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"" {TEXT -1 20 "Sequences and series" }}{PARA 0 " " 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 " " 0 "" {TEXT -1 19 "Version: 12.1.2008" }}{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 49 "Convergence of an infinite sequence . . example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "C onsider the sequence:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1,1/2,1/3,1/4,1/5,1/6,` . . . `,1/n,` . . . `;" "6+\"\"\"*&F#F# \"\"#!\"\"*&F#F#\"\"$F&*&F#F#\"\"%F&*&F#F#\"\"&F&*&F#F#\"\"'F&%(~.~.~. ~G*&F#F#%\"nGF&F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 34 "The general form of a sequence is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "a[1],a[2],a[3],a[4],` . . . `,a[n],` . . . `;" "6)&%\"a G6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%%(~.~.~.~G&F$6#%\"nGF0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "In the example just given we h ave " }{XPPEDIT 18 0 "a[n]=1/n" "6#/&%\"aG6#%\"nG*&\"\"\"F)F'!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 80 "The terms of this sequen ce are approaching or getting progressively closer to 0." }}{PARA 0 " " 0 "" {TEXT -1 25 "We say that the sequence " }{TEXT 269 9 "converges " }{TEXT -1 18 " to 0, or has the " }{TEXT 269 5 "limit" }{TEXT -1 13 " 0 and write:" }}{PARA 256 "" 0 "" {TEXT -1 4 " As " }{XPPEDIT 18 0 " n -> infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F *" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "1/n;" "6#*&\"\"\"F$%\"nG!\"\"" } {TEXT -1 8 " -> 0," }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(a[n],n=infinity)=Limit(1/n ,n=infinity)" "6#/-%&LimitG6$&%\"aG6#%\"nG/F*%)infinityG-F%6$*&\"\"\"F 0F*!\"\"/F*F," }{TEXT -1 6 " = 0. " }}{PARA 0 "" 0 "" {TEXT -1 93 "The following picture illustrates the first few terms of this sequence by plotting the points" }{XPPEDIT 18 0 "``(n,1/n);" "6#-%!G6$%\"nG*&\"\" \"F(F&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "pts := [seq([n,1/n],n=1..10 )];\nplot([pts,1/x],x=0..10,style=[point,line],color=[red,gray],\n l inestyle=2,symbol=circle,view=[0..10,0..1],labels=[``,``]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7,7$\"\"\"F'7$\"\"##F'F)7$\"\"$#F'F,7 $\"\"%#F'F/7$\"\"&#F'F27$\"\"'#F'F57$\"\"(#F'F87$\"\")#F'F;7$\"\"*#F'F >7$\"#5#F'FA" }}{PARA 13 "" 1 "" {GLPLOT2D 507 256 256 {PLOTDATA 2 "6( -%'CURVESG6%7,7$$\"\"\"\"\"!F(7$$\"\"#F*$\"1+++++++]!#;7$$\"\"$F*$\"1L LLLLLLLF07$$\"\"%F*$\"1+++++++DF07$$\"\"&F*$\"1+++++++?F07$$\"\"'F*$\" 1nmmmmmm;F07$$\"\"(F*$\"1G9dG9dG9F07$$\"\")F*$\"1++++++]7F07$$\"\"*F*$ \"166666666F07$$\"#5F*$\"1+++++++5F0-%'COLOURG6&%$RGBG$\"*++++\"!\")F* F*-%&STYLEG6#%&POINTG-F$6%7gn7$$\"1+++;arz@F0$\"19I8j[v(e%!#:7$$\"1+++ P.D)H#F0$\"1-/X'4O6N%Ffo7$$\"1+++e_y;CF0$\"1^Nv')zsPTFfo7$$\"1+++z,KND F0$\"1yWQ+]FWRFfo7$$\"1+++*4bQl#F0$\"1:#=6C.\"oPFfo7$$\"1+++T\\#4*GF0$ \"1d3_J05fMFfo7$$\"1+++\"y%*z7$F0$\"1Y-J!GOp>$Ffo7$$\"1+++kW8-OF0$\"13 :t!zJhx#Ffo7$$\"1+++XTFwSF0$\"1pQns1A`CFfo7$$\"1+++3Q\\4YF0$\"1W6e2cVp @Ffo7$$\"1+++pMrU^F0$\"1$H'4n()\\W>Ffo7$$\"1,++JJ$fn&F0$\"1@wXV[#=w\"F fo7$$\"1+++\"z_\"4iF0$\"1#3EmnD0h\"Ffo7$$\"1+++m6m#G(F0$\"15l:NW7t8Ffo 7$$\"1+++S&phN)F0$\"1C2+t0s'>\"Ffo7$$\"1+++*=)H\\5Ffo$\"1>E/9%z,`*F07$ $\"1+++[!3uC\"Ffo$\"1;Z`hHi;!)F07$$\"1+++J$RDX\"Ffo$\"1f]9k^\\%)oF07$$ \"1+++)R'ok;Ffo$\"1^$QwWPr+'F07$$\"1+++1J:w=Ffo$\"1]:F.a0I`F07$$\"1+++ 3En$4#Ffo$\"1I>&3&fHwZF07$$\"1+++/RE&G#Ffo$\"1d_kGA'eP%F07$$\"1+++D.&4 ]#Ffo$\"1Qcxd+[)*RF07$$\"1+++vB_vmtT$F07$$\"1+++(*ev:JFfo$\"1:B$f/%\\4KF07$$\"1+++347TLFfo$\"1 T+9>v+$*HF07$$\"1+++LY.KNFfo$\"1uWa;$H7$GF07$$\"1+++\"o7Tv$Ffo$\"1*Q?u HXPm#F07$$\"1+++$Q*o]RFfo$\"1A'[`!Q?JDF07$$\"1+++\"=lj;%Ffo$\"1U+JoO<+ CF07$$\"1+++V&R+# F07$$\"1+++(RQb@&Ffo$\"1&)Qf%fZt\">F07$$\"1+++=>Y2aFfo$\"1MDCEkH\\=F07 $$\"1+++yXu9cFfo$\"1p3r<\\-\"y\"F07$$\"1+++\\y))GeFfo$\"1_1qQJf:MX(=2c;F07$$\"1+++!y%3TiFfo$\"1p#=lb&G-;F07$$\"1++ +O![hY'Ffo$\"17)*4%y:la\"F07$$\"1+++#Qx$omFfo$\"1@cz(=:'*\\\"F07$$\"1+ ++u.I%)oFfo$\"1+6OI/e_9F07$$\"1+++(pe*zqFfo$\"1/!\\$=wV79F07$$\"1+++C \\'QH(Ffo$\"1r#3UC:5P\"F07$$\"1+++8S8&\\(Ffo$\"1>bIf*)>M8F07$$\"1+++0# =bq(Ffo$\"1\"F07$$\"1***** HvJga)Ffo$\"1\"*\\hGP8q6F07$$\"1+++8tOc()Ffo$\"1$Hd&)4E?9\"F07$$\"1*** ***[Qk\\*)Ffo$\"1J " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The \+ notion of convergence can be made more precise as follows." }}{PARA 0 "" 0 "" {TEXT -1 37 "If we prescribe some small distance " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 135 " from zero, it is always possible to go far enough along the sequence to find a term of the se quence within this prescribed distance " }{XPPEDIT 18 0 "epsilon" "6# %(epsilonG" }{TEXT -1 74 " from 0. Then all the subsequent terms will also be within the distance " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG " }{TEXT -1 12 " from zero." }}{PARA 0 "" 0 "" {TEXT -1 12 "Indeed, i f " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 36 " is a smal l positive number, then " }{XPPEDIT 18 0 "1/epsilon;" "6#*&\"\"\"F$%( epsilonG!\"\"" }{TEXT -1 46 " is large, but we can always find an int eger " }{TEXT 262 1 "N" }{TEXT -1 22 " which is larger than " } {XPPEDIT 18 0 "1/epsilon;" "6#*&\"\"\"F$%(epsilonG!\"\"" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "a[N];" "6#&% \"aG6#%\"NG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/N < epsilon;" "6#2*& \"\"\"F%%\"NG!\"\"%(epsilonG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[n] < 1/N;" "6#2&%\"aG6#%\"nG*&\"\"\"F)%\"NG!\"\"" }{TEXT -1 18 " for all integers " }{TEXT 263 1 "n" }{TEXT -1 7 " with " }{XPPEDIT 18 0 "n < N;" "6#2%\"nG%\"NG" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "For example, if " }{XPPEDIT 18 0 "ep silon = 1/(Pi^100);" "6#/%(epsilonG*&\"\"\"F&*$%#PiG\"$+\"!\"\"" } {TEXT -1 1 " " }{TEXT 260 2 "~ " }{XPPEDIT 18 0 "1.9276*10^(-50)" "6#* &-%&FloatG6$\"&w#>!\"%\"\"\")\"#5,$\"#]!\"\"F)" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "1/epsilon = Pi^100;" "6#/*&\"\"\"F%%(epsilonG!\"\"*$%# PiG\"$+\"" }{TEXT -1 1 " " }{TEXT 261 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "0.51878*10^50" "6#*&-%&FloatG6$\"&y=&!\"&\"\"\"*$\"#5\"#]F)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "We can find the next pos itive integer " }{TEXT 264 1 "N" }{TEXT -1 11 " following " }{XPPEDIT 18 0 "Pi^100" "6#*$%#PiG\"$+\"" }{TEXT -1 10 " by using " }{TEXT 0 4 " ceil" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Digits := 55;\nN := ceil(Pi^100);\nDigits := 10;;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"S1/ognoiN,.jC:E'3#>8'>VJ[y=&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 93 "The 51878483143196131920 862615246303013562686760680406 th term of the sequence is less than \+ " }{XPPEDIT 18 0 "epsilon = 1/(Pi^100);" "6#/%(epsilonG*&\"\"\"F&*$%#P iG\"$+\"!\"\"" }{TEXT -1 35 " , and so are all subsequent terms." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Convergence of an inf inite sequence . . example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the sequence:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2,3/4,7/8,15/16,31/32,` . . . `,1-1/(2^n),` . . . `;" "6**&\"\"\"F$\"\"#!\"\"*&\"\"$F$\"\"%F&*& \"\"(F$\"\")F&*&\"#:F$\"#;F&*&\"#JF$\"#KF&%(~.~.~.~G,&F$F$*&F$F$)F%%\" nGF&F&F3" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 287 1 "n" }{TEXT -1 9 " th term " }{XPPEDIT 18 0 "a[n]" "6#&%\"a G6#%\"nG" }{TEXT -1 29 " of the sequence is given by " }{XPPEDIT 18 0 "a[n] = 1-1/(2^n);" "6#/&%\"aG6#%\"nG,&\"\"\"F)*&F)F))\"\"#F'!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "This sequence converges to 1." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a := n -> 1-1/2^n;\nLimit(a(n),n=infinity);\nvalue(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%& arrowGF(,&\"\"\"F-*&F-F-)\"\"#9$!\"\"F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,&\"\"\"F'*&F'F')\"\"#%\"nG!\"\"F,/F+%)infin ityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(a[n],n = infinity) = Limit(``(1-1 /(2^n)),n = infinity);" "6#/-%&LimitG6$&%\"aG6#%\"nG/F*%)infinityG-F%6 $-%!G6#,&\"\"\"F3*&F3F3)\"\"#F*!\"\"F7/F*F," }{TEXT -1 6 " = 1. " }} {PARA 0 "" 0 "" {TEXT -1 93 "The following picture illustrates the fir st few terms of this sequence by plotting the points" }{XPPEDIT 18 0 " ``(n,1-1/(2^n));" "6#-%!G6$%\"nG,&\"\"\"F(*&F(F()\"\"#F&!\"\"F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 176 "pts := [seq([n,1-1/2^n],n=1..8)];\nplot([1-1/ 2^x,1,pts],x=0..8,style=[line$2,point],\n color=[gray$2,red],linestyl e=[2,3],symbol=circle,\n view=[0..8,0..1.1],labels=[``,``]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7*7$\"\"\"#F'\"\"#7$F)#\"\"$\" \"%7$F,#\"\"(\"\")7$F-#\"#:\"#;7$\"\"&#\"#J\"#K7$\"\"'#\"#j\"#k7$F0#\" $F\"\"$G\"7$F1#\"$b#\"$c#" }}{PARA 13 "" 1 "" {GLPLOT2D 499 238 238 {PLOTDATA 2 "6(-%'CURVESG6&7U7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>$\"3)Hj8q ahW'eF-7$$\"3ELLLLBxV5E$F3$\"3$3B2\"=76B?F37$$\"3MLLLLAKn\\F3$\"3+Y257x*G \"HF37$$\"3=LLLLc$\\o'F3$\"3J)G\"3WpO3PF37$$\"3)emmm^&Q%R)F3$\"3u\"fyQ $)*Q6WF37$$\"3wKLL$Qk#z**F3$\"3i:4&fT3G*\\F37$$\"3))*****\\YJ?;\"!#<$ \"3Ev6(Qlr6`&F37$$\"3?LLL=\"\\S2q=Pr-'F37$$\"3\")*****\\[A 4]\"FW$\"3#=BW]0EnY'F37$$\"3wmmm'3Q\\n\"FW$\"3;mc:0X?ooF37$$\"3OLLLB6@ G=FW$\"3s.!e42kQ=(F37$$\"3&)******f-w+?FW$\"3PP_7$3<8](F37$$\"3%****** ***y,u@FW$\"3[Iz#fCtSy(F37$$\"3)*******RP)4M#FW$\"35R<+QICE!)F37$$\"3I LLL=Zg#\\#FW$\"3;\"3rf<[JA)F37$$\"3cmmmEn*Gn#FW$\"3*37=q*\\)=V)F37$$\" 3Tmmm1xiDGFW$\"31&)RAm$3%*e)F37$$\"3!)*****\\9!H.IFW$\"3KR9IXu%Gv)F37$ $\"3Immm1:bgJFW$\"3k*4m?EY;)))F37$$\"3<+++X@4LLFW$\"3r3j1m%3x+*F37$$\" 31+++N;R(\\$FW$\"3xPPjLq^9\"*F37$$\"3wmmm;4#)oOFW$\"3B#3'Q(HCP@*F37$$ \"3jmmm6lCEQFW$\"354!*zNm+&H*F37$$\"3ELLL$G^g*RFW$\"3-61m0qGt$*F37$$\" 3oKLL=2VsTFW$\"3Me#yG^3aW*F37$$\"3f*****\\`pfK%FW$\"3(e)>S'p)R,&*F37$$ \"3!HLLLm&z\"\\%FW$\"3y,O3iy`b&*F37$$\"3s******z-6jYFW$\"3il%zD!HI0'*F 37$$\"3<******4#32$[FW$\"3mi7**p `G^fo*F37$$\"3G******H%=H<&FW$\"3FDm2#)zzA(*F37$$\"35mmm1>qM`FW$\"3'*) 42Bw.Av*F37$$\"3%)*******HSu]&FW$\"3#3F!R$zl,y*F37$$\"3'HLL$ep'Rm&FW$ \"3rx^*GgoF!)*F37$$\"3')******R>4NeFW$\"3'y4bR'*G[#)*F37$$\"3#emm;@2h* fFW$\"3Yv^JUHG3c()*F37$$\"3j*****\\iN7]'FW$\"3!z&oQ_\"4'*))*F37$$\"3a LLLt>:nmFW$\"3=l1itD\"**F37$$\"3am mm^Q40qFW$\"3)Rd&[c.:A**F37$$\"3y******z]rfrFW$\"3u@wgbB1I**F37$$\"3gm mmc%GpL(FW$\"3NJ[[uk9Q**F37$$\"3/LLL8-V&\\(FW$\"3U5z1A?eW**F37$$\"3=++ +XhUkwFW$\"3e2K7Uyq]**F37$$\"3=+++:o " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 19 "Divergent sequences" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 60 "A sequence which do es not converge is said to diverge or be " }{TEXT 269 9 "divergent" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "For example, the sequenc es" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1,-1,1,-1,1,-1, ` . . . `,(-1)^(n-1),` . . . `;" "6+\"\"\",$F#!\"\"F#,$F#F%F#,$F#F%%(~ .~.~.~G),$F#F%,&%\"nGF#F#F%F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2, 4,6,8,10,` . . . `,2*n,` . . . `;" "6*\"\"#\"\"%\"\"'\"\")\"#5%(~.~.~. ~G*&F#\"\"\"%\"nGF*F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 14 "are divergent." }}{PARA 0 "" 0 "" {TEXT -1 8 "Since 2 " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF* F*F*" }{TEXT -1 5 ", as " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7 \"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 43 ", the secon d of these sequences is said to " }{TEXT 269 19 "diverge to infinity" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The sequence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1],a[2],a[3],` . . . `,a[ n],` . . . `;" "6(&%\"aG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G&F$6#%\" nGF-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "a[n]=ln(1/n)" "6#/&%\"aG6#%\"nG-%#lnG6#*&\"\"\"F,F'!\" \"" }{TEXT -1 29 ", diverges to minus infinity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Limit(ln(1/n ),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG 6$-%#lnG6#*&\"\"\"F*%\"nG!\"\"/F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 32 "Series and convergence of series" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 40 "Consider the infinite decimal expansion:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/3 = 0;" "6#/*&\"\"\"F%\"\"$!\"\"\"\"!" }{TEXT -1 1 ". " }{XPPEDIT 18 0 "3333333*` . . . `" "6#*&\"(LLL$\"\"\"%(~.~.~.~GF%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "Underlying this equatio n is the sequence " }}{PARA 256 "" 0 "" {TEXT -1 27 " 0.3, 0.33, 0.333 , 0.3333, " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3/10,33/100,333/1000,3333/10000,` . . . `;" " 6'*&\"\"$\"\"\"\"#5!\"\"*&\"#LF%\"$+\"F'*&\"$L$F%\"%+5F'*&\"%LLF%\"&++ \"F'%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "This s equence converges to " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 63 "Each new term can be obt ained by adding a fraction of the form " }{XPPEDIT 18 0 "3/(10^i);" "6 #*&\"\"$\"\"\")\"#5%\"iG!\"\"" }{TEXT -1 43 " to the preceding term, s o the sequence is:" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "3/10" "6#*&\"\"$\"\"\"\"#5!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "3/10+3/100" "6#,&*&\"\"$\"\"\" \"#5!\"\"F&*&F%F&\"$+\"F(F&" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "3/10+3/100+3/1000" "6#,(*&\"\"$\"\"\"\"# 5!\"\"F&*&F%F&\"$+\"F(F&*&F%F&\"%+5F(F&" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "3/10+3/100+3/1000+3/10000" "6 #,**&\"\"$\"\"\"\"#5!\"\"F&*&F%F&\"$+\"F(F&*&F%F&\"%+5F(F&*&F%F&\"&++ \"F(F&" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 " " } {XPPEDIT 18 0 "3/10+3/100+3/1000+3/10000+3/100000" "6#,,*&\"\"$\"\"\" \"#5!\"\"F&*&F%F&\"$+\"F(F&*&F%F&\"%+5F(F&*&F%F&\"&++\"F(F&*&F%F&\"'++ 5F(F&" }{TEXT -1 9 ", . . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "Here we have an example of an " }{TEXT 269 15 "infinite series" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1]+a[2]+a[3]+a[4]" "6#,*&%\"aG6#\"\"\"F'&F%6#\"\"#F'&F%6#\"\"$ F'&F%6#\"\"%F'" }{TEXT -1 13 " + . . . . + " }{XPPEDIT 18 0 "a[n]" "6# &%\"aG6#%\"nG" }{TEXT -1 13 " + . . . . = " }{XPPEDIT 18 0 "Sum(a[i],i = 1 .. infinity);" "6#-%$SumG6$&%\"aG6#%\"iG/F);\"\"\"%)infinityG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 55 "For any infinite series \+ we can consider the associated " }{TEXT 269 24 "sequence of partial su ms" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S[1]=a[1]" "6#/&%\"SG6#\"\"\"&%\"aG6#F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S[2] = a[1] + a[2] " "6#/&%\"SG6#\"\"#,&&%\"aG6#\"\"\"F,&F*6#F'F," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S[3] = a[1] + a[2] + a[3]" "6#/&%\"SG6#\"\"$,(&%\"aG6#\"\"\"F,&F*6#\"\"#F,&F*6#F'F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S [4] = a[1] + a[2] + a[3] + a[4]" "6#/&%\"SG6#\"\"%,*&%\"aG6#\"\"\"F,&F *6#\"\"#F,&F*6#\"\"$F,&F*6#F'F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 272 5 ". . ." }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 273 5 ". . ." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S[n] = a[1] + a[2] + a[3] + a[4]" "6#/&%\"SG6#%\"nG,*&%\"aG6#\"\"\"F,&F*6#\"\"#F,&F*6#\"\"$F,&F*6#\"\"%F ," }{TEXT -1 13 " + . . . . + " }{XPPEDIT 18 0 "a[n] = Sum(a[i],i = 1 \+ .. n);" "6#/&%\"aG6#%\"nG-%$SumG6$&F%6#%\"iG/F-;\"\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "For example," }}{PARA 0 "" 0 "" {TEXT -1 61 " 0.3 + 0.03 + 0.003 + 0.000 3 + 0.00003 + 0.000003 + . . . + " }{XPPEDIT 18 0 "3/(10^i);" "6#*&\" \"$\"\"\")\"#5%\"iG!\"\"" }{TEXT -1 11 " + . . . = " }{XPPEDIT 18 0 "S um(3/(10^i),i = 1 .. infinity);" "6#-%$SumG6$*&\"\"$\"\"\")\"#5%\"iG! \"\"/F+;F(%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 " The sequence of partial sums is:" }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 18 0 "S[1]" "6#&%\"SG6#\"\"\"" }{TEXT -1 6 " = 0.3" }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S[2]" "6#&%\"SG6# \"\"#" }{TEXT -1 28 " = 0.3 + 0.03 = 0.33 \n " }{XPPEDIT 18 0 "S[ 3]" "6#&%\"SG6#\"\"$" }{TEXT -1 35 " = 0.3 + 0.03 + 0.003 = 0.333 \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S[4]" "6#&%\" SG6#\"\"%" }{TEXT -1 39 " = 0.3 + 0.03 + 0.003 + 0.0003 = 0.3333" }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S[5]" "6#&%\"SG6# \"\"&" }{TEXT -1 50 " = 0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 = 0.3333 3" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 274 5 ". . ." }} {PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 275 5 ". . ." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S[n" "6#&%\"S G6#%\"nG" }{TEXT -1 51 " = 0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 + . . . + " }{XPPEDIT 18 0 "3/10^n" "6#*&\"\"$\"\"\")\"#5%\"nG!\"\"" } {TEXT -1 21 " = 0.333 . . . 333 ( " }{TEXT 305 1 "n" }{TEXT -1 12 " th rees) = " }{XPPEDIT 18 0 "Sum(3/(10^i),i = 1 .. n);" "6#-%$SumG6$*&\" \"$\"\"\")\"#5%\"iG!\"\"/F+;F(%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 25 "Given an infinite series " }{XPPEDIT 18 0 "Sum(a[i],i = 1 .. infinity);" "6#-%$SumG6$&%\"aG6#%\"iG/F);\"\"\"%)infinityG" } {TEXT -1 25 ", we say that the series " }{TEXT 269 20 "converges to th e sum" }{TEXT -1 57 " S provided that the associated sequence of parti al sums " }{XPPEDIT 18 0 "S[1],S[2],S[3]" "6%&%\"SG6#\"\"\"&F$6#\"\"#& F$6#\"\"$" }{TEXT -1 15 ", . . ., where " }{XPPEDIT 18 0 "S[n]=Sum(a[i ],i=1..n)" "6#/&%\"SG6#%\"nG-%$SumG6$&%\"aG6#%\"iG/F.;\"\"\"F'" } {TEXT -1 17 ", converges to S." }}{PARA 0 "" 0 "" {TEXT -1 8 "We write " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(Sum(a[i],i= 1..n),n=infinity) = Sum(a[i],i=1..infinity)" "6#/-%&LimitG6$-%$SumG6$& %\"aG6#%\"iG/F-;\"\"\"%\"nG/F1%)infinityG-F(6$&F+6#F-/F-;F0F3" }{TEXT -1 5 " = S." }}{PARA 0 "" 0 "" {TEXT -1 20 " Thus, for example, " } {XPPEDIT 18 0 "Sum(3/10^i,i=1..infinity) = 1/3" "6#/-%$SumG6$*&\"\"$\" \"\")\"#5%\"iG!\"\"/F,;F)%)infinityG*&F)F)F(F-" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 3 "sum" }{TEXT -1 53 " \"knows\" how to find the sum of some infinite series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Sum(3/10^i,i=1..infinity);\n value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$*&\"\"\"F()\"# 5%\"iG!\"\"\"\"$/F+;F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6## \"\"\"\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Geometr ic series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 39 "A finite geometric series has the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n] = a+a*r+a*r^2+a*r^3+` \+ . . .`+a*r^(n-1);" "6#/&%\"SG6#%\"nG,.%\"aG\"\"\"*&F)F*%\"rGF*F**&F)F* *$F,\"\"#F*F**&F)F**$F,\"\"$F*F*%'~.~.~.GF**&F)F*)F,,&F'F*F*!\"\"F*F* " }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 3 "In " } {XPPEDIT 18 0 "Sigma" "6#%&SigmaG" }{TEXT -1 10 " notation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n] = Sum(a*r^(i-1),i = 1 \+ .. n);" "6#/&%\"SG6#%\"nG-%$SumG6$*&%\"aG\"\"\")%\"rG,&%\"iGF-F-!\"\"F -/F1;F-F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 68 "Each term i s obtained from the preceding term by multiplying by the " }{TEXT 269 12 "common ratio" }{TEXT -1 1 " " }{TEXT 304 1 "r" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Now" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r*S[n] = a*r+a*r^2+a* r^3+a*r^4+` . . . `+a*r^(n-1)+a*r^n;" "6#/*&%\"rG\"\"\"&%\"SG6#%\"nGF& ,0*&%\"aGF&F%F&F&*&F-F&*$F%\"\"#F&F&*&F-F&*$F%\"\"$F&F&*&F-F&*$F%\"\"% F&F&%(~.~.~.~GF&*&F-F&)F%,&F*F&F&!\"\"F&F&*&F-F&)F%F*F&F&" }{TEXT -1 15 " ------- (ii), " }}{PARA 0 "" 0 "" {TEXT -1 40 "so, by subtracting (ii) from (i) we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1-r)*S[n]=a - a*r^n" "6#/*&,&\"\"\"F&%\"rG!\"\"F&&%\"SG6#%\"nGF &,&%\"aGF&*&F.F&)F'F,F&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "It follows that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n]=a*(1-r^n)/(1-r)" "6#/&%\"SG6#% \"nG*(%\"aG\"\"\",&F*F*)%\"rGF'!\"\"F*,&F*F*F-F.F." }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{MPLTEXT 1 0 3 "sum" }{TEXT -1 58 " applies this formula for summing finite geometric series." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Sum(a*r^(i-1),i = 1 .. n );\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*& %\"aG\"\"\")%\"rG,&%\"iGF(!\"\"F(F(/F,;F(%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&%\"aG\"\"\",&)%\"rG%\"nGF&!\"\"F&F&F&,&F)F&F+F&F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We also wish to consider \+ infinite geometric series " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "Sum(a*r^(i-1),i = 1 .. infinity) =a+a*r+a*r^2+a*r^3+` . . . ` + a*r^(n-1) + ` . . . `" "6#/-%$SumG6$*&%\"aG\"\"\")%\"rG,&%\"i GF)F)!\"\"F)/F-;F)%)infinityG,0F(F)*&F(F)F+F)F)*&F(F)*$F+\"\"#F)F)*&F( F)*$F+\"\"$F)F)%(~.~.~.~GF)*&F(F))F+,&%\"nGF)F)F.F)F)F:F)" }{TEXT -1 18 " --------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 11 "The series " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(3/(10^i),i = 1 .. infinity)=3/10+3/100+3/1000+` . . . `+3/(10^n)+` . . . `" "6#/-%$SumG 6$*&\"\"$\"\"\")\"#5%\"iG!\"\"/F,;F)%)infinityG,.*&F(F)F+F-F)*&F(F)\"$ +\"F-F)*&F(F)\"%+5F-F)%(~.~.~.~GF)*&F(F))F+%\"nGF-F)F7F)" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 100 "considered in the last section, p rovides an example of an infinite geometric series with first term " } {XPPEDIT 18 0 "a=3/10" "6#/%\"aG*&\"\"$\"\"\"\"#5!\"\"" }{TEXT -1 18 " and common ratio " }{XPPEDIT 18 0 "r=1/10" "6#/%\"rG*&\"\"\"F&\"#5!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "The partial sum " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "S[n] = 3/10+3/100+3/1000;" "6#/&%\" SG6#%\"nG,(*&\"\"$\"\"\"\"#5!\"\"F+*&F*F+\"$+\"F-F+*&F*F+\"%+5F-F+" } {TEXT -1 11 " + . . . + " }{XPPEDIT 18 0 "3/10^n = Sum(3/10^i,i=1..n) " "6#/*&\"\"$\"\"\")\"#5%\"nG!\"\"-%$SumG6$*&F%F&)F(%\"iGF*/F0;F&F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "is a finite geometric se ries." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "S[n]=a*(1-r^n)/(1-r)" "6#/&%\"SG6#%\"nG*(%\" aG\"\"\",&F*F*)%\"rGF'!\"\"F*,&F*F*F-F.F." }{TEXT -1 3 " = " } {XPPEDIT 18 0 "3/10" "6#*&\"\"$\"\"\"\"#5!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(1-1/(10^n))/``(1-1/10);" "6#*&-%!G6#,&\"\"\"F(*&F(F( )\"#5%\"nG!\"\"F-F(-F%6#,&F(F(*&F(F(F+F-F-F-" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(1-1/(10^n));" "6#-%!G6#,&\"\"\"F'*&F'F')\"#5%\"nG!\" \"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "n := 'n':\nSum(3/10^i,i = 1 .. n);\nsimpl ify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$*&\"\"\"F ()\"#5%\"iG!\"\"\"\"$/F+;F(%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& )\"#5,$%\"nG!\"\"#F(\"\"$#\"\"\"F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)ope ratorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " ``(1/(10^n));" "6#-%!G6#*&\"\"\"F')\"#5%\"nG!\"\"" }{TEXT -1 16 " tend s to 0, so " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{XPPEDIT 18 0 "``(1-1/(10^n))" "6#-%!G6#,&\"\"\"F'*&F'F')\"#5%\"nG!\"\"F," } {TEXT -1 10 " tends to " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(3/10^i,i=1..infinity)=1 /3" "6#/-%$SumG6$*&\"\"$\"\"\")\"#5%\"iG!\"\"/F,;F)%)infinityG*&F)F)F( F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "In general:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 259 34 "______________________ ____________" }{TEXT -1 1 " " }{TEXT 277 0 "" }}{PARA 256 "" 0 "" {TEXT -1 33 " The infinite geometric series " }{XPPEDIT 18 0 "Sum(a* r^(i-1),i=1..infinity)" "6#-%$SumG6$*&%\"aG\"\"\")%\"rG,&%\"iGF(F(!\" \"F(/F,;F(%)infinityG" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "a /(1-r)" "6#*&%\"aG\"\"\",&F%F%%\"rG!\"\"F(" }{TEXT -1 15 " provided th at " }{XPPEDIT 18 0 "abs(r)<1" "6#2-%$absG6#%\"rG\"\"\"" }{TEXT -1 1 " ." }{TEXT 276 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 278 34 " __________________________________" }{TEXT -1 1 " " }{TEXT 279 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "This foll ows from the formula " }{XPPEDIT 18 0 "Sum( a*r^(i-1),i = 1 ..n)= a*( 1-r^n)/(1-r)" "6#/-%$SumG6$*&%\"aG\"\"\")%\"rG,&%\"iGF)F)!\"\"F)/F-;F) %\"nG*(F(F),&F)F))F+F1F.F),&F)F)F+F.F." }{TEXT -1 21 ", and the fact \+ that " }{XPPEDIT 18 0 "Limit(r^n,n=infinity)=0" "6#/-%&LimitG6$)%\"rG% \"nG/F)%)infinityG\"\"!" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "abs(r)< 1" "6#2-%$absG6#%\"rG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Even if " }{TEXT 280 1 "r" }{TEXT -1 17 " is close to 1 or" } {XPPEDIT 18 0 " ``-1" "6#,&%!G\"\"\"F%!\"\"" }{TEXT -1 13 ", as long a s " }{XPPEDIT 18 0 "abs(r)<1" "6#2-%$absG6#%\"rG\"\"\"" }{TEXT -1 27 " , it is possible to choose " }{TEXT 281 1 "n" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "r^n" "6#)%\"rG%\"nG" }{TEXT -1 27 " is arbitrarily clo se to 0." }}{PARA 0 "" 0 "" {TEXT -1 22 "Compare the following." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "0.999^10000;\n1.001^10000;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ )fMt^%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M\"o;>#!\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Examples of geometric s eries " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "Consider the infinite series: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-1/2+1/4-1/8+1/16-1/ 32+` . . . `=Sum((-1/2)^(i-1),i = 1 .. infinity)" "6#/,0\"\"\"F%*&F%F% \"\"#!\"\"F(*&F%F%\"\"%F(F%*&F%F%\"\")F(F(*&F%F%\"#;F(F%*&F%F%\"#KF(F( %(~.~.~.~GF%-%$SumG6$),$*&F%F%F'F(F(,&%\"iGF%F%F(/F9;F%%)infinityG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "This series is geometri c with first term " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"" }{TEXT -1 18 " and common ratio " }{XPPEDIT 18 0 "r= -1/2" "6#/%\"rG,$*&\"\"\"F' \"\"#!\"\"F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "abs(r)=1/2" "6#/-%$absG6#%\"rG*&\"\"\"F)\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 26 ", th e series converges to " }{XPPEDIT 18 0 " a/(1-r)=1/(1-(-1/2)" "6#/*&% \"aG\"\"\",&F&F&%\"rG!\"\"F)*&F&F&,&F&F&,$*&F&F&\"\"#F)F)F)F)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Sum((-1/2)^(i-1),i=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$)#!\"\"\"\"#,&%\"iG\"\"\"F(F,/F+;F, %)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 290 1 "n" }{TEXT -1 33 " th partial sum of the series is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "S[n]=Sum((-1/2)^(i-1),i=1..infinity)" "6#/&%\"SG6#%\"nG -%$SumG6$),$*&\"\"\"F.\"\"#!\"\"F0,&%\"iGF.F.F0/F2;F.%)infinityG" }} {PARA 256 "" 0 "" {TEXT -1 7 " = " }{XPPEDIT 18 0 "(1-(-1/2)^n)/(1 -(-1/2)) = 2/3;" "6#/*&,&\"\"\"F&),$*&F&F&\"\"#!\"\"F+%\"nGF+F&,&F&F&, $*&F&F&F*F+F+F+F+*&F*F&\"\"$F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1-( -1/2)^n);" "6#-%!G6#,&\"\"\"F'),$*&F'F'\"\"#!\"\"F,%\"nGF," }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "which approaches " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF *F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "S[n] = PIECEWISE([``(2/3 )*(1-1/(2^n)), ` if n is even`],[``(2/3)*(1+1/(2^n)), ` if n is odd`]) " "6#/&%\"SG6#%\"nG-%*PIECEWISEG6$7$*&-%!G6#*&\"\"#\"\"\"\"\"$!\"\"F2, &F2F2*&F2F2)F1F'F4F4F2%.~if~n~is~evenG7$*&-F.6#*&F1F2F3F4F2,&F2F2*&F2F 2)F1F'F4F2F2%-~if~n~is~oddG" }{TEXT -1 71 " , so that the partial sums are alternately greater than and less than " }{XPPEDIT 18 0 "2/3" "6# *&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Sum((-1/2)^(i-1),i=1..n);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$)#!