\"F-%\"nG!\"\"F-)*&\"\"*F-\"#5F/F.F-F//F.;F-%)infinityG,.*&F2F-F3F /F/*&-F*6#*&F-F-\"\"#F/F-*$*&F2F-F3F/F=F-F/*&-F*6#*&F-F-\"\"$F/F-*$*&F 2F-F3F/FDF-F/*&-F*6#*&F-F-\"\"%F/F-*$*&F2F-F3F/FKF-F/*&-F*6#*&F-F-\"\" &F/F-*$*&F2F-F3F/FRF-F/%'.~.~.~GF/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We can check for " } {TEXT 259 20 "absolute convergence" }{TEXT -1 59 " by considering the \+ corresponding series of positive terms:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(``(1/n)*(9/10)^n,n = 1 .. infinity) = 9/10+ ``(1/2)*(9/10)^2+``(1/3)*(9/10)^3+``(1/4)*(9/10)^4+``(1/5)*(9/10)^5+`. . . `;" "6#/-%$SumG6$*&-%!G6#*&\"\"\"F,%\"nG!\"\"F,)*&\"\"*F,\"#5F.F- F,/F-;F,%)infinityG,.*&F1F,F2F.F,*&-F)6#*&F,F,\"\"#F.F,*$*&F1F,F2F.F1" "6#2\"\"\"-%$ab sG6#%\"xG" }{TEXT -1 17 ", note that when " }{XPPEDIT 18 0 "x = -1" "6 #/%\"xG,$\"\"\"!\"\"" }{TEXT -1 13 " it becomes: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(-1/n,n = 1 .. infinity) = -1-1/2- 1/3-1/4-1/5-1/6-`. . . `;" "6#/-%$SumG6$,$*&\"\"\"F)%\"nG!\"\"F+/F*;F) %)infinityG,0F)F+*&F)F)\"\"#F+F+*&F)F)\"\"$F+F+*&F)F)\"\"%F+F+*&F)F)\" \"&F+F+*&F)F)\"\"'F+F+%'.~.~.~GF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 67 "which is the negative of the harmonic series and so it di verges to " }{XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "If we take a negative value fo r " }{TEXT 308 1 "x" }{TEXT -1 16 " just less than " }{XPPEDIT 18 0 "- 1" "6#,$\"\"\"!\"\"" }{TEXT -1 126 ", this makes all the terms larger \+ in the negative direction, so it is immediately clear that the series \+ will still diverge to " }{XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 137 "Without considerin g the argument developed in this section, it may not be quite so obvio us that the series must also fail to converge if " }{TEXT 309 1 "x" } {TEXT -1 25 " is just greater than +1." }}{PARA 0 "" 0 "" {TEXT -1 104 "However it is not possible for the series to converge in this sit uation because, if it did converge for " }{XPPEDIT 18 0 "x = 1 + epsil on" "6#/%\"xG,&\"\"\"F&%(epsilonGF&" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 82 " is some small positive nu mber, then it would have to converge absolutely for any " }{TEXT 310 1 "x" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "abs(x) < 1+epsilon" "6#2-%$ absG6#%\"xG,&\"\"\"F)%(epsilonGF)" }{TEXT -1 17 ", which includes " } {XPPEDIT 18 0 "x = -1-epsilon/2" "6#/%\"xG,&\"\"\"!\"\"*&%(epsilonGF& \"\"#F'F'" }{TEXT -1 52 ". We have just observed that the series diver ges to " }{XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT -1 21 " for such a value of " }{TEXT 311 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }