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" }}{PARA 0 "" 0 "" {TEXT -1 16 "Then the series " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity)" "6# -%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 1 " " }{TEXT 270 9 "converges" }{TEXT -1 5 " if " }{XPPEDIT 18 0 "rho < 1" "6#2%$r hoG\"\"\"" }{TEXT -1 5 " and " }{TEXT 270 8 "diverges" }{TEXT -1 4 " i f " }{XPPEDIT 18 0 "rho > 1" "6#2\"\"\"%$rhoG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "rho=1" "6#/%$rhoG\"\" \"" }{TEXT -1 19 ", then the test is " }{TEXT 270 12 "inconclusive" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 273 28 "_____ _______________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "First note in the special case that \+ " }{XPPEDIT 18 0 "Sum(a[n],n=1..infinity)" "6#-%$SumG6$&%\"aG6#%\"nG/F );\"\"\"%)infinityG" }{TEXT -1 41 " is a geometric series with common \+ ratio " }{TEXT 297 1 "r" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "a[n+1]/ a[n]=r" "6#/*&&%\"aG6#,&%\"nG\"\"\"F*F*F*&F&6#F)!\"\"%\"rG" }{TEXT -1 9 " for all " }{TEXT 275 1 "n" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "Lim it(a[n+1]/a[n],n = infinity) = rho;" "6#/-%&LimitG6$*&&%\"aG6#,&%\"nG \"\"\"F-F-F-&F)6#F,!\"\"/F,%)infinityG%$rhoG" }{XPPEDIT 18 0 " ``= r" "6#/%!G%\"rG" }{TEXT -1 30 ", and we have convergence for " }{XPPEDIT 18 0 "rho < 1;" "6#2%$rhoG\"\"\"" }{TEXT -1 20 " and divergence for " }{XPPEDIT 18 0 "1 < rho;" "6#2\"\"\"%$rhoG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "If a series satis fies the condition for convergence given by the ratio test, that is, \+ " }{XPPEDIT 18 0 "Limit(a[n+1]/a[n],n = infinity) = rho;" "6#/-%&Limit G6$*&&%\"aG6#,&%\"nG\"\"\"F-F-F-&F)6#F,!\"\"/F,%)infinityG%$rhoG" } {XPPEDIT 18 0 "``< 1" "6#2%!G\"\"\"" }{TEXT -1 126 ", then, if we go s ufficiently far along the series, it is eventually very close to being a geometric series with common ratio " }{XPPEDIT 18 0 "rho" "6#%$rhoG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "In more detail, if " }{XPPEDIT 18 0 "rho<1" "6#2%$rhoG\" \"\"" }{TEXT -1 18 ", choose a number " }{TEXT 276 1 "r" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "rho=N" "6#1%\"NG%\"nG" } {TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "a[n+1]=0" "6#1\"\"!%\"kG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "The series " }{XPPEDIT 18 0 "Sum(a[n],n=N..infinity)" "6#-%$SumG6$&%\"aG6#%\"nG/F);%\"NG%)inf inityG" }{TEXT -1 29 " can now be seen to converge " }{TEXT 270 43 "by comparison with the the geometric series" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[N]*r^k,k=0..infinity)" "6#-%$SumG6$*&&%\"aG6#%\"NG\"\"\")% \"rG%\"kGF+/F.;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "By adding the finite sum " }{XPPEDIT 18 0 "Sum(a[n],n=1.. N-1)" "6#-%$SumG6$&%\"aG6#%\"nG/F);\"\"\",&%\"NGF,F,!\"\"" }{TEXT -1 14 ", we see that " }{XPPEDIT 18 0 "Sum(a[n],n=1..infinity)" "6#-%$Sum G6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 11 " converges." }} {PARA 0 "" 0 "" {TEXT -1 22 "On the other hand, if " }{XPPEDIT 18 0 "r ho > 1" "6#2\"\"\"%$rhoG" }{TEXT -1 20 " then the condition " } {XPPEDIT 18 0 "Limit(a[n+1]/a[n],n = infinity) = rho" "6#/-%&LimitG6$* &&%\"aG6#,&%\"nG\"\"\"F-F-F-&F)6#F,!\"\"/F,%)infinityG%$rhoG" }{TEXT -1 12 " means that " }{XPPEDIT 18 0 "a[n+1]/a[n] > 1" "6#2\"\"\"*&&%\" aG6#,&%\"nGF$F$F$F$&F'6#F*!\"\"" }{TEXT -1 6 " when " }{TEXT 280 1 "n " }{TEXT -1 53 " is sufficiently large, that is, there is an integer \+ " }{TEXT 281 1 "N" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "a[n+1]/a [n] > 1" "6#2\"\"\"*&&%\"aG6#,&%\"nGF$F$F$F$&F'6#F*!\"\"" }{TEXT -1 6 " for " }{TEXT 282 1 "n" }{TEXT -1 12 " such that " }{XPPEDIT 18 0 " n >= N" "6#1%\"NG%\"nG" }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "a[n+1] > a[n]" "6#2&%\"aG6#%\"nG&F%6#,&F'\"\"\"F+F+" }{TEXT -1 5 " for " } {TEXT 283 1 "n" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "N>=n" "6#1% \"nG%\"NG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "This means t hat " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 22 " cannot ap proach 0 as " }{TEXT 284 1 "n" }{TEXT -1 23 " tends to infinity, so " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity)" "6#-%$SumG6$&%\"aG6#%\"nG /F);\"\"\"%)infinityG" }{TEXT -1 22 " diverges to infinity." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Ratio test examples" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exa mple 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the series " }{XPPEDIT 18 0 "Sum(n/(2^n),n = \+ 1 .. infinity)=1/2+2/4+3/8+4/16+5/32+6/64+7/128+` . . . `" "6#/-%$SumG 6$*&%\"nG\"\"\")\"\"#F(!\"\"/F(;F)%)infinityG,2*&F)F)F+F,F)*&F+F)\"\"% F,F)*&\"\"$F)\"\")F,F)*&F3F)\"#;F,F)*&\"\"&F)\"#KF,F)*&\"\"'F)\"#kF,F) *&\"\"(F)\"$G\"F,F)%(~.~.~.~GF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a[n]=n/2^n" "6#/&%\"aG6#%\"nG*&F'\" \"\")\"\"#F'!\"\"" }{TEXT -1 8 ". Then " }{XPPEDIT 18 0 "a[n+1]/a[n] \+ = ``((n+1)/(2^(n+1)))/``(n/(2^n));" "6#/*&&%\"aG6#,&%\"nG\"\"\"F*F*F*& F&6#F)!\"\"*&-%!G6#*&,&F)F*F*F*F*)\"\"#,&F)F*F*F*F-F*-F06#*&F)F*)F5F)F -F-" }{XPPEDIT 18 0 "`` = ``((n+1)/(2^(n+1)))*``(2^n/n);" "6#/%!G*&-F$ 6#*&,&%\"nG\"\"\"F+F+F+)\"\"#,&F*F+F+F+!\"\"F+-F$6#*&)F-F*F+F*F/F+" } {XPPEDIT 18 0 "``= (n+1)/(2*n)" "6#/%!G*&,&%\"nG\"\"\"F(F(F(*&\"\"#F(F 'F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "rho=Limit(a[n+1]/a[n],n=infinity)" "6#/%$rhoG-%&LimitG6 $*&&%\"aG6#,&%\"nG\"\"\"F.F.F.&F*6#F-!\"\"/F-%)infinityG" }{XPPEDIT 18 0 "`` = Limit((n+1)/(2*n),n = infinity);" "6#/%!G-%&LimitG6$*&,&%\" nG\"\"\"F+F+F+*&\"\"#F+F*F+!\"\"/F*%)infinityG" }{XPPEDIT 18 0 "``= 1/ 2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "rh o<1" "6#2%$rhoG\"\"\"" }{TEXT -1 30 ", the series converges by the " } {TEXT 270 10 "ratio test" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "a := n -> n/2^n;\nLim it(a(n+1)/a(n),n=infinity);\nsimplify(%);\nvalue(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&9$\"\" \")\"\"#F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*& *&,&%\"nG\"\"\"F*F*F*)\"\"#F)F*F**&)F,F(F*F)F*!\"\"/F)%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,&%\"nG\"\"\"F*F*F*F)! \"\"#F*\"\"#/F)%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\" \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "In fact " }{XPPEDIT 18 0 "Sum(n/(2^n),n=1..infinity)=2" "6#/-%$SumG6$*&%\"nG\"\"\")\"\"#F( !\"\"/F(;F)%)infinityGF+" }{TEXT -1 62 ". An explanation for this resu lt is given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Sum(n/2^n,n=1..infinity);\nv alue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&%\"nG\"\"\")\" \"#F'!\"\"/F';F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The co nvergence is quite fast as is shown by the sequence of partial sums." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Sum(i/2^i,i=1..n):\nPS := unapply(value(%),n);\nseq(evalf(PS(n)) ,n=1..35);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PSGf*6#%\"nG6\"6$%)op eratorG%&arrowGF(,(*&)#\"\"\"\"\"#,&9$F0F0F0F0F2F0!\"#F.F4F1F0F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6E$\"+++++]!#5$\"\"\"\"\"!$\"++++v8!\" *$\"++++D;F+$\"+++D\"y\"F+$\"++++v=F+$\"++voH>F+$\"++v$4'>F+$\"+Dc^y>F +$\"+]7G))>F+$\"+WBl$*>F+$\"+J?e'*>F+$\"+X*o\")*>F+$\"+QM-**>F+$\"+,7[ **>F+$\"+U`s**>F+$\"+U]&)**>F+$\"+1P#***>F+$\"+Y*f***>F+$\"+>!z***>F+$ \"+L!*)***>F+$\"+yU****>F+$\"+?q****>F+$\"+]%)****>F+$\"+&>*****>F+$\" +$e*****>F+$\"+%y*****>F+$\"+)))*****>F+$\"+U******>F+$\"+q******>F+$ \"+&)******>F+$\"+#*******>F+$\"+'*******>F+$\"+)*******>F+$\"+******* *>F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 " The following picture shows the first few terms of the sequence of par tial sums." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 225 "Sum(i/2^i,i=1..n):\nPS := unapply(value(%),n):\npt s := [seq([n,PS(n)],n=1..15)]:\nplot([PS(x),2,pts],x=0..15,style=[line $2,point],\n color=[gray,brown,red],linestyle=[2,3],symbol=circle,\n view=[0..15,0..2.5],labels=[`n`,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 618 230 230 {PLOTDATA 2 "6(-%'CURVESG6&7S7$$\"\"!F)F(7$$\"3' )*****\\7t&pK!#=$\"3[_]$Q[A\"\\9F-7$$\"3$****\\(=7T9hF-$\"3\"**Qsl#y\"F=7$$\"3i******\\>0)H&F=$\"38@$)zlS]9=F=7$$\" 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F)$\"3+++vVBl$*>F=7$$\"#7F)$\"3+++DJ?e'*>F=7$$\"#8F)$\"3++DJX*o\")*>F= 7$$\"#9F)$\"3+++]PM-**>F=Fbz-Fhz6&Fjz$\"*++++\"F][lF(F(-F_[l6#%&POINTG Fb[l-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$%\"nG%!G-%%VIEWG6$;F(Fcz;F($ \"#D!\"\"" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" }}}}{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 81 "We can use this example to illustrate the argument given in the previous section." }}{PARA 0 "" 0 "" {TEXT -1 10 "First let " }{XPPEDIT 18 0 "r = 3/4" "6#/%\"rG*&\"\"$\"\"\"\"\" %!\"\"" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "rho < r " "6#2%$rhoG %\"rG" }{XPPEDIT 18 0 "``< 1" "6#2%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 " Now " }{XPPEDIT 18 0 "(n+1)/(2*n)<3/4" "6#2*&,& %\"nG\"\"\"F'F'F'*&\"\"#F'F&F'!\"\"*&\"\"$F'\"\"%F*" }{TEXT -1 7 " wh en " }{XPPEDIT 18 0 "4*n+4<6*n" "6#2,&*&\"\"%\"\"\"%\"nGF'F'F&F'*&\"\" 'F'F(F'" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "4<2*n" "6#2\"\"%*& \"\"#\"\"\"%\"nGF'" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "n>2" "6#2\"\" #%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(seq((n+1)/(2*n),n=1..10));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6,$\"\"\"\"\"!$\"+++++v!#5$\"+nmmmmF($ \"++++]iF($\"+++++gF($\"+LLLLeF($\"+9dG9dF($\"++++DcF($\"+cbbbbF($\"++ +++bF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The series " } {XPPEDIT 18 0 "Sum(a[n],n=3..infinity)=Sum(n/2^n,n=3..infinity)" "6#/- %$SumG6$&%\"aG6#%\"nG/F*;\"\"$%)infinityG-F%6$*&F*\"\"\")\"\"#F*!\"\"/ F*;F-F." }{TEXT -1 34 " converges by comparison with the " }{TEXT 256 16 "geometric series" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[3]*(3/4)^k ,k = 0 .. infinity) = Sum(``(3/8)*(3/4)^k,k = 0 .. infinity);" "6#/-%$ SumG6$*&&%\"aG6#\"\"$\"\"\")*&F+F,\"\"%!\"\"%\"kGF,/F1;\"\"!%)infinity G-F%6$*&-%!G6#*&F+F,\"\")F0F,)*&F+F,F/F0F1F,/F1;F4F5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 75 "In fact the first series converges to 1 and the second series converges to " }{XPPEDIT 18 0 "``(3/8)/``(1-3/4 ) = 3/2;" "6#/*&-%!G6#*&\"\"$\"\"\"\"\")!\"\"F*-F&6#,&F*F**&F)F*\"\"%F ,F,F,*&F)F*\"\"#F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "Sum((n+2)/2^(n+2),n=1..infin ity);\nvalue(%);\nSum(3/8*(3/4)^(n-1),n=1..infinity);\nvalue(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&,&%\"nG\"\"\"\"\"#F)F))F*F' !\"\"/F(;F)%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$)#\"\"$\"\"%,&%\"nG\"\"\"! \"\"F-#F)\"\")/F,;F-%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\" \"$\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "We can set up functions to provide values for partial sums of thes e two series and draw a picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Sum((i+2)/2^(i+2),i=1..n); \nPSB := unapply(value(%),n);\nSum(3/8*(3/4)^(i-1),i=1..n);\nPSA := un apply(value(%),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&,&%\" iG\"\"\"\"\"#F)F))F*F'!\"\"/F(;F)%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PSBGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,(*&#\"\"$\"\"#\"\"\") #F1F0,&9$F1F1F1F1!\"\"*&#F1F0F1*&F2F1F4F1F1F6F1F1F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$SumG6$,$*(\"\"$\"\"\"\"\")!\"\")#F(\"\"%,&%\" iGF)F)F+F)F)/F0;F)%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PSAGf*6# %\"nG6\"6$%)operatorG%&arrowGF(,&*&\"\"#\"\"\")#\"\"$\"\"%,&9$F/F/F/F/ !\"\"#F2F.F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "ptsA := [seq([n,PSA(n)],n=1..15)]:\nptsB : = [seq([n,PSB(n)],n=1..15)]:\nplot([PSA(x),PSB(x),3/2,1,ptsA,ptsB],x=0 ..15,style=[line$4,point$2],\n color=[gray$2,brown,cyan,red,blue],li nestyle=[2,2,3,3],\n symbol=[circle,box],view=[0..15,0..2],labels=[` n`,``]);" }}{PARA 13 "" 1 "" 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first two terms of the series which are both " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 13 ", the series " } {XPPEDIT 18 0 "Sum(n/(2^n),n = 1 .. infinity)" "6#-%$SumG6$*&%\"nG\"\" \")\"\"#F'!\"\"/F';F(%)infinityG" }{TEXT -1 16 " converges to 2." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Sum(n/2^n,n=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&%\"nG\"\"\")\"\"#F'!\"\"/F';F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 34 "Hybrid arithmetic/geometric series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 118 "The series in example 1 as a sort of hybrid arithmetic/geometric series for whic h we can develop the following theory." }}{PARA 0 "" 0 "" {TEXT -1 68 "Using the formula for the sum of a finite geometric series we have: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a+a*r+a*r^2+a*r^3 +a*r^4+` . . . `+a*r^n = a*(1-r^(n+1))/(1-r);" "6#/,0%\"aG\"\"\"*&F%F& %\"rGF&F&*&F%F&*$F(\"\"#F&F&*&F%F&*$F(\"\"$F&F&*&F%F&*$F(\"\"%F&F&%(~. ~.~.~GF&*&F%F&)F(%\"nGF&F&*(F%F&,&F&F&)F(,&F5F&F&F&!\"\"F&,&F&F&F(F:F: " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "Differentiating this equation with respect to " }{TEXT 285 1 "r" }{TEXT -1 8 " gives: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a+2*a*r+3*a*r^2+4*a*r ^3+` . . . `+n*a*r^(n-1) = a*(-(n+1)*r^n*(1-r)+1-r^(n+1))/((1-r)^2);" "6#/,.%\"aG\"\"\"*(\"\"#F&F%F&%\"rGF&F&*(\"\"$F&F%F&F)F(F&*(\"\"%F&F%F &F)F+F&%(~.~.~.~GF&*(%\"nGF&F%F&)F),&F0F&F&!\"\"F&F&*(F%F&,(*(,&F0F&F& F&F&)F)F0F&,&F&F&F)F3F&F3F&F&)F),&F0F&F&F&F3F&*$,&F&F&F)F3F(F3" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "which gives the formula " }}{PARA 256 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "Sum(i*a*r^(i -1),i = 1 .. n) = a*(n*r^(n+1)-(n+1)*r^n+1)/((1-r)^2);" "6#/-%$SumG6$* (%\"iG\"\"\"%\"aGF))%\"rG,&F(F)F)!\"\"F)/F(;F)%\"nG*(F*F),(*&F1F))F,,& F1F)F)F)F)F)*&,&F1F)F)F)F))F,F1F)F.F)F)F)*$,&F)F)F,F.\"\"#F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Sum(i*a*r^(i-1),i=1..n);\nmap(simplify,normal(map(sim plify,value(%))));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(%\" iG\"\"\"%\"aGF()%\"rG,&F'F(F(!\"\"F(/F';F(%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"aG\"\"\",**&)%\"rG,&%\"nGF%F%F%F%F+F%F%*&)F)F+F%F+ F%!\"\"F-F.F%F%F%,&F)F%F%F.!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Note that, if " }{XPPEDIT 18 0 "abs(r)<1" "6#2-%$absG6#%\"rG\"\" \"" }{TEXT -1 22 ", the infinite series " }{XPPEDIT 18 0 "Sum(n*a*r^(n -1),n = 1 .. infinity);" "6#-%$SumG6$*(%\"nG\"\"\"%\"aGF()%\"rG,&F'F(F (!\"\"F(/F';F(%)infinityG" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "a/(1-r)^2" "6#*&%\"aG\"\"\"*$,&F%F%%\"rG!\"\"\"\"#F)" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 16 " Multiplying by " }{TEXT 286 1 "r" } {TEXT -1 6 " gives" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(n*a*r^n,n = 1 .. infinity) = a*r/((1-r)^2);" "6#/-%$SumG6$*(%\" nG\"\"\"%\"aGF))%\"rGF(F)/F(;F)%)infinityG*(F*F)F,F)*$,&F)F)F,!\"\"\" \"#F3" }{TEXT -1 8 ", when " }{XPPEDIT 18 0 "abs(r)<1" "6#2-%$absG6#% \"rG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{MPLTEXT 1 0 3 "sum" }{TEXT -1 25 " pro cedure assumes that " }{XPPEDIT 18 0 "abs(r)<1" "6#2-%$absG6#%\"rG\" \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Sum(n*a*r^n,n=1..infinity);\nvalue(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(%\"nG\"\"\"%\"aGF()%\"rGF' F(/F';F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"aG\"\"\"% \"rGF%,&F&F%F%!\"\"!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 11 "The series " }{XPPEDIT 18 0 "Sum(n/(2^n),n = 1 .. \+ infinity)" "6#-%$SumG6$*&%\"nG\"\"\")\"\"#F'!\"\"/F';F(%)infinityG" } {TEXT -1 27 " of example 1 has the form " }{XPPEDIT 18 0 "Sum(n*a*r^(n -1),n = 1 .. infinity);" "6#-%$SumG6$*(%\"nG\"\"\"%\"aGF()%\"rG,&F'F(F (!\"\"F(/F';F(%)infinityG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a=1 " "6#/%\"aG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r=1/2" "6#/%\"r G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 30 ", so it converges to the sum " } {XPPEDIT 18 0 "``(1/2)/((1/2)^2) = 2;" "6#/*&-%!G6#*&\"\"\"F)\"\"#!\" \"F)*$*&F)F)F*F+F*F+F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the series " }{XPPEDIT 18 0 "Sum(n!/(2*n)!,n=1..infinity)" "6#-%$SumG6$*&-%*factorialG6#%\"nG \"\"\"-F(6#*&\"\"#F+F*F+!\"\"/F*;F+%)infinityG" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a[n]=n!/(2*n)!" "6#/ &%\"aG6#%\"nG*&-%*factorialG6#F'\"\"\"-F*6#*&\"\"#F,F'F,!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "a[n+1]/a [n] = ``((n+1)!/(2*n+2)!)/``(n!/(2*n)!);" "6#/*&&%\"aG6#,&%\"nG\"\"\"F *F*F*&F&6#F)!\"\"*&-%!G6#*&-%*factorialG6#,&F)F*F*F*F*-F46#,&*&\"\"#F* F)F*F*F;F*F-F*-F06#*&-F46#F)F*-F46#*&F;F*F)F*F-F-" }{XPPEDIT 18 0 "`` \+ = ``((n+1)!/(2*n+2)!)*``((2*n)!/n!);" "6#/%!G*&-F$6#*&-%*factorialG6#, &%\"nG\"\"\"F.F.F.-F*6#,&*&\"\"#F.F-F.F.F3F.!\"\"F.-F$6#*&-F*6#*&F3F.F -F.F.-F*6#F-F4F." }{XPPEDIT 18 0 " ``= (n+1)/((2*n+2)*(2*n+1))" "6#/%! G*&,&%\"nG\"\"\"F(F(F(*&,&*&\"\"#F(F'F(F(F,F(F(,&*&F,F(F'F(F(F(F(F(!\" \"" }{XPPEDIT 18 0 "`` = 1/(2*(2*n+1));" "6#/%!G*&\"\"\"F&*&\"\"#F&,&* &F(F&%\"nGF&F&F&F&F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "rho=Limit (a[n+1]/a[n],n=infinity)" "6#/%$rhoG-%&LimitG6$*&&%\"aG6#,&%\"nG\"\"\" F.F.F.&F*6#F-!\"\"/F-%)infinityG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "L imit(1/(2*(2*n+1)),n = infinity) = 0;" "6#/-%&LimitG6$*&\"\"\"F(*&\"\" #F(,&*&F*F(%\"nGF(F(F(F(F(!\"\"/F-%)infinityG\"\"!" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "rho<1" "6#2%$rhoG \"\"\"" }{TEXT -1 41 ", the series converges by the ratio test." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "a := n -> n!/(2*n)!;\nLimit(a(n+1)/a(n),n=infinity);\nsimplify(%); \nvalue(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$% )operatorG%&arrowGF(*&-%*factorialG6#9$\"\"\"-F.6#,$F0\"\"#!\"\"F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&*&-%*factorialG6#,&% \"nG\"\"\"F-F-F--F)6#,$F,\"\"#F-F-*&-F)6#,&F,F1F1F-F--F)6#F,F-!\"\"/F, %)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&\"\"\"F( ,&%\"nG\"\"#F(F(!\"\"#F(F+/F*%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 3 "sum" }{TEXT -1 57 " can find \+ an explicit formula for the sum of this series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Sum(n!/(2*n) !,n=1..infinity);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%*factorialG6#%\"nG\"\"\"-F(6#,$*&\"\"#F+F*F+F+!\"\" /F*;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F &**F'#F&\"\"#-%$expG6##F&F'F&%#PiGF),&F&!\"\"-%%erfcG6#F)F&F&F&F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+l`'H#f!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "The convergence is so fas t that the 9 th partial sum agrees with this last value to 10 digits. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "Sum(i!/(2*i)!,i=1..n):\nPS := unapply(value(%),n):\nseq(evalf( evalf(PS(n),14)),n=1..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$\"+++++ ]!#5$\"+LLLLeF%$\"+nmm;fF%$\"+[!>E#fF%$\"+N(\\H#fF%$\"+nZ'H#fF%$\"+X`' H#fF%$\"+k`'H#fF%$\"+l`'H#fF%F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the series " }{XPPEDIT 18 0 "Sum(5^n/n!,n = 1 .. infinity);" "6#-%$SumG6$*&)\"\"&%\"nG\"\"\"-%*f actorialG6#F)!\"\"/F);F*%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a[n] = 5^n/n!;" "6#/&%\"aG6#%\"nG* &)\"\"&F'\"\"\"-%*factorialG6#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "a[n+1]/a[n] = ``(5^(n+1)/(n+1)!) /``(5^n/n!);" "6#/*&&%\"aG6#,&%\"nG\"\"\"F*F*F*&F&6#F)!\"\"*&-%!G6#*&) \"\"&,&F)F*F*F*F*-%*factorialG6#,&F)F*F*F*F-F*-F06#*&)F4F)F*-F76#F)F-F -" }{XPPEDIT 18 0 "`` = ``(5^(n+1)/(n+1)!)*``(n!/(5^n));" "6#/%!G*&-F$ 6#*&)\"\"&,&%\"nG\"\"\"F-F-F--%*factorialG6#,&F,F-F-F-!\"\"F--F$6#*&-F /6#F,F-)F*F,F2F-" }{XPPEDIT 18 0 "``= 5/(n+1)" "6#/%!G*&\"\"&\"\"\",&% \"nGF'F'F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "rho=Limit(a[n+1]/a[ n],n=infinity)" "6#/%$rhoG-%&LimitG6$*&&%\"aG6#,&%\"nG\"\"\"F.F.F.&F*6 #F-!\"\"/F-%)infinityG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit(5/(n+ 1),n = infinity) = 0;" "6#/-%&LimitG6$*&\"\"&\"\"\",&%\"nGF)F)F)!\"\"/ F+%)infinityG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Sin ce " }{XPPEDIT 18 0 "rho<1" "6#2%$rhoG\"\"\"" }{TEXT -1 41 ", the seri es converges by the ratio test." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "a := n -> 5^n/n!;\nLimit(a(n +1)/a(n),n=infinity);\nsimplify(%);\nvalue(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&)\"\"&9$\"\" \"-%*factorialG6#F/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&L imitG6$*&*&)\"\"&,&%\"nG\"\"\"F,F,F,-%*factorialG6#F+F,F,*&-F.6#F*F,)F )F+F,!\"\"/F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6 $,$*&\"\"\"F(,&%\"nGF(F(F(!\"\"\"\"&/F*%)infinityG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 3 "sum" }{TEXT -1 57 " can find an explicit formula for the sum of this series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Sum( 5^n/n!,n=1..infinity);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)\"\"&%\"nG\"\"\"-%*factorialG6#F)!\"\"/F);F *%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#\"\"&\"\"\" F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"fJTZ\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The convergence is quite fast." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 83 "Sum(5^i/i!,i=1..n):\nPS := unapply(value(%),n) :\nseq(evalf(evalf(PS(n),14)),n=1..24);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6:$\"+++++]!\"*$\"++++]w7F1$\"+x;2t8F1$\"+mX*oU\"F1$\"+5g!QX\"F1$\"+&[QgY\"F1 $\"+]`8r9F1$\"+#o&4t9F1$\"+,ezt9F1$\"+u\"HSZ\"F1$\"+.@5u9F1$\"+_N7u9F1 $\"+6&HTZ\"F1$\"+z58u9F1$\"+r98u9F1$\"+k:8u9F1$\"+&eJTZ\"F1$\"+!fJTZ\" F1$\"+\"fJTZ\"F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "PS := unapply(sum(5^i/i!,i=1..n),n):\npts := [seq([n,PS(n)],n=1..15)]:\nplot([PS(x),147.4131591,pts],x=0..15,style =[line$2,point],\n color=[gray,brown,red],linestyle=[2,3],symbol=cir cle,\n view=[0..15,0..150],labels=[`n`,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 498 232 232 {PLOTDATA 2 "6(-%'CURVESG6&7S7$$\"\"!F)F(7$$\"+D JdpK!#5$\"+g\"pY+\"!\"*7$$\"+>7T9hF-$\"+!eGCL#F07$$\"+)=HPJ*F-$\"+ga9 \\WF07$$\"+JaU`7F0$\"+wA?dtF07$$\"+%GZRd\"F0$\"+Yu666!\")7$$\"+s?6r=F0 $\"+f@zR:FE7$$\"+(**3)y@F0$\"+a3\\l?FE7$$\"+(fHq\\#F0$\"+-:X#p#FE7$$\" +f'HU\"GF0$\"+7b&HR$FE7$$\"+7*309$F0$\"+1'R*yTFE7$$\"+ce*yU$F0$\"+F8&H \"\\FE7$$\"+)[D9v$F0$\"+C(Q$pdFE7$$\"+iNGwSF0$\"+!fEGk'FE7$$\"+7XM*Q%F 0$\"+D`*)zuFE7$$\"+ZQjtYF0$\"+hVwA#)FE7$$\"+i8o6]F0$\"+Lk9q!*FE7$$\"+] >0)H&F0$\"+*y%\\Z(*FE7$$\"+A!p6j&F0$\"+1k%z/\"!\"(7$$\"+vS.EfF0$\"+()* zr5\"Faq7$$\"+sxa\\iF0$\"+@e#e;\"Faq7$$\"+;$4wb'F0$\"+7OT:7Faq7$$\"+># R!zoF0$\"+(*4wg7Faq7$$\"+4A@urF0$\"+(*4!pH\"Faq7$$\"+chf#\\(F0$\"+#)GO I8Faq7$$\"+(f2L#yF0$\"+gRdf8Faq7$$\"+yG>6\")F0$\"+[,(3Q\"Faq7$$\"+po6A %)F0$\"+%R`+S\"Faq7$$\"+v1\"FE$\"+$GXDY\"Faq7 $$\"+R(zS4\"FE$\"+\"HnbY\"Faq7$$\"+-,FC6FE$\"+X6tn9Faq7$$\"+Jx#e:\"FE$ \"+*yV%p9Faq7$$\"+\"3\"o'=\"FE$\"+Jvpq9Faq7$$\"+!o\")*=7FE$\"+*yp;Z\"F aq7$$\"+&*44]7FE$\"+foNs9Faq7$$\"+jZ!>G\"FE$\"+E+(GZ\"Faq7$$\"+(4bMJ\" FE$\"+3\"QKZ\"Faq7$$\"+ylWU8FE$\"+%[%[t9Faq7$$\"+'3ucP\"FE$\"+'R(ot9Fa q7$$\"+lJR09FE$\"+]h\"QZ\"Faq7$$\"+-*zqV\"FE$\"+#*Q\"RZ\"Faq7$$\"+`\"3 uY\"FE$\"+$zzRZ\"Faq7$$\"#:F)$\"+u\"HSZ\"Faq-%'COLOURG6&%$RGBG$\")=THv FEF\\[lF\\[l-%&STYLEG6#%%LINEG-%*LINESTYLEG6#\"\"#-F$6&7S7$F($\"3-+++ \"fJTZ\"!#:7$$\"3')*****\\7t&pK!#=Fj[l7$$\"3$****\\(=7T9hF`\\lFj[l7$$ \"3X****\\(=HPJ*F`\\lFj[l7$$\"3;++DJaU`7!#0)H&Fj\\lFj[l7$$\"3Y**\\(=-p6j&Fj\\lFj[l7$ $\"3d*****\\2Mg#fFj\\lFj[l7$$\"35+](=xZ&\\iFj\\lFj[l7$$\"3;+]i:$4wb'Fj \\lFj[l7$$\"3-++v=#R!zoFj\\lFj[l7$$\"3q+]P4A@urFj\\lFj[l7$$\"3I++Dchf# \\(Fj\\lFj[l7$$\"3))**\\(of2L#yFj\\lFj[l7$$\"3M**\\7yG>6\")Fj\\lFj[l7$ $\"3w++voo6A%)Fj\\lFj[l7$$\"3q*****\\xJLu)Fj\\lFj[l7$$\"3W++v$*ydd!*Fj \\lFj[l7$$\"3#***\\(=1\"F_b lFj[l7$$\"3=++vQ(zS4\"F_blFj[l7$$\"3***\\(=-,FC6F_blFj[l7$$\"33+v$4tFe :\"F_blFj[l7$$\"3!****\\73\"o'=\"F_blFj[l7$$\"3-+voz;)*=7F_blFj[l7$$\" 31+++&*44]7F_blFj[l7$$\"35+]7jZ!>G\"F_blFj[l7$$\"34+v=(4bMJ\"F_blFj[l7 $$\"3;++]xlWU8F_blFj[l7$$\"39+]i&3ucP\"F_blFj[l7$$\"3\"******\\;$R09F_ blFj[l7$$\"38+v=-*zqV\"F_blFj[l7$$\"33+D\"G:3uY\"F_blFj[l7$FdzFj[l-Fiz 6&F[[l$\")#)eqkFE$\"))eqk\"FEFbelF^[l-Fc[l6#\"\"$-F$6&717$$\"\"\"F)$\" \"&F)7$$Fe[lF)$\"3+++++++]w/>w7F\\\\l7$$\"\")F)$\"3],tew;2t8F\\\\l7$$\"\"*F)$\"37 \\mplX*oU\"F\\\\l7$$\"#5F)$\"3yA8D5g!QX\"F\\\\l7$$\"#6F)$\"3!=!*[][QgY \"F\\\\l7$$\"#7F)$\"3)G*ya\\`8r9F\\\\l7$$\"#8F)$\"3!\\]Z?o&4t9F\\\\l7$ $\"#9F)$\"3Emtz+ezt9F\\\\l7$Fdz$\"3+()Rrt\"HSZ\"F\\\\l-Fiz6&F[[l$\"*++ ++\"FEF(F(-F_[l6#%&POINTGFb[l-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$%\"n G%!G-%%VIEWG6$;F(Fdz;F($\"$]\"F)" 1 2 4 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the s eries " }{XPPEDIT 18 0 "Sum((n/3)^(n-1)*``(1/(n-1)!),n = 1 .. infinity );" "6#-%$SumG6$*&)*&%\"nG\"\"\"\"\"$!\"\",&F)F*F*F,F*-%!G6#*&F*F*-%*f actorialG6#,&F)F*F*F,F,F*/F);F*%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a[n] = (n/3)^(n-1)*``(1/(n -1)!);" "6#/&%\"aG6#%\"nG*&)*&F'\"\"\"\"\"$!\"\",&F'F+F+F-F+-%!G6#*&F+ F+-%*factorialG6#,&F'F+F+F-F-F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "a[n+1]/a[n] = ((n+1)/3)^n*``(1/n!)/ ((n/3)^(n-1)*``(1/(n-1)!));" "6#/*&&%\"aG6#,&%\"nG\"\"\"F*F*F*&F&6#F)! \"\"*()*&,&F)F*F*F*F*\"\"$F-F)F*-%!G6#*&F*F*-%*factorialG6#F)F-F**&)*& F)F*F2F-,&F)F*F*F-F*-F46#*&F*F*-F86#,&F)F*F*F-F-F*F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``= ((n+1)/3)^n*``((n-1)!/n!)*(3/n)^(n-1)" "6#/%!G*()* &,&%\"nG\"\"\"F*F*F*\"\"$!\"\"F)F*-F$6#*&-%*factorialG6#,&F)F*F*F,F*-F 16#F)F,F*)*&F+F*F)F,,&F)F*F*F,F*" }{XPPEDIT 18 0 " ``= ``(1/3)*((n+1)/ n)^n" "6#/%!G*&-F$6#*&\"\"\"F)\"\"$!\"\"F))*&,&%\"nGF)F)F)F)F/F+F/F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "rho=Limit(a[n+1]/a[n],n=infinity)" "6#/%$rhoG-%&LimitG6$*&&%\"aG6#,&%\"nG\"\"\"F.F.F.&F*6#F-!\"\"/F-%)inf inityG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit(``(1/3)*((n+1)/n)^n,n = infinity) = e/3;" "6#/-%&LimitG6$*&-%!G6#*&\"\"\"F,\"\"$!\"\"F,)*&, &%\"nGF,F,F,F,F2F.F2F,/F2%)infinityG*&%\"eGF,F-F." }{TEXT -1 1 " " } {TEXT 259 1 "~" }{TEXT -1 14 " 0.9060939428." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "rho<1" "6#2%$rhoG\"\"\"" }{TEXT -1 41 ", the series converges by the ratio test." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "a := n -> (n/3)^(n-1 )/(n-1)!;\nLimit(a(n+1)/a(n),n=infinity);\nsimplify(%);\nvalue(%);\nev alf(evalf(%,13));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG 6\"6$%)operatorG%&arrowGF(*&),$*&#\"\"\"\"\"$F19$F1F1,&F3F1F1!\"\"F1-% *factorialG6#F4F5F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$ **),&*&\"\"$!\"\"%\"nG\"\"\"F-#F-F*F-F,F--%*factorialG6#F,F+),$*&F*F+F ,F-F-,&F,F-F-F+F+-F06#F5F-/F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*(\"\"$!\"\")%\"nG,$F+F)\"\"\"),&F+F-F-F-F+F-F-/F+% )infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&-%$exp G6#F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G%R41*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "We can estimate the sum of this series by calculating some partial sums." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "Sum( (i/3)^(i-1)/(i-1)!,i =1..n);\nPS := unapply(value(%),n):\nevalf(PS(10) );\nevalf(PS(100));\nevalf(PS(200));\nevalf(PS(300));\nevalf(PS(400)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&),$%\"iG#\"\"\"\"\"$,&F )F+!\"\"F+F+-%*factorialG6#F-F./F);F+%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V>.nQ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+q^ Av[!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+XDGv[!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+ZDGv[!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+ZDGv[!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "It is time consuming to calculate large partial sums usin g " }{TEXT 0 3 "sum" }{TEXT -1 75 ", but the following procedure calcu lates the partial sums more efficiently." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "PS := proc(n)\n lo cal i,sum,term;\n sum := 1.;\n term := 1.;\n Digits := 15;\n f or i from 1 to n-1 do\n term := term*((1+1./i)^i)/3;\n sum : = sum+term;\n end do;\n sum;\nend:\nevalf(evalhf(PS(10)));\nevalf( evalhf(PS(100)));\nevalf(evalhf(PS(200)));\nevalf(evalhf(PS(300)));\ne valf(evalhf(PS(400)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V>.nQ!\" *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+q^Av[!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+XDGv[!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ ZDGv[!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ZDGv[!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "These computati ons show that " }{XPPEDIT 18 0 "Sum((n/3)^(n-1)*``(1/(n-1)!),n = 1 .. infinity);" "6#-%$SumG6$*&)*&%\"nG\"\"\"\"\"$!\"\",&F)F*F*F,F*-%!G6#* &F*F*-%*factorialG6#,&F)F*F*F,F,F*/F);F*%)infinityG" }{TEXT -1 1 " " } {TEXT 260 1 "~" }{TEXT -1 13 " 4.875282547." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "pts := [seq([n,PS (n)],n=1..30)]:\nplot([pts,4.875282547,pts],x=0..30,style=[line$2,poin t],\n color=[gray,brown,red],linestyle=[2,3],symbol=circle,\n view =[0..30,0..5],labels=[`n`,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 451 242 242 {PLOTDATA 2 "6(-%'CURVESG6&7@7$$\"\"\"\"\"!F(7$$\"\"#F*$\"31qmmmmm m;!#<7$$\"\"$F*$\"31qmmmmmm@F07$$\"\"%F*$\"3+I<1&RGr*)\\JF07$$\"\"(F*$\"3;S'=doTSP$F07$ $\"\")F*$\"3#**[4D/.Vc$F07$$\"\"*F*$\"3<+_zci-FPF07$$\"#5F*$\"3#)R@/V> .nQF07$$\"#6F*$\"31]o1u\"y!))RF07$$\"#7F*$\"3'*RjZKV:$4%F07$$\"#8F*$\" 3m\\Z]-pn%=%F07$$\"#9F*$\"3I+TEX\\ikUF07$$\"#:F*$\"3))R-D?ojMVF07$$\"# ;F*$\"3w\\mzy53'R%F07$$\"#F*$\"32S$R.1y'RXF07$$\"#?F*$\"3m\\$['p2wwXF07$$\"#@F*$\"3$ )f][Dzb4YF07$$\"#AF*$\"3W!)=M=sfQYF07$$\"#BF*$\"3Qg]\")>aLkYF07$$\"#CF *$\"3?S)pi1pro%F07$$\"#DF*$\"3')R*o))HVuq%F07$$\"#EF*$\"3d4v[F#fas%F07 $$\"#FF*$\"36Ik*>5![TZF07$$\"#GF*$\"3!*HD.7ltbZF07$$\"#HF*$\"3Aq))\\*y I%oZF07$$\"#IF*$\"3q>`Q_2uzZF0-%'COLOURG6&%$RGBG$\")=THv!\")F[uF[u-%&S TYLEG6#%%LINEG-%*LINESTYLEG6#F--F$6&7S7$$F*F*$\"3I+++ZDGv[F07$$\"3s*** ***\\i9Rl!#=Fju7$$\"3*****\\PC#)GA\"F0Fju7$$\"3))****\\Peui=F0Fju7$$\" 3L++]i3&o]#F0Fju7$$\"3'****\\(oX*y9$F0Fju7$$\"3o***\\P9CAu$F0Fju7$$\"3 #****\\P*zhdVF0Fju7$$\"31++v$>fS*\\F0Fju7$$\"30++v=$f%GcF0Fju7$$\"3;++ +Dy,\"G'F0Fju7$$\"3I++]7&)\\\"F]yFju7$$\"3)***\\P> :mk:F]yFju7$$\"3()**\\iv&QAi\"F]yFju7$$\"3;++vtLU%o\"F]yFju7$$\"3%**** **\\Nm'[F]yFju7$$\"3z*****\\@80+#F]yFju7$$\"3)*****\\7,Hl?F]yFju7$$\"3() **\\P4w)R7#F]yFju7$$\"3O++]x%f\")=#F]yFju7$$\"3)***\\P/-a[AF]yFju7$$\" 38+](=Yb;J#F]yFju7$$\"3y****\\i@OtBF]yFju7$$\"3/+]PfL'zV#F]yFju7$$\"35 +++!*>=+DF]yFju7$$\"3?++DE&4Qc#F]yFju7$$\"3=+]P%>5pi#F]yFju7$$\"3K+++b J*[o#F]yFju7$$\"3E++Dr\"[8v#F]yFju7$$\"3#)******Hjy5GF]yFju7$$\"3E+]P/ )fT(GF]yFju7$$\"3;+]i0j\"[$HF]yFju7$FctFju-Fht6&Fjt$\")#)eqkF]u$\"))eq k\"F]uF`_lF^u-Fcu6#F3-F$6&F&-Fht6&Fjt$\"*++++\"F]uFiuFiu-F_u6#%&POINTG Fbu-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$%\"nG%!G-%%VIEWG6$;FiuFct;FiuF <" 1 2 4 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Raabe's test" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 270 12 "Raabe's test" }{TEXT -1 108 " is somewhat similar to the ratio test in that it also involves the ratio of successive terms of the se ries." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "Sum(a[n],n= 1..infinity)" "6#-%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 42 " be a series of positive terms, such that " }{XPPEDIT 18 0 "Lim it(n*``(a[n]/a[n+1]-1),n = infinity) = sigma;" "6#/-%&LimitG6$*&%\"nG \"\"\"-%!G6#,&*&&%\"aG6#F(F)&F06#,&F(F)F)F)!\"\"F)F)F5F)/F(%)infinityG %&sigmaG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "Then the ser ies " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity)" "6#-%$SumG6$&%\"aG6 #%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 15 " converges if " }{XPPEDIT 18 0 "1 < sigma;" "6#2\"\"\"%&sigmaG" }{TEXT -1 17 " and diverges if \+ " }{XPPEDIT 18 0 "sigma < 1;" "6#2%&sigmaG\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "sigma = 1;" "6#/%&sig maG\"\"\"" }{TEXT -1 32 ", then the test is inconclusive." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 274 28 "____________________________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "1 < sigma;" "6#2\"\"\"%&sigmaG" } {TEXT -1 18 ", choose a number " }{TEXT 287 1 "d" }{TEXT -1 11 " such \+ that " }{XPPEDIT 18 0 "1 < 1+d;" "6#2\"\"\",&F$F$%\"dGF$" }{XPPEDIT 18 0 "`` < sigma;" "6#2%!G%&sigmaG" }{TEXT -1 29 ". For example, we c an take " }{XPPEDIT 18 0 "d = (sigma-1)/2;" "6#/%\"dG*&,&%&sigmaG\"\" \"F(!