\"\"\"\"#,&%\"iG\"\"\"F(F,/F+;F,%\"nG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&)#!\"\"\"\"#,&%\"nG\"\"\"F*F*#\" \"%\"\"$#F'F-F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "The following picture illustrates the first few partial s ums of this series by plotting the points" }{XPPEDIT 18 0 "``(n,S[n]); " "6#-%!G6$%\"nG&%\"SG6#F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "The points lie alternately on the curves " }{XPPEDIT 18 0 "y= \+ 2/3" "6#/%\"yG*&\"\"#\"\"\"\"\"$!\"\"" }{XPPEDIT 18 0 "``(1+1/(2^x))" "6#-%!G6#,&\"\"\"F'*&F'F')\"\"#%\"xG!\"\"F'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y= 2/3" "6#/%\"yG*&\"\"#\"\"\"\"\"$!\"\"" }{XPPEDIT 18 0 "``(1-1/(2^x))" "6#-%!G6#,&\"\"\"F'*&F'F')\"\"#%\"xG!\"\"F," }{TEXT -1 50 ", both of which approach the horizontal asymptote " }{XPPEDIT 18 0 "y=2/3" "6#/%\"yG*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "pts := [seq([n,4/3*(-1/2)^(n+1)+2/3],n=1..10)];\nf := x -> 2/3+ 2/3*1/2^x;\ng := x -> 2/3-2/3*1/2^x;\nplot([f(x),g(x),2/3,pts],x=0..10 ,style=[line$3,point],\n color=[gray$2,green,red],linestyle=[2,2,3],s ymbol=circle,\n view=[0..10,0..1.4],labels=[``,``]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$ptsG7,7$\"\"\"F'7$\"\"##F'F)7$\"\"$#F,\"\"%7$ F.#\"\"&\"\")7$F1#\"#6\"#;7$\"\"'#\"#@\"#K7$\"\"(#\"#V\"#k7$F2#\"#&)\" $G\"7$\"\"*#\"$r\"\"$c#7$\"#5#\"$T$\"$7&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&#\"\"#\"\"$\"\"\"*&F- F0*&F0F0)F.9$!\"\"F0F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG f*6#%\"xG6\"6$%)operatorG%&arrowGF(,&#\"\"#\"\"$\"\"\"*&#F.F/F0*&F0F0) F.9$!\"\"F0F6F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 487 243 243 {PLOTDATA 2 "6)-%'CURVESG6&7W7$$\"\"!F)$\"3ELLLLLLL8!#<7$$\"3GLLL3x&)* 3\"!#=$\"3UbE6hk#[G\"F,7$$\"3emmm;arz@F0$\"3OF4(Q'*[)R7F,7$$\"3v***\\7 y%*z7$F0$\"3?J$*Q_`Q.7F,7$$\"3[LL$e9ui2%F0$\"31?:(HTT#p6F,7$$\"3z***\\ (oMrU^F0$\"3897zu4VL6F,7$$\"3nmmm\"z_\"4iF0$\"3ZB*[>%p()F07$$\" 33++D1J:w=F,$\"3m7U/Zus#[)F07$$\"3oLLL3En$4#F,$\"3Lb[#>\"obG#)F07$$\"3 #pmmT!RE&G#F,$\"3+M]3[\">V.)F07$$\"3D+++D.&4]#F,$\"3$R$f3@>SWyF07$$\"3 ;+++vB_f\\.\\B!o(F07$$\"33+++v'Hi#HF,$\"3[4n))e'>Pa(F07$$\"3 &om;z*ev:JF,$\"3#f.b0')[dV(F07$$\"3_LLL347TLF,$\"31JXX.J_CtF07$$\"3nLL LLY.KNF,$\"3\\dXAgA)HC(F07$$\"33++D\"o7Tv$F,$\"3;lGk:\"f2;(F07$$\"3?LL L$Q*o]RF,$\"31)=>y(4#y4(F07$$\"3m++D\"=lj;%F,$\"3+v'*oY?&z.(F07$$\"3S+ +vV&Rj_(o))pF07$$\"3CML$e9Ege%F,$\"3Z(=]ZsRU%pF07$$\"3]LL eR\"3Gy%F,$\"3$[=pRkZ)3pF07$$\"3emm;/T1&*\\F,$\"35&>.?*RrvoF07$$\"3=nm \"zRQb@&F,$\"39yz>z%)3YoF07$$\"3:++v=>Y2aF,$\"3o@47#HRP#oF07$$\"3Znm;z Xu9cF,$\"36:,w7yr-oF07$$\"34+++]y))GeF,$\"3Mi]FO1&Ry'F07$$\"3H++]i_QQg F,$\"3u<$QVN)4onF07$$\"3b++D\"y%3TiF,$\"3#=l$e2I![v'F07$$\"3+++]P![hY' F,$\"3=%fr'F07$$\"3;,++D\\'QH (F,$\"3+N7nj<:4nF07$$\"3%HL$e9S8&\\(F,$\"3PE`Gr&>Oq'F07$$\"3s++D1#=bq( F,$\"3CpT2A`g)p'F07$$\"3\"HLL$3s?6zF,$\"3ojf@c9O%p'F07$$\"3a***\\7`Wl7 )F,$\"39#R2!G9_!p'F07$$\"3enmmm*RRL)F,$\"3K3f(=EFto'F07$$\"3%zmmTvJga) F,$\"3cOO,TE]%o'F07$$\"3]MLe9tOc()F,$\"3u'p9W%H3#o'F07$$\"31,++]Qk\\*) F,$\"3!*)GTo+],o'F07$$\"3![LL3dg6<*F,$\"3y^KeK3BymF07$$\"3%ymmmw(Gp$*F ,$\"3wNu\"[$puwmF07$$\"3C++D\"oK0e*F,$\"3%*Hp(p\"RPvmF07$$\"35,+v=5s#y *F,$\"3g;OCz_BumF07$$\"#5F)$\"3gKLL$3xJn'F0-%'COLOURG6&%$RGBG$\")=THv! \")F`\\lF`\\l-%&STYLEG6#%%LINEG-%*LINESTYLEG6#\"\"#-F$6&7W7$F(F(7$F.$ \"3O%yn?A(o][!#>7$F4$\"3S,1CYpV[$*Fb]l7$F9$\"3g?+W4)z%*H\"F07$F>$\"3/L \"=O?>4k\"F07$FC$\"3G\">@aeB!**>F07$FH$\"3V)4WQh6;L#F07$FM$\"38qy#[stC k#F07$FR$\"3*)QA#*\\U/JHF07$FW$\"3ggOKJAJXMF07$Ffn$\"3Y'>h!H#Q'eQF07$F [o$\"3'yAr<.:3B%F07$F`o$\"3U'f_X&R\"Rc%F07$Feo$\"3%*>\"*G')eg][F07$Fjo $\"3Ex%39_wZ5&F07$F_p$\"3e)H[_=9!*H&F07$Fdp$\"3m)RZATJ*)[&F07$Fip$\"3= 8u$)H%)4`cF07$F^q$\"37BmWuOh*y&F07$Fcq$\"3m'HyFZ%e(*eF07$Fhq$\"3`,)y)H -\")3gF07$F]r$\"35v(3J2^.4'F07$Fbr$\"3Wn/pwX'F07$Fet$\"3Ya`8a[C(['F07$Fjt$\"3#4T 77/%f4lF07$F_u$\"3[\"Gk'F07$F`y$\"3GCuXrg+YmF07$Fey$\"3/'p>BpI)[mF07$Fjy$\"3'ej=*)Q] 7l'F07$F_z$\"3qV?\\EL=`mF07$Fdz$\"3!33]2]-^l'F07$Fiz$\"3%o*e^)R'ecmF07 $F^[l$\"3m-kN;%fzl'F07$Fc[l$\"3+;(*3a!)4fmF07$Fh[l$\"3++++]i:gmF0F\\\\ lFc\\lFg\\l-F$6&7S7$F($\"3ImmmmmmmmF07$F4F`gl7$F>F`gl7$FHF`gl7$FRF`gl7 $FWF`gl7$FfnF`gl7$F[oF`gl7$F`oF`gl7$FeoF`gl7$FjoF`gl7$F_pF`gl7$FdpF`gl 7$FipF`gl7$F^qF`gl7$FcqF`gl7$FhqF`gl7$F]rF`gl7$FbrF`gl7$FgrF`gl7$F\\sF `gl7$FasF`gl7$FfsF`gl7$F[tF`gl7$F`tF`gl7$FetF`gl7$FjtF`gl7$F_uF`gl7$Fd uF`gl7$FiuF`gl7$F^vF`gl7$FcvF`gl7$FhvF`gl7$F]wF`gl7$FbwF`gl7$FgwF`gl7$ F\\xF`gl7$FaxF`gl7$FfxF`gl7$F[yF`gl7$F`yF`gl7$FeyF`gl7$FjyF`gl7$F_zF`g l7$FdzF`gl7$FizF`gl7$F^[lF`gl7$Fc[lF`gl7$Fh[lF`gl-F]\\l6&F_\\lF($\"*++ ++\"Fb\\lF(Fc\\l-Fh\\l6#\"\"$-F$6&7,7$$\"\"\"F)F][m7$$Fj\\lF)$\"3+++++ +++]F07$$FhjlF)$\"3++++++++vF07$$\"\"%F)$\"3+++++++]iF07$$\"\"&F)$\"3+ ++++++voF07$$\"\"'F)$\"3++++++]ilF07$$\"\"(F)$\"3++++++v=nF07$$\"\")F) $\"3+++++]iSmF07$$\"\"*F)$\"3+++++vozmF0Fifl-F]\\l6&F_\\lFdjlF(F(-Fd\\ l6#%&POINTGFg\\l-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$%!GFa^m-%%VIEWG6$ ;F(Fh[l;F($\"#9!\"\"" 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" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Consider the infinite ser ies:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "5+3+9/5+27/25 +81/125+` . . . `;" "6#,.\"\"&\"\"\"\"\"$F%*&\"\"*F%F$!\"\"F%*&\"#FF% \"#DF)F%*&\"#\")F%\"$D\"F)F%%(~.~.~.~GF%" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 41 "This series is geometric with first term " } {XPPEDIT 18 0 "a = 5;" "6#/%\"aG\"\"&" }{TEXT -1 18 " and common ratio " }{XPPEDIT 18 0 "r = 3/5;" "6#/%\"rG*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "abs(r) \+ = 3/5;" "6#/-%$absG6#%\"rG*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 26 ", the series converg es to " }{XPPEDIT 18 0 "a/(1-r) = 5/(1-3/5);" "6#/*&%\"aG\"\"\",&F&F&% \"rG!\"\"F)*&\"\"&F&,&F&F&*&\"\"$F&F+F)F)F)" }{XPPEDIT 18 0 "`` = 25/( 5-3);" "6#/%!G*&\"#D\"\"\",&\"\"&F'\"\"$!\"\"F+" }{XPPEDIT 18 0 "`` = \+ 25/2;" "6#/%!G*&\"#D\"\"\"\"\"#!\"\"" }{XPPEDIT 18 0 "`` = 12;" "6#/%! G\"#7" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 293 4 "Note" }{TEXT -1 6 ": The " }{TEXT 292 1 "n" }{TEXT -1 19 " th partial sum is " }{XPPEDIT 18 0 "S[n] = a*(1-r^n)/(1-r);" "6#/&%\"SG6#%\"nG*(%\"aG\"\"\",&F*F*)%\"rGF' !\"\"F*,&F*F*F-F.F." }{XPPEDIT 18 0 "`` = 25/2;" "6#/%!G*&\"#D\"\"\"\" \"#!\"\"" }{XPPEDIT 18 0 "``(1-(3/5)^n)" "6#-%!G6#,&\"\"\"F')*&\"\"$F' \"\"&!\"\"%\"nGF," }{TEXT -1 31 ", and the first few terms are: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "5, 8, 49/5, 272/25, 1 441/125, 7448/625, 37969/3125, 192032/15625,` . . . `" "6+\"\"&\"\")*& \"#\\\"\"\"F#!\"\"*&\"$s#F'\"#DF(*&\"%T9F'\"$D\"F(*&\"%[uF'\"$D'F(*&\" &pz$F'\"%DJF(*&\"'K?>F'\"&Dc\"F(%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "These partial sums can be pictured as follows. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "pts := [seq([n,25/2*(1-(3/5)^n)],n=1..8)];\nplot([pts,25/2*(1 -(3/5)^x),25/2],x=0..8,style=[point,line$2],\n color=[red,gray,green] ,linestyle=2,symbol=circle,\n view=[0..8,0..13],labels=[``,``],yti ckmarks=13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7*7$\"\"\"\"\"& 7$\"\"#\"\")7$\"\"$#\"#\\F(7$\"\"%#\"$s#\"#D7$F(#\"%T9\"$D\"7$\"\"'#\" %[u\"$D'7$\"\"(#\"&pz$\"%DJ7$F+#\"'K?>\"&Dc\"" }}{PARA 13 "" 1 "" {GLPLOT2D 438 259 259 {PLOTDATA 2 "6*-%'CURVESG6%7*7$$\"\"\"\"\"!$\"\" &F*7$$\"\"#F*$\"\")F*7$$\"\"$F*$\"3q+++++++)*!#<7$$\"\"%F*$\"33++++++) 3\"!#;7$F+$\"31+++++!G:\"F=7$$\"\"'F*$\"3/+++++o\">\"F=7$$\"\"(F*$\"34 ++++!3]@\"F=7$F0$\"31++++[+H7F=-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*FU -%&STYLEG6#%&POINTG-F$6%7S7$FUFU7$$\"3ELLLLBxV5E$F[o$\"35N8J:V2=>F77$$\"3MLLLLAKn\\F[o$\"3/\")4YdbO,GF77$$ \"3=LLLLc$\\o'F[o$\"3pqCzF81;OF77$$\"3)emmm^&Q%R)F[o$\"3aR.?Xv#*eVF77$ $\"3wKLL$Qk#z**F[o$\"3'zP2zd^?*\\F77$$\"3))*****\\YJ?;\"F7$\"3%fp>ORxd f&F77$$\"3?LLL=\"\\E$p'F77$$\"3wmmm'3Q\\n\"F7$\"3s&\\Twkbr=(F77$$\"3OLLLB6@G=F7$ \"3y[CiY8E(e(F77$$\"3&)******f-w+?F7$\"3wG)*)>GZ<+)F77$$\"3%*********y ,u@F7$\"3i!p\"G$e`FQ)F77$$\"3)*******RP)4M#F7$\"3zFuhdnN>()F77$$\"3ILL L=Zg#\\#F7$\"33SL'[@A6+*F77$$\"3cmmmEn*Gn#F7$\"38TwM\"en*3$*F77$$\"3Tm mm1xiDGF7$\"3oqC#yIk%[&*F77$$\"3!)*****\\9!H.IF7$\"30Wuw_S`/)*F77$$\"3 Immm1:bgJF7$\"30^Gk4+E,5F=7$$\"3<+++X@4LLF7$\"3#\\2;:7XA-\"F=7$$\"31++ +N;R(\\$F7$\"3SMd&)\\+eS5F=7$$\"3wmmm;4#)oOF7$\"3a@zW5!R\"e5F=7$$\"3jm mm6lCEQF7$\"3a([0(GQ'H2\"F=7$$\"3ELLL$G^g*RF7$\"3!\\O'4**Gn(3\"F=7$$\" 3oKLL=2VsTF7$\"3e#Ryn!*e;5\"F=7$$\"3f*****\\`pfK%F7$\"3O*)\\pn*[G6\"F= 7$$\"3!HLLLm&z\"\\%F7$\"3R.QjJ$))R7\"F=7$$\"3s******z-6jYF7$\"37N6#[5Z X8\"F=7$$\"3<******4#32$[F7$\"3`ZMs;,-W6F=7$$\"3O*****\\#y'G*\\F7$\"3C ktdD_W_6F=7$$\"3G******H%=H<&F7$\"31$)[`:v,h6F=7$$\"35mmm1>qM`F7$\"3'f 6p%Qc2o6F=7$$\"3%)*******HSu]&F7$\"3'y4f!=[*\\<\"F=7$$\"3'HLL$ep'Rm&F7 $\"3#oG!\\q&e2=\"F=7$$\"3')******R>4NeF7$\"3SPga/Vb'=\"F=7$$\"3#emm;@2 h*fF7$\"3#4n&>8Rc\">\"F=7$$\"3]*****\\c9W;'F7$\"3?/dK\"3yj>\"F=7$$\"3L mmmmd'*GjF7$\"3xilgS6q+7F=7$$\"3j*****\\iN7]'F7$\"30>C)o-a[?\"F=7$$\"3 aLLLt>:nmF7$\"3$GL#H!oA&37F=7$$\"35LLL.a#o$oF7$\"3qu*4kFm>@\"F=7$$\"3a mmm^Q40qF7$\"3#*)=tK$*)4:7F=7$$\"3y******z]rfrF7$\"3aZ'*e?&\\x@\"F=7$$ \"3gmmmc%GpL(F7$\"38V5Bb2a?7F=7$$\"3/LLL8-V&\\(F7$\"3g'4Ms'>$GA\"F=7$$ \"3=+++XhUkwF7$\"3dwalJ*y]A\"F=7$$\"3=+++:o " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "We can use the formula for the sum of an infinite geo metric series to express an infinite repeating decimal as a ratio of t wo integers." }}{PARA 0 "" 0 "" {TEXT -1 75 "For example, consider the infinite repeating decimal: 0.123123123123 . . . " }}{PARA 0 "" 0 "" {TEXT -1 26 "This number has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "123/(10^3)+123/(10^6)+123/(10^9)+123/(10^12)+` . . . `;" "6#,,*&\"$B\"\"\"\"*$\"#5\"\"$!\"\"F&*&F%F&*$F(\"\"'F*F&*&F%F &*$F(\"\"*F*F&*&F%F&*$F(\"#7F*F&%(~.~.~.~GF&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 53 "This is an infinite geometric series with first term " }{XPPEDIT 18 0 "a=123/10^3" "6#/%\"aG*&\"$B\"\"\"\"*$\"# 5\"\"$!\"\"" }{TEXT -1 18 " and common ratio " }{XPPEDIT 18 0 "r=1/10^ 3" "6#/%\"rG*&\"\"\"F&*$\"#5\"\"$!\"\"" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 46 "The sum of this infinite geometric series is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a/(1-r)=``(123/10^3)/ (1-1/10^3)" "6#/*&%\"aG\"\"\",&F&F&%\"rG!\"\"F)*&-%!G6#*&\"$B\"F&*$\"# 5\"\"$F)F&,&F&F&*&F&F&*$F1F2F)F)F)" }{XPPEDIT 18 0 "``=123/(10^3-1)" " 6#/%!G*&\"$B\"\"\"\",&*$\"#5\"\"$F'F'!\"\"F," }{XPPEDIT 18 0 "``=123/9 99" "6#/%!G*&\"$B\"\"\"\"\"$***!\"\"" }{XPPEDIT 18 0 "``=41/333" "6#/% !G*&\"#T\"\"\"\"$L$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 65 "As a (partial) check, a calculator working with ten digits gives \+ " }{XPPEDIT 18 0 "41/333" "6#*&\"#T\"\"\"\"$L$!\"\"" }{TEXT -1 1 " " } {TEXT 291 1 "~" }{TEXT -1 15 " 0.1231231231. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Conve rgence of bounded monotonic sequences" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 11 "A sequence " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1],a[2],` . . . `,a [n],` . . . `;" "6'&%\"aG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nGF*" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 64 "which steadily increase s (or at least never decreases), that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[n]<=a[n+1]" "6#1&%\"aG6#%\"nG&F%6#,&F'\"\" \"F+F+" }{TEXT -1 9 " for all " }{TEXT 282 1 "n" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 12 "is called a " }{TEXT 269 20 "monotonic in creasing" }{TEXT 258 1 " " }{TEXT -1 9 "sequence." }}{PARA 0 "" 0 "" {TEXT -1 22 "Similarly, a sequence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "a[1],a[2],` . . . `,a[n],` . . . `;" "6'&%\"aG6#\"\" \"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nGF*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 64 "which steadily decreases (or at least never increases), that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[n]>=a[ n+1]" "6#1&%\"aG6#,&%\"nG\"\"\"F)F)&F%6#F(" }{TEXT -1 9 " for all " } {TEXT 283 1 "n" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 12 "is call ed a " }{TEXT 269 20 "monotonic decreasing" }{TEXT -1 10 " sequence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "A usefu l result concerning monotonic sequences is the following:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "If a monotonic inc reasing sequence " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 4 " is " }{TEXT 269 13 "bounded above" }{TEXT -1 16 " by some number \+ " }{TEXT 303 1 "M" }{TEXT -1 10 ", that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[n] <= M" "6#1&%\"aG6#%\"nG%\"MG" } {TEXT -1 10 " for all " }{TEXT 285 1 "n" }{TEXT -1 1 "," }}{PARA 0 " " 0 "" {TEXT -1 5 "then " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" } {TEXT -1 1 " " }{TEXT 269 20 "converges to a limit" }{TEXT -1 1 " " } {TEXT 297 1 "L" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "L <= M" "6#1%\"L G%\"MG" }{TEXT -1 2 ". 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We then call " }{TEXT 298 1 "L" }{TEXT -1 3 " a " }{TEXT 269 17 "least upper bound" }{TEXT -1 18 " for the sequence." }}{PARA 0 "" 0 "" {TEXT -1 45 "It turns out that the sequence must approach " }{TEXT 299 1 "L" }{TEXT -1 13 " as \+ a limit. " }}{PARA 0 "" 0 "" {TEXT -1 99 "This is because there must b e terms of the sequence which are within any small prescribed distance " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 6 " from " } {TEXT 300 1 "L" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "Indeed , suppose we pick a specific positive number " }{XPPEDIT 18 0 "epsilon " "6#%(epsilonG" }{TEXT -1 27 ". Then no matter how small " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 56 " is, if we go far enough a long the sequence ( by taking " }{TEXT 284 1 "n" }{TEXT -1 53 " suffic iently large ) we must be able to find a term " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 21 " of the sequence with" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L-epsilon " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 22 "Consider the sequence " }{XPPEDIT 18 0 " a[1],a[2],` . . . `,a[n],` . . . `;" "6'&%\"aG6#\"\"\"&F$6#\"\"#%(~.~. ~.~G&F$6#%\"nGF*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a[n] = 1+1/1!+1/2!+1/3!+1/4!+` . . . `+1/n!;" "6#/& %\"aG6#%\"nG,0\"\"\"F)*&F)F)-%*factorialG6#F)!\"\"F)*&F)F)-F,6#\"\"#F. F)*&F)F)-F,6#\"\"$F.F)*&F)F)-F,6#\"\"%F.F)%(~.~.~.~GF)*&F)F)-F,6#F'F.F )" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "a[1] = 1 + 1" "6#/&%\"aG6#\"\"\",&F' F'F'F'" }{TEXT -1 5 " = 2 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "a[2] = 1+1+1/2" "6#/&%\"aG6#\"\"#,(\"\"\"F)F)F)*&F)F)F' !\"\"F)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "5/2 = 2.5;" "6#/*&\"\"&\"\" \"\"\"#!\"\"-%&FloatG6$\"#DF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[3] = 1+1+1/2+1/6" "6#/&%\"aG6#\"\"$, *\"\"\"F)F)F)*&F)F)\"\"#!\"\"F)*&F)F)\"\"'F,F)" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "8/3" "6#*&\"\")\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {TEXT 265 1 "~" }{TEXT -1 13 " 2.666666667 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[4] = 1+1+1/2+1/6+1/24" "6#/&%\"aG6#\"\"%,, \"\"\"F)F)F)*&F)F)\"\"#!\"\"F)*&F)F)\"\"'F,F)*&F)F)\"#CF,F)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "65/24" "6#*&\"#l\"\"\"\"#C!\"\"" }{TEXT -1 1 " " }{TEXT 266 1 "~" }{TEXT -1 12 " 2.70833333 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[5] = 1+1+1/2+1/6+1/24+1/120" "6#/&% \"aG6#\"\"&,.\"\"\"F)F)F)*&F)F)\"\"#!\"\"F)*&F)F)\"\"'F,F)*&F)F)\"#CF, F)*&F)F)\"$?\"F,F)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "163/60" "6#*&\"$ j\"\"\"\"\"#g!\"\"" }{TEXT -1 1 " " }{TEXT 267 1 "~" }{TEXT -1 13 " 2. 716666667 " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 286 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "This sequ ence is monotonic increasing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "In fact the terms are partial sums of the series " }{XPPEDIT 18 0 "Sum(1/n!,n = 0 .. infinity)=1+1/1!+1/2!+1/3! +1/4!+` . . . `+1/n!+` . . . `" "6#/-%$SumG6$*&\"\"\"F(-%*factorialG6# %\"nG!\"\"/F,;\"\"!%)infinityG,2F(F(*&F(F(-F*6#F(F-F(*&F(F(-F*6#\"\"#F -F(*&F(F(-F*6#\"\"$F-F(*&F(F(-F*6#\"\"%F-F(%(~.~.~.~GF(*&F(F(-F*6#F,F- F(FBF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 "Considering th e terms of the series successively we have: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/2! = 1/2" "6#/*&\"\"\"F%-%*factorial G6#\"\"#!\"\"*&F%F%F)F*" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "1/3! = 1 /(``(3)*``(2));" "6#/*&\"\"\"F%-%*factorialG6#\"\"$!\"\"*&F%F%*&-%!G6# F)F%-F.6#\"\"#F%F*" }{XPPEDIT 18 0 "``< 1/2^2" "6#2%!G*&\"\"\"F&*$\"\" #F(!\"\"" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "1/4! = 1/(``(4)*``(3)*`` (2));" "6#/*&\"\"\"F%-%*factorialG6#\"\"%!\"\"*&F%F%*(-%!G6#F)F%-F.6# \"\"$F%-F.6#\"\"#F%F*" }{XPPEDIT 18 0 "``< 1/2^3" "6#2%!G*&\"\"\"F&*$ \"\"#\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "In gen eral, " }{XPPEDIT 18 0 "1/n! < 1/2^(n-1)" "6#2*&\"\"\"F%-%*factorialG6 #%\"nG!\"\"*&F%F%)\"\"#,&F)F%F%F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 6 "Hence " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[n] = 1+1/1!+1/2!+1/3!+1/4!" "6#/&%\"aG6#%\"nG,,\"\"\"F)*&F)F)-%*fac torialG6#F)!\"\"F)*&F)F)-F,6#\"\"#F.F)*&F)F)-F,6#\"\"$F.F)*&F)F)-F,6# \"\"%F.F)" }{TEXT -1 11 " + . . . + " }{XPPEDIT 18 0 "1/n! < 1 + 1 + 1 /2 + 1/2^2 + 1/2^3" "6#2*&\"\"\"F%-%*factorialG6#%\"nG!\"\",,F%F%F%F%* &F%F%\"\"#F*F%*&F%F%*$F-F-F*F%*&F%F%*$F-\"\"$F*F%" }{TEXT -1 11 " + . \+ . . + " }{XPPEDIT 18 0 "1/2^(n-1)" "6#*&\"\"\"F$)\"\"#,&%\"nGF$F$!\"\" F)" }{TEXT -1 13 " ------- (i)" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1+``(1-1/(2^n))/``(1-1/2);" "6#,&\"\"\"F$*&-%!G6#,&F$F $*&F$F$)\"\"#%\"nG!\"\"F.F$-F'6#,&F$F$*&F$F$F,F.F.F.F$" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 73 "since, after the first term, the seri es on the right of (i) is geometric." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "a[n] < 1 + 1/(1-1/2)" "6#2&%\"aG6#%\"nG,&\"\"\"F)*&F)F),&F)F)*&F)F)\"\"#!\"\"F.F .F)" }{TEXT -1 56 " = 3 for all n, that is, the sequence with general \+ term " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 25 " has 3 as an upper bound." }}{PARA 0 "" 0 "" {TEXT -1 57 "It follows that the s equence converges to a limit L with " }{XPPEDIT 18 0 "L <= 3" "6#1%\"L G\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "In other words, the series " }{XPPEDIT 18 0 "Sum(1/n!,n=0..infinity)" "6#-%$SumG6$*& \"\"\"F'-%*factorialG6#%\"nG!\"\"/F+;\"\"!%)infinityG" }{TEXT -1 37 " \+ converges to a sum which is less 3." }}{PARA 0 "" 0 "" {TEXT -1 8 "In fact " }{XPPEDIT 18 0 "Sum(1/n!,n=0..infinity)=exp(1)" "6#/-%$SumG6$* &\"\"\"F(-%*factorialG6#%\"nG!\"\"/F,;\"\"!%)infinityG-%$expG6#F(" } {TEXT -1 1 " " }{TEXT 268 1 "~" }{TEXT -1 13 " 2.718281828." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "a \+ := n -> Sum(1/i!,i=0..n);\nLimit(a(n),n=infinity);\nevalf(%);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrow GF(-%$SumG6$*&\"\"\"F0-%*factorialG6#%\"iG!\"\"/F4;\"\"!9$F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%$SumG6$*&\"\"\"F*-%*facto rialG6#%\"iG!\"\"/F.;\"\"!%\"nG/F3%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "The following picture illustrates the con vergence of the sequence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 281 "a := n -> Sum(1/i!,i=0..n):\nf := \+ unapply(sum(1/i!,i=0..x),x): # for dotted curve\npts := [seq([n,a(n)], n=1..8)]:\nplot([f(x),exp(1),3,pts],x=0..8,style=[line$3,point],\n co lor=[gray,blue,COLOR(RGB,0,.7,0),red],linestyle=[2,3,3],\n symbol=cir cle,view=[0..8,1.5..3.2],labels=[`n`,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 499 255 255 {PLOTDATA 2 "6)-%'CURVESG6&7X7$$\"\"!F)$\"+***** *****!#57$$\"+q;')=()!#6$\"+9f&=5\"!\"*7$$\"+MBxV5E$F,$\"+&[A@P\"F37$$\"+v2<9TF,$\"+[x$RY \"F37$$\"+MAKn\\F,$\"+Z>j_:F37$$\"+N*Gh#eF,$\"+oEXQ;F37$$\"+Nc$\\o'F,$ \"+egb?F37$$\"+l9.i6F3$\"+kTQ9@F37$$\"+>\"\\-]#F37$$\"++z,u@F3$\"+FfhXDF37$$\"+SP)4 M#F3$\"+$\\M8e#F37$$\"+>Zg#\\#F3$\"+2VsTF3$\"+3l#4r#F37$$\"+O&pfK%F3$\"+jTo7FF37$$\"+kcz\"\\%F3$\"+e<89FF 37$$\"+\"G5Jm%F3$\"+3#\\_r#F37$$\"+6#32$[F3$\"+)Rggr#F37$$\"+Ey'G*\\F3 $\"+#pWmr#F37$$\"+J%=H<&F3$\"+C07qM`F3$\"+IAV4NeF3$\"+/g'zr#F37$$ \"+8s5'*fF3$\"+eP0=FF37$$\"+mXTkhF3$\"+<,7=FF37$$\"+od'*GjF3$\"+%om\"= FF37$$\"+EcB,lF3$\"+E9?