\"\"F(\"\"#F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "In \+ order to prove that the series " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. inf inity)" "6#-%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 71 " converges, it is sufficient to show that the sequence of partial sums " }{XPPEDIT 18 0 "s[n]=Sum(a[i],i=1..n)" "6#/&%\"sG6#%\"nG-%$SumG6$&% \"aG6#%\"iG/F.;\"\"\"F'" }{TEXT -1 34 " is bounded above by some numb er " }{TEXT 288 1 "K" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "T he condition " }{XPPEDIT 18 0 "Limit(n*``(a[n]/a[n+1]-1),n = infinity) = sigma;" "6#/-%&LimitG6$*&%\"nG\"\"\"-%!G6#,&*&&%\"aG6#F(F)&F06#,&F( F)F)F)!\"\"F)F)F5F)/F(%)infinityG%&sigmaG" }{TEXT -1 26 " means that \+ we can make " }{XPPEDIT 18 0 "n*``(a[n]/a[n+1]-1);" "6#*&%\"nG\"\"\"- %!G6#,&*&&%\"aG6#F$F%&F,6#,&F$F%F%F%!\"\"F%F%F1F%" }{TEXT -1 24 " as c lose as we like to " }{XPPEDIT 18 0 "sigma;" "6#%&sigmaG" }{TEXT -1 11 " by taking " }{TEXT 290 1 "n" }{TEXT -1 20 " sufficiently large." }}{PARA 0 "" 0 "" {TEXT -1 26 "Hence there is an integer " }{TEXT 289 1 "N" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "1+d=N" "6#1%\"NG%\"nG" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " } {XPPEDIT 18 0 "a[n+1]+d*a[n+1]=N" "6#1 %\"NG%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Now we write down this last inequality for " } {XPPEDIT 18 0 "n = N,N+1,N+2,` . . . `,m-1;" "6'/%\"nG%\"NG,&F%\"\"\"F 'F',&F%F'\"\"#F'%(~.~.~.~G,&%\"mGF'F'!\"\"" }{TEXT -1 9 ", where " } {XPPEDIT 18 0 "m>N" "6#2%\"NG%\"mG" }{TEXT -1 2 ": " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d*a [N+1] 0" "6#2\"\"!%\"dG" }{TEXT -1 11 " such that " } {XPPEDIT 18 0 "sigma < 1-d;" "6#2%&sigmaG,&\"\"\"F&%\"dG!\"\"" } {XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\"" }{TEXT -1 29 ". For example, w e can take " }{XPPEDIT 18 0 "r = (1-sigma)/2;" "6#/%\"rG*&,&\"\"\"F'% &sigmaG!\"\"F'\"\"#F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 " Then we can choose an integer " }{TEXT 293 1 "N" }{TEXT -1 11 " such t hat " }{XPPEDIT 18 0 "n*``(a[n]/a[n+1]-1) < 1-d;" "6#2*&%\"nG\"\"\"-%! G6#,&*&&%\"aG6#F%F&&F-6#,&F%F&F&F&!\"\"F&F&F2F&,&F&F&%\"dGF2" }{TEXT -1 10 " for all " }{TEXT 264 1 "n" }{TEXT -1 11 " such that " } {XPPEDIT 18 0 "n >=N" "6#1%\"NG%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "n*a[n]-n*a[n+1]=N" "6#1%\"NG%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Hence " }{XPPEDIT 18 0 "n*a[n] < (n+1)*a[n+1];" "6#2*&%\"nG\"\"\"&%\"aG6#F%F&*&,&F%F&F&F&F& &F(6#,&F%F&F&F&F&" }{TEXT -1 10 " for all " }{TEXT 266 1 "n" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "n >=N" "6#1%\"NG%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "I t follows that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "N*a [N] < (N+1)*a[N+1];" "6#2*&%\"NG\"\"\"&%\"aG6#F%F&*&,&F%F&F&F&F&&F(6#, &F%F&F&F&F&" }{XPPEDIT 18 0 "`` < (N+2)*a[N+2];" "6#2%!G*&,&%\"NG\"\" \"\"\"#F(F(&%\"aG6#,&F'F(F)F(F(" }{XPPEDIT 18 0 "`` < ` . . . `;" "6#2 %!G%(~.~.~.~G" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "N*a[N] <= i*a[i] ;" "6#1*&%\"NG\"\"\"&%\"aG6#F%F&*&%\"iGF&&F(6#F+F&" }{TEXT -1 9 " for \+ all " }{TEXT 294 1 "i" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "N <= i;" "6#1%\"NG%\"iG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "or \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "N*a[N]/i <= a[i]; " "6#1*(%\"NG\"\"\"&%\"aG6#F%F&%\"iG!\"\"&F(6#F*" }{TEXT -1 9 " for al l " }{TEXT 267 1 "i" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "N <= i ;" "6#1%\"NG%\"iG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "Then , for all " }{TEXT 268 1 "m" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "N < m;" "6#2%\"NG%\"mG" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i],i = 1 .. m) = Sum(a[i],i = 1 .. N-1) +Sum(a[i],i = N .. m);" "6#/-%$SumG6$&%\"aG6#%\"iG/F*;\"\"\"%\"mG,&-F% 6$&F(6#F*/F*;F-,&%\"NGF-F-!\"\"F--F%6$&F(6#F*/F*;F7F.F-" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 4 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i],i = 1 .. N-1)+N*a[N]*Sum(1/i,i = N .. m ) <= Sum(a[i],i = 1 .. m);" "6#1,&-%$SumG6$&%\"aG6#%\"iG/F+;\"\"\",&% \"NGF.F.!\"\"F.*(F0F.&F)6#F0F.-F&6$*&F.F.F+F1/F+;F0%\"mGF.F.-F&6$&F)6# F+/F+;F.F:" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " } {XPPEDIT 18 0 "Sum(1/i,i = N+1 .. m)" "6#-%$SumG6$*&\"\"\"F'%\"iG!\"\" /F(;,&%\"NGF'F'F'%\"mG" }{TEXT -1 33 " is a partial sum of the series " }{XPPEDIT 18 0 "Sum(1/i,i = N+1 .. infinity);" "6#-%$SumG6$*&\"\"\" F'%\"iG!\"\"/F(;,&%\"NGF'F'F'%)infinityG" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 94 "Since this last series is a tail of the harmonic s eries, we know that it diverges to infinity." }}{PARA 0 "" 0 "" {TEXT -1 73 "Thus the left hand side of the last inequality increases indefi nitely as " }{TEXT 295 1 "m" }{TEXT -1 79 " increases. This means that the right hand side also increases indefinitely as " }{TEXT 296 1 "m " }{TEXT -1 32 " increases, that is, the series " }{XPPEDIT 18 0 "Sum( a[n],n = 1 .. infinity)" "6#-%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinity G" }{TEXT -1 22 " diverges to infinity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 21 "Raabe's test examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the series " }{XPPEDIT 18 0 "Sum(1/(n^2),n = 1 .. infi nity) = 1+1/4+1/9+1/16+1/25+` . . . `;" "6#/-%$SumG6$*&\"\"\"F(*$%\"nG \"\"#!\"\"/F*;F(%)infinityG,.F(F(*&F(F(\"\"%F,F(*&F(F(\"\"*F,F(*&F(F( \"#;F,F(*&F(F(\"#DF,F(%(~.~.~.~GF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a[n] = 1/(n^2);" "6#/&%\"aG6#%\"nG *&\"\"\"F)*$F'\"\"#!\"\"" }{TEXT -1 8 ". Then " }{XPPEDIT 18 0 "n*``( a[n]/a[n+1]-1) = n*``((n+1)^2/(n^2)-1)" "6#/*&%\"nG\"\"\"-%!G6#,&*&&% \"aG6#F%F&&F-6#,&F%F&F&F&!\"\"F&F&F2F&*&F%F&-F(6#,&*&,&F%F&F&F&\"\"#*$ F%F9F2F&F&F2F&" }{XPPEDIT 18 0 " ``=n*``(((n+1)^2-n^2)/(n^2))" "6#/%! G*&%\"nG\"\"\"-F$6#*&,&*$,&F&F'F'F'\"\"#F'*$F&F.!\"\"F'*$F&F.F0F'" } {XPPEDIT 18 0 "``= (2*n+1)/n" "6#/%!G*&,&*&\"\"#\"\"\"%\"nGF)F)F)F)F)F *!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "sigma = Limit(n*[a[n]/a[n+1]-1],n = infinity);" "6#/%&s igmaG-%&LimitG6$*&%\"nG\"\"\"7#,&*&&%\"aG6#F)F*&F/6#,&F)F*F*F*!\"\"F*F *F4F*/F)%)infinityG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit((2*n+1)/ n,n = infinity) = 2;" "6#/-%&LimitG6$*&,&*&\"\"#\"\"\"%\"nGF+F+F+F+F+F ,!\"\"/F,%)infinityGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "1 < sigma;" "6#2\"\"\"%&sigmaG" }{TEXT -1 26 ", the series converges by " }{TEXT 270 12 "Raabe's test" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "a := n -> 1/n^2;\nLimit(n*(a(n)/a(n+1)-1),n=infinity) ;\nsimplify(%);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf* 6#%\"nG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-*$)9$\"\"#F-!