=FF37$$\"+v>:nmF3$\"+J^A=FF37$$\"+0a#o$oF3$\"+$ HU#=FF37$$\"+`Q40qF3$\"+sUD=FF37$$\"+\"3:(frF3$\"+1@E=FF37$$\"+e%GpL(F 3$\"+I%o#=FF37$$\"+:-V&\\(F3$\"+xBF=FF37$$\"+ZhUkwF3$\"+K`F=FF37$$\"+< o5E$F[^lFe]l7$$\"3MLLLLAKn\\F[^ lFe]l7$$\"3=LLLLc$\\o'F[^lFe]l7$$\"3)emmm^&Q%R)F[^lFe]l7$$\"3wKLL$Qk#z **F[^lFe]l7$$\"3))*****\\YJ?;\"Fg]lFe]l7$$\"3?LLL=\"\\qM`Fg]lFe]l7$$\"3%)*******HSu]& Fg]lFe]l7$$\"3'HLL$ep'Rm&Fg]lFe]l7$$\"3')******R>4NeFg]lFe]l7$$\"3#emm ;@2h*fFg]lFe]l7$$\"3]*****\\c9W;'Fg]lFe]l7$$\"3Lmmmmd'*GjFg]lFe]l7$$\" 3j*****\\iN7]'Fg]lFe]l7$$\"3aLLLt>:nmFg]lFe]l7$$\"35LLL.a#o$oFg]lFe]l7 $$\"3ammm^Q40qFg]lFe]l7$$\"3y******z]rfrFg]lFe]l7$$\"3gmmmc%GpL(Fg]lFe ]l7$$\"3/LLL8-V&\\(Fg]lFe]l7$$\"3=+++XhUkwFg]lFe]l7$$\"3=+++:o " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Ta sks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 32 "Find the lim it of the sequence " }{XPPEDIT 18 0 "a[1],a[2],` . . . `,a[n],` . . . `" "6'&%\"aG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nGF*" }{TEXT -1 32 " \+ in each of the following cases." }}{PARA 0 "" 0 "" {TEXT -1 113 "It ma y be useful to perform a numerical investigation involving the computa tion of various terms of the sequence." }}{PARA 0 "" 0 "" {TEXT -1 9 " (a) " }{XPPEDIT 18 0 "2*n/(5*n-3)" "6#*(\"\"#\"\"\"%\"nGF%,&*&\" \"&F%F&F%F%\"\"$!\"\"F+" }{TEXT -1 29 " (b) " }{XPPEDIT 18 0 "a[n] = 1-(9/10)^n" "6#/&%\"aG6#%\"nG,&\"\"\"F))*&\"\"* F)\"#5!\"\"F'F." }{TEXT -1 17 " (c) " }{XPPEDIT 18 0 "a[n] =(1+1/n)^n" "6#/&%\"aG6#%\"nG),&\"\"\"F**&F*F*F'!\"\"F*F'" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " (d) " }{XPPEDIT 18 0 "a[1 ] = 1, a[n+1] = 1/2" "6$/&%\"aG6#\"\"\"F'/&F%6#,&%\"nGF'F'F'*&F'F'\"\" #!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(a[n]+2/a[n]),1 <= n;" "6$-% !G6#,&&%\"aG6#%\"nG\"\"\"*&\"\"#F+&F(6#F*!\"\"F+1F+F*" }{TEXT -1 25 " \+ (e) " }{XPPEDIT 18 0 "a[n] = n*sin(1/n)" "6#/&%\"a G6#%\"nG*&F'\"\"\"-%$sinG6#*&F)F)F'!\"\"F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "_________________ ___________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 29 "____________ ________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 69 " In each of the following cases, find the sum of the infinite seri es " }{XPPEDIT 18 0 "Sum(a[n],n=1..infinity)" "6#-%$SumG6$&%\"aG6#%\"n G/F);\"\"\"%)infinityG" }{TEXT -1 29 ", provided that it converges." } }{PARA 0 "" 0 "" {TEXT -1 10 " (a) " }{XPPEDIT 18 0 "1+1/3+1/9+1/ 27+` . . . `" "6#,,\"\"\"F$*&F$F$\"\"$!\"\"F$*&F$F$\"\"*F'F$*&F$F$\"#F F'F$%(~.~.~.~GF$" }{TEXT -1 7 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " (b) " }{XPPEDIT 18 0 "1-1/3+1/9-1/27+1/81-` . . . `;" "6#,.\"\" \"F$*&F$F$\"\"$!\"\"F'*&F$F$\"\"*F'F$*&F$F$\"#FF'F'*&F$F$\"#\")F'F$%(~ .~.~.~GF'" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " (c) " }{XPPEDIT 18 0 "1/3+2/9+4/27+8/81+` . . . `" "6#,,*&\"\"\"F%\"\"$! \"\"F%*&\"\"#F%\"\"*F'F%*&\"\"%F%\"#FF'F%*&\"\")F%\"#\")F'F%%(~.~.~.~G F%" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " (d) " } {XPPEDIT 18 0 "1+1/2+1/3+1/4+1/5+1/6+` . . . `" "6#,0\"\"\"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 29 "_________ ___________________ " }}{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 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 29 "__ __________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 75 "Express the following infinite decimals as fractions in t heir lowest terms." }}{PARA 0 "" 0 "" {TEXT -1 41 "(a) 0.44444444 . . \+ . " }}{PARA 0 "" 0 "" {TEXT -1 23 "(b) 0.474747474 7 . . . " }}{PARA 0 "" 0 "" {TEXT -1 29 "____________________________ \+ " }}{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 29 "____________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Find the \+ sum of the series" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(1/(n*(n+1)),n=1..infinity)=1/(1*`.`*2)+1/(2*`.`*3)+1/(3*`.`*4)+1/( 4*`.`*5)+` . . . `" "6#/-%$SumG6$*&\"\"\"F(*&%\"nGF(,&F*F(F(F(F(!\"\"/ F*;F(%)infinityG,,*&F(F(*(F(F(%\".GF(\"\"#F(F,F(*&F(F(*(F4F(F3F(\"\"$F (F,F(*&F(F(*(F7F(F3F(\"\"%F(F,F(*&F(F(*(F:F(F3F(\"\"&F(F,F(%(~.~.~.~GF (" }{TEXT -1 14 "------- (A) " }}{PARA 0 "" 0 "" {TEXT -1 12 " as fo llows." }}{PARA 0 "" 0 "" {TEXT -1 17 " (a) Check that " }{XPPEDIT 18 0 "1/(n*(n+1))=1/n-1/(n+1)" "6#/*&\"\"\"F%*&%\"nGF%,&F'F%F%F%F%!\" \",&*&F%F%F'F)F%*&F%F%,&F'F%F%F%F)F)" }{TEXT -1 27 " for any positive integer " }{TEXT 288 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 " (b) Use the formula in (a) to find " }{XPPEDIT 18 0 "S[10] = \+ ``(1/1-1/2)+``(1/2-1/3)+``(1/3-1/4)+` . . . `+``(1/10-1/11);" "6#/&%\" SG6#\"#5,,-%!G6#,&*&\"\"\"F.F.!\"\"F.*&F.F.\"\"#F/F/F.-F*6#,&*&F.F.F1F /F.*&F.F.\"\"$F/F/F.-F*6#,&*&F.F.F7F/F.*&F.F.\"\"%F/F/F.%(~.~.~.~GF.-F *6#,&*&F.F.F'F/F.*&F.F.\"#6F/F/F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 " (c) Find a formula for the " }{TEXT 289 1 "n" }{TEXT -1 16 " th partial sum " }{XPPEDIT 18 0 "S[n] = Sum(1/(i*(i+1)),i = 1 .. \+ n);" "6#/&%\"SG6#%\"nG-%$SumG6$*&\"\"\"F,*&%\"iGF,,&F.F,F,F,F,!\"\"/F. ;F,F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 " (d) Find " } {XPPEDIT 18 0 "Limit(S[n],n=infinity)" "6#-%&LimitG6$&%\"SG6#%\"nG/F)% )infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "___________ _________________ " }}{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 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 29 "____________ ________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the series " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(1/(n^2),n = 1 .. infinity) = 1/(1*`.`*1)+1/(2*`.`*2)+1/(3*`.`*3)+1/(4*`.`*4)+` . \+ . . `" "6#/-%$SumG6$*&\"\"\"F(*$%\"nG\"\"#!\"\"/F*;F(%)infinityG,,*&F( F(*(F(F(%\".GF(F(F(F,F(*&F(F(*(F+F(F3F(F+F(F,F(*&F(F(*(\"\"$F(F3F(F8F( F,F(*&F(F(*(\"\"%F(F3F(F;F(F,F(%(~.~.~.~GF(" }{TEXT -1 13 " ------- (B )." }}{PARA 0 "" 0 "" {TEXT -1 29 " (a) Omitting the first term " } {XPPEDIT 18 0 "1/(1*`.`*1)=1" "6#/*&\"\"\"F%*(F%F%%\".GF%F%F%!\"\"F%" }{TEXT -1 43 ", the series formed by the remaining terms " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(1/(n^2),n = 2 .. infin ity) = 1/(2*`.`*2)+1/(3*`.`*3)+1/(4*`.`*4)+1/(5*`.`*5)+` . . . `" "6#/ -%$SumG6$*&\"\"\"F(*$%\"nG\"\"#!\"\"/F*;F+%)infinityG,,*&F(F(*(F+F(%\" .GF(F+F(F,F(*&F(F(*(\"\"$F(F3F(F6F(F,F(*&F(F(*(\"\"%F(F3F(F9F(F,F(*&F( F(*(\"\"&F(F3F(F " 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 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 29 "____________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 16 "Code for picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 379 "pts := [seq([n,1-1/n^1.5],n=2..20) ]:\np1:=plot([1-1/x^1.5,1,1.2,pts],x=0..20,y=0..1.25,style=[line$3,poi nt],\n color=[gray,blue,COLOR(RGB,0,.7,0),red],linestyle=[2,3,3],\n \+ symbol=circle,view=[1..20,.3..1.25],labels=[``,``],\n tickmarks=[0,0] ):\nt1 := plots[textplot]([2,1.25,`M`],color=COLOR(RGB,0,.7,0)):\nt2 : = plots[textplot]([2,1.05,`L`],color=blue):\nplots[display]([p1,t1,t2] );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 }