\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&%\"nG\"\"\",&*&*$),&F'F(F (F(\"\"#F(F(*$)F'F.F(!\"\"F(F1F(F(/F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&%\"nG\"\"#\"\"\"F*F*F(!\"\"/F(%)infinity G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Example 2 .. p-series " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Consider \+ the p-series " }{XPPEDIT 18 0 "Sum(1/(n^p),n = 1 .. infinity);" "6#-%$ SumG6$*&\"\"\"F')%\"nG%\"pG!\"\"/F);F'%)infinityG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a[n] = 1/(n^2);" "6# /&%\"aG6#%\"nG*&\"\"\"F)*$F'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "n*``(a[n]/a[n+1]-1) = n*``((n+ 1)^p/(n^p)-1)" "6#/*&%\"nG\"\"\"-%!G6#,&*&&%\"aG6#F%F&&F-6#,&F%F&F&F&! \"\"F&F&F2F&*&F%F&-F(6#,&*&),&F%F&F&F&%\"pGF&)F%F:F2F&F&F2F&" }{TEXT -1 3 " " }{XPPEDIT 18 0 "`` = n*``(((n+1)^p-n^p)/(n^p));" "6#/%!G*&% \"nG\"\"\"-F$6#*&,&),&F&F'F'F'%\"pGF')F&F.!\"\"F')F&F.F0F'" }{XPPEDIT 18 0 "``= n*``((1+1/n)^p-1)" "6#/%!G*&%\"nG\"\"\"-F$6#,&),&F'F'*&F'F'F &!\"\"F'%\"pGF'F'F.F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=n*``(``(1+p/n+ p*(p-1)/(2!*n^2)+p*(p-1)*(p-2)/(3!*n^3)+` . . . `)-1)" "6#/%!G*&%\"nG \"\"\"-F$6#,&-F$6#,,F'F'*&%\"pGF'F&!\"\"F'*(F/F',&F/F'F'F0F'*&-%*facto rialG6#\"\"#F'*$F&F7F'F0F'**F/F',&F/F'F'F0F',&F/F'F7F0F'*&-F56#\"\"$F' *$F&F?F'F0F'%(~.~.~.~GF'F'F'F0F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ p+p*(p-1)/(2!*n)+p*(p-1)*(p-2)/(3!*n^2)+` . . . `;" "6#/%!G,*%\"pG\"\" \"*(F&F',&F&F'F'!\"\"F'*&-%*factorialG6#\"\"#F'%\"nGF'F*F'**F&F',&F&F' F'F*F',&F&F'F/F*F'*&-F-6#\"\"$F'*$F0F/F'F*F'%(~.~.~.~GF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "The n " }{XPPEDIT 18 0 "sigma = Limit(n*``(a[n]/a[n+1]-1),n = infinity);" "6#/%&sigmaG-%&LimitG6$*&%\"nG\"\"\"-%!G6#,&*&&%\"aG6#F)F*&F16#,&F)F*F *F*!\"\"F*F*F6F*/F)%)infinityG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "p; " "6#%\"pG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 270 12 "Raabe's t est" }{TEXT -1 42 " now shows that the series converges when " } {XPPEDIT 18 0 "1 < p;" "6#2\"\"\"%\"pG" }{TEXT -1 26 ", and diverges w hen p < 1." }}{PARA 0 "" 0 "" {TEXT -1 18 "In the case that " } {XPPEDIT 18 0 "p = 1" "6#/%\"pG\"\"\"" }{TEXT -1 76 " we get the harmo nic series, for which Raabe's test provides no information." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " } {TEXT 0 5 "Limit" }{TEXT -1 37 " procedure can handle specific cases. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a := n -> 1/n^1.01;\nLimit(n*(a(n)/a(n+1)-1),n=infinity);\nsim plify(%);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"n G6\"6$%)operatorG%&arrowGF(*&\"\"\"F-*$)9$$\"$,\"!\"#F-!\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&%\"nG\"\"\",&*&F'$!$,\"! \"#,&F'F(F(F($\"$,\"F-F(F(!\"\"F(/F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++55!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++ +55!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the series " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((2*n-1)!/(2^(2*n-1)*n!*(n-1)!*(2*n+1)),n = 1 .. in finity)=1/6+3/40+5/112+35/1152+63/2816+231/13312+143/10240+` . . . `" "6#/-%$SumG6$*&-%*factorialG6#,&*&\"\"#\"\"\"%\"nGF.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/F/; F.%)infinityG,2*&F.F.\"\"'F0F.*&\"\"$F.\"#SF0F.*&\"\"&F.\"$7\"F0F.*&\" #NF.\"%_6F0F.*&\"#jF.\"%;GF0F.*&\"$J#F.\"&7L\"F0F.*&\"$V\"F.\"&S-\"F0F .%(~.~.~.~GF." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "a[n] = (2*n-1)!/(2^(2*n-1)*n!*(n-1)!*(2*n+1));" "6#/&% \"aG6#%\"nG*&-%*factorialG6#,&*&\"\"#\"\"\"F'F/F/F/!\"\"F/**)F.,&*&F.F /F'F/F/F/F0F/-F*6#F'F/-F*6#,&F'F/F/F0F/,&*&F.F/F'F/F/F/F/F/F0" } {XPPEDIT 18 0 "`` = (2*n-1)*(2*n-3)*` . . . `*``(3)*``(2)*``(1)/(2^n*n !*(2*n+1));" "6#/%!G*0,&*&\"\"#\"\"\"%\"nGF)F)F)!\"\"F),&*&F(F)F*F)F) \"\"$F+F)%(~.~.~.~GF)-F$6#F.F)-F$6#F(F)-F$6#F)F)*()F(F*F)-%*factorialG 6#F*F),&*&F(F)F*F)F)F)F)F)F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "a[n]/a[n+1] = ``((2*n-1)*(2*n-3)*` . \+ . . `*``(3)*``(2)*``(1)/(2^n*n!*(2*n+1)))*``(2^(n+1)*(n+1)!*(2*n+3)/(( 2*n+1)*(2*n-1)*` . . . `*``(3)*``(2)*``(1)));" "6#/*&&%\"aG6#%\"nG\"\" \"&F&6#,&F(F)F)F)!\"\"*&-%!G6#*0,&*&\"\"#F)F(F)F)F)F-F),&*&F5F)F(F)F) \"\"$F-F)%(~.~.~.~GF)-F06#F8F)-F06#F5F)-F06#F)F)*()F5F(F)-%*factorialG 6#F(F),&*&F5F)F(F)F)F)F)F)F-F)-F06#**)F5,&F(F)F)F)F)-FC6#,&F(F)F)F)F), &*&F5F)F(F)F)F8F)F)*.,&*&F5F)F(F)F)F)F)F),&*&F5F)F(F)F)F)F-F)F9F)-F06# F8F)-F06#F5F)-F06#F)F)F-F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*(n+1)* (2*n+3)/((2*n+1)^2)" "6#/%!G**\"\"#\"\"\",&%\"nGF'F'F'F',&*&F&F'F)F'F' \"\"$F'F'*$,&*&F&F'F)F'F'F'F'F&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "n*``(a[n]/a[n+1]-1) = n*``(2*(n+1) *(2*n+3)/((2*n+1)^2)-1);" "6#/*&%\"nG\"\"\"-%!G6#,&*&&%\"aG6#F%F&&F-6# ,&F%F&F&F&!\"\"F&F&F2F&*&F%F&-F(6#,&**\"\"#F&,&F%F&F&F&F&,&*&F8F&F%F&F &\"\"$F&F&*$,&*&F8F&F%F&F&F&F&F8F2F&F&F2F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "n*``((2*(n+1)*(2*n+3)-(2*n+1)^2)/((2*n+1)^2));" "6#*&%\"nG\"\"\" -%!G6#*&,&*(\"\"#F%,&F$F%F%F%F%,&*&F,F%F$F%F%\"\"$F%F%F%*$,&*&F,F%F$F% F%F%F%F,!\"\"F%*$,&*&F,F%F$F%F%F%F%F,F4F%" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= n*(6*n+5)/((2*n+1)^2)" "6#/%!G*(%\"nG\"\"\",&*&\"\"'F'F&F'F'\" \"&F'F'*$,&*&\"\"#F'F&F'F'F'F'F/!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "sigma = Limit(n*``(a[n]/a[n+1]-1),n = infinity);" "6#/%&sigmaG-%&Li mitG6$*&%\"nG\"\"\"-%!G6#,&*&&%\"aG6#F)F*&F16#,&F)F*F*F*!\"\"F*F*F6F*/ F)%)infinityG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit(n*(6*n+5)/((2* n+1)^2),n = infinity) = 3/2;" "6#/-%&LimitG6$*(%\"nG\"\"\",&*&\"\"'F)F (F)F)\"\"&F)F)*$,&*&\"\"#F)F(F)F)F)F)F1!\"\"/F(%)infinityG*&\"\"$F)F1F 2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "1 < sigma;" "6#2\"\"\"%&sigmaG" } {TEXT -1 26 ", the series converges by " }{TEXT 270 12 "Raabe's test" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "a := n -> (2*n-1)!/(2^(2*n-1)*n!*(n-1)!*(2*n+1) );\nn*(a(n)/a(n+1)-1);\nsimplify(%);\nLimit(%,n=infinity);\nvalue(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&a rrowGF(*&-%*factorialG6#,&9$\"\"#!\"\"\"\"\"F4**)F2F0F4-F.6#F1F4-F.6#, &F1F4F3F4F4,&F1F2F4F4F4F3F(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)*$),&F(\"\"#F)F)F0F)!\" \"/F(%)infinityG" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Note that the ratio test does not give any information concerning the convergence or divergenc e of the series since " }{XPPEDIT 18 0 "Limit(a[n+1]/a[n],n = infinity ) = 1;" "6#/-%&LimitG6$*&&%\"aG6#,&%\"nG\"\"\"F-F-F-&F)6#F,!\"\"/F,%)i nfinityGF-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "a := n -> (2*n-1)!/(2^(2*n-1)*n!*( n-1)!*(2*n+1));\na(n+1)/a(n);\nsimplify(%);\nLimit(%,n=infinity);\nval ue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operat orG%&arrowGF(*&-%*factorialG6#,&9$\"\"#!\"\"\"\"\"F4**)F2F0F4-F.6#F1F4 -F.6#,&F1F4F3F4F4,&F1F2F4F4F4F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&**-%*factorialG6#,&%\"nG\"\"#\"\"\"F+F+)F*,&F)F*!\"\"F+F+-F&6#,&F )F+F.F+F+F(F+F+**)F*F(F+-F&6#,&F)F+F+F+F+,&F)F*\"\"$F+F+-F&6#F-F+F." } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$),&%\"nG\"\"#\"\"\"F*F)F*F**&,& F(F*F*F*F*,&F(F)\"\"$F*F*!\"\"#F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%&LimitG6$,$*&*$),&%\"nG\"\"#\"\"\"F-F,F-F-*&,&F+F-F-F-F-,&F+F,\"\"$ F-F-!\"\"#F-F,/F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The \+ series converges to " }{XPPEDIT 18 0 "arcsin(1)-1=Pi/2-1" "6#/,&-%'arc sinG6#\"\"\"F(F(!\"\",&*&%#PiGF(\"\"#F)F(F(F)" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "a := n -> (2*n-1)!/(2^(2*n-1)*n!*(n-1)!*(2*n+1));\nSum(a(n),n=1.. infinity);\nevalf(%);\nevalf(Pi/2-1,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*,-%*factorialG6#,&*&\" \"#\"\"\"9$F3F3F3!\"\"F3)F2F0F5-F.6#F4F5-F.6#,&F4F3F3F5F5,&*&F2F3F4F3F 3F3F3F5F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,-%*factori alG6#,&*&\"\"#\"\"\"%\"nGF-F-F-!\"\"F-)F,F*F/-F(6#F.F/-F(6#,&F.F-F-F/F /,&*&F,F-F.F-F-F-F-F//F.;F-%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+oK'zq&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+oK'zq&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 11 "T he series " }{XPPEDIT 18 0 "Sum(1/(n*log(n)^2),n = 2 .. infinity);" "6 #-%$SumG6$*&\"\"\"F'*&%\"nGF'*$-%$logG6#F)\"\"#F'!\"\"/F);F.%)infinity G" }{TEXT -1 18 " converges by the " }{TEXT 270 13 "integral test" } {TEXT -1 6 ", but " }{TEXT 270 12 "Raabe's test" }{TEXT -1 17 " is inc onclusive." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a[n] = 1/(n*log(n)^2);" "6#/&%\"aG6#%\"nG*&\"\"\"F)*&F'F)*$-%$logG6#F'\"\"#F )!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "n*``(a[n]/a[n+1]-1) = n*``((n+1)*ln(n+1)^2/(n*ln(n)^2)- 1);" "6#/*&%\"nG\"\"\"-%!G6#,&*&&%\"aG6#F%F&&F-6#,&F%F&F&F&!\"\"F&F&F2 F&*&F%F&-F(6#,&*(,&F%F&F&F&F&*$-%#lnG6#,&F%F&F&F&\"\"#F&*&F%F&*$-F;6#F %F>F&F2F&F&F2F&" }{XPPEDIT 18 0 "`` = ((n+1)*ln(n+1)^2-n*ln(n)^2)/(ln( n)^2);" "6#/%!G*&,&*&,&%\"nG\"\"\"F*F*F**$-%#lnG6#,&F)F*F*F*\"\"#F*F** &F)F**$-F-6#F)F0F*!\"\"F**$-F-6#F)F0F5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "sigma = Limit(n*``(a[n]/a[n+1] -1),n = infinity);" "6#/%&sigmaG-%&LimitG6$*&%\"nG\"\"\"-%!G6#,&*&&%\" aG6#F)F*&F16#,&F)F*F*F*!\"\"F*F*F6F*/F)%)infinityG" }{TEXT -1 7 " = 1 . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "a := n -> 1/(n*log(n)^2);\nLimit(n*(a(n)/a(n+1)-1),n= infinity);\nsimplify(%);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-*&9$F-)-%$logG6#F/ \"\"#F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&%\" nG\"\"\",&*&*&,&F'F(F(F(F()-%#lnG6#F,\"\"#F(F(*&F'F()-F/6#F'F1F(!\"\"F (F6F(F(/F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*& ,(*&)-%#lnG6#,&%\"nG\"\"\"F/F/\"\"#F/F.F/F/*$F)F/F/*&F.F/)-F+6#F.F0F/! \"\"F/*$)F4F0F/F6/F.%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " Raabe's test gives no information." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 45 "Use the ratio test to verify that the series " }{XPPEDIT 18 0 "Sum(exp(n)/(exp(2*n)+1),n=1..infinity)" "6#-%$SumG6$*& -%$expG6#%\"nG\"\"\",&-F(6#*&\"\"#F+F*F+F+F+F+!\"\"/F*;F+%)infinityG" }{TEXT -1 11 " converges." }}{PARA 0 "" 0 "" {TEXT -1 52 "Find an appr oximate value for the sum of the series." }}{PARA 0 "" 0 "" {TEXT -1 43 "___________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "__ _________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 45 "Use the ratio test to verify that the ser ies " }{XPPEDIT 18 0 "Sum(2^n/(2+3^n),n = 1 .. infinity);" "6#-%$SumG6 $*&)\"\"#%\"nG\"\"\",&F(F*)\"\"$F)F*!\"\"/F);F*%)infinityG" }{TEXT -1 11 " converges." }}{PARA 0 "" 0 "" {TEXT -1 52 "Find an approximate va lue for the sum of the series." }}{PARA 0 "" 0 "" {TEXT -1 43 "_______ ____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "____________________ _______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 45 "Use the ratio test to verify that the series " }{XPPEDIT 18 0 " Sum(n^n/n!,n = 1 .. infinity);" "6#-%$SumG6$*&)%\"nGF(\"\"\"-%*factori alG6#F(!\"\"/F(;F)%)infinityG" }{TEXT -1 22 " diverges to infinity." } }{PARA 0 "" 0 "" {TEXT -1 43 "________________________________________ ___" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "___________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 59 "Can the ratio test be used to determine whether the series " }{XPPEDIT 18 0 "Sum(n^n/(exp(n )*n!),n = 1 .. infinity)" "6#-%$SumG6$*&)%\"nGF(\"\"\"*&-%$expG6#F(F)- %*factorialG6#F(F)!\"\"/F(;F)%)infinityG" }{TEXT -1 23 " converges or \+ diverges?" }}{PARA 0 "" 0 "" {TEXT -1 69 "Use Raabe's test to decide w hether this series converges or diverges." }}{PARA 0 "" 0 "" {TEXT -1 43 "___________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "__ _________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }} {PARA 0 "" 0 "" {TEXT -1 59 "Can the ratio test be used to determine w hether the series " }{XPPEDIT 18 0 "Sum((2*n)!/(2^(2*n)*n!^2),n = 1 .. infinity);" "6#-%$SumG6$*&-%*factorialG6#*&\"\"#\"\"\"%\"nGF,F,*&)F+* &F+F,F-F,F,*$-F(6#F-F+F,!\"\"/F-;F,%)infinityG" }{TEXT -1 23 " converg es or diverges?" }}{PARA 0 "" 0 "" {TEXT -1 69 "Use Raabe's test to de cide whether this series converges or diverges." }}{PARA 0 "" 0 "" {TEXT -1 43 "___________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "___________________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 59 "Can the ratio test be used to determi ne whether the series " }{XPPEDIT 18 0 "Sum(2^(2*n)*(n-1)!^2/(2*n)!,n \+ = 1 .. infinity);" "6#-%$SumG6$*()\"\"#*&F(\"\"\"%\"nGF*F**$-%*factori alG6#,&F+F*F*!\"\"F(F*-F.6#*&F(F*F+F*F1/F+;F*%)infinityG" }{TEXT -1 23 " converges or diverges?" }}{PARA 0 "" 0 "" {TEXT -1 70 "Use Raabe' s test to show that this series converges, and find its sum." }}{PARA 0 "" 0 "" {TEXT -1 43 "___________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 43 "___________________________________________" }}{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 "" }}{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 }