{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 257 "" 0 "" {TEXT -1 43 "Lesson 16: Introduction to Infinite Series" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 "Infinite Series" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 215 "What does it mean to add up a sequence of numb ers? (Remember that sequences are always infinite, so this is a quest ion about adding up an infinite set of numbers.) As explained in Sect ion 10.2, to compute the sum " }{XPPEDIT 18 0 "sum(a_n,n = 1 .. infini ty);" "6#-%$sumG6$%$a_nG/%\"nG;\"\"\"%)infinityG" }{TEXT -1 29 ", we f irst form the sequence " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 4 " o f " }{TEXT 256 13 "partial sums:" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s_1 = a_1;" "6#/%$s_1G%$a_1G" }{TEXT -1 5 " , " } {XPPEDIT 18 0 "s_2 = a_1+a_2;" "6#/%$s_2G,&%$a_1G\"\"\"%$a_2GF'" } {TEXT -1 5 " , " }{XPPEDIT 18 0 "s_3 = a_1+a_2+a_3;" "6#/%$s_3G,(%$a _1G\"\"\"%$a_2GF'%$a_3GF'" }{TEXT -1 10 " , ... ," }}{PARA 0 "" 0 " " {TEXT -1 21 "whose general term is" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s_n;" "6#%$s_nG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "sum(a_i,i = 1 \+ .. n);" "6#-%$sumG6$%$a_iG/%\"iG;\"\"\"%\"nG" }{TEXT -1 7 " = " } {XPPEDIT 18 0 "a_1;" "6#%$a_1G" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "a_2 ;" "6#%$a_2G" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "a_3;" "6#%$a_3G" } {TEXT -1 10 " + ... + " }{XPPEDIT 18 0 "a_n;" "6#%$a_nG" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 27 "and then compute the limit " } {XPPEDIT 18 0 "s = limit(s_n,n = infinity);" "6#/%\"sG-%&limitG6$%$s_n G/%\"nG%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 174 "This procedure can be very hard to do d irectly by hand---a large part of Chapter 10 is devoted to showing you ways of doing it indirectly---but it is sometimes feasible with " } {TEXT 257 5 "Maple" }{TEXT -1 35 ". Let's look at summing the series " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sum((5/9)^n,n = 0 .. infinity);" "6#-%$sumG6$)*&\"\"&\"\"\"\"\"*!\"\"%\"nG/F,;\"\"!%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 34 "We begin by defining the sequ ence " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 42 " of terms of the se ries, and the sequence " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 82 " \+ of partial sums. To save some typing, we will use the function syntax to define " }{XPPEDIT 18 0 "a_n;" "6#%$a_nG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "s_n;" "6#%$s_nG" }{TEXT -1 17 " as functions of " } {XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 49 " . Note that both sequence s actually start with " }{XPPEDIT 18 0 "n = 0;" "6#/%\"nG\"\"!" } {TEXT -1 15 ", rather than " }{XPPEDIT 18 0 "n = 1;" "6#/%\"nG\"\"\" " }{TEXT -1 37 ", since the first term in the sum is " }{XPPEDIT 18 0 "n = 0;" "6#/%\"nG\"\"!" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a := proc(n) (5/9)^n end:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "s := proc(n) sum(a(i), i=0..n) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "We can now easily print out the first few terms; for example," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a(n ) $n=0..10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"\"#\"\"&\"\"*#\"#D \"#\")#\"$D\"\"$H(#\"$D'\"%hl#\"%DJ\"&\\!f#\"&Dc\"\"'T9`#\"&D\"y\"(pHy %#\"'D1R\")@n/V#\"(DJ&>\"**[?uQ#\"(Dcw*\"+,Wy'[$" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 47 "Similarly, here are the first few partial sums:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s(n) $n=0..10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"\"#\"#9\"\"*#\"$^\"\"#\")#\"%%[\"\"$H(# \"&\")R\"\"%hl#\"'a*G\"\"&\\!f#\"(6i<\"\"'T9`#\")CSm5\"(pHy%#\")ToO'* \")@n/V#\"*%pa#p)\"**[?uQ#\"+ry0Ly\"+,Wy'[$" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 258 48 "Notice that these two sequences are dif ferent!!!" }{TEXT -1 121 " The second sequence consists of sums of t he terms in the first sequence; for example, the first three partial s ums are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "a(0); a(0) + a(1 ); a(0) + a(1) + a(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"#9\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$^\"\"#\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 " In case you still need convincing, here are plots of the two sequences . First is a plot of the sequence of terms; next is a plot of the seq uence of partial sums." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "p lot([[n,a(n)] $n=0..50], x=0..50, style=point);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F) 7$F*$\"3!ebbbbbbb&!#=7$$\"\"#F)$\"3e>k3`(>k3$F/7$$\"\"$F)$\"3fmNgSwn9< F/7$$\"\"%F)$\"3*f.UA*o)f_*!#>7$$\"\"&F)$\"3okW8S\\@#H&F?7$$\"\"'F)$\" 3f8e=6%>,%HF?7$$\"\"(F)$\"3Z25)Gn*RL;F?7$$\"\")F)$\"3a4n6FEWu!*!#?7$$ \"\"*F)$\"30Q[1:qNT]FT7$$\"#5F)$\"3TV#es*Qv+GFT7$$\"#6F)$\"3'3pltQufb \"FT7$$\"#7F)$\"39:;._@IW')!#@7$$\"#8F)$\"3f3?Y%3!R-[Fco7$$\"#9F)$\"3W #yn8\\%*zm#Fco7$$\"#:F)$\"3Q!*4(=;>A[\"Fco7$$\"#;F)$\"3xzKG**3bM#)!#A7 $$\"#F)$\"3<.8uI +'>T\"Fhp7$$\"#?F)$\"3/Ss+$RAU%y!#B7$$\"#@F)$\"39m%[%H8!zN%F]r7$$\"#AF )$\"3yO!QTHc5U#F]r7$$\"#BF)$\"3]J6_u7.X8F]r7$$\"#CF)$\"3KT=cpfRsu!#C7$ $\"#DF)$\"3!R-,U4J8:%Fbs7$$\"#EF)$\"38-RL_]H1BFbs7$$\"#FF)$\"3')*Qu!H] F\"G\"Fbs7$$\"#GF)$\"3%=$)ofg%>=r!#D7$$\"#HF)$\"3wt:()eDbaRFgt7$$\"#IF )$\"3/Tv\"Qktp>#Fgt7$$\"#JF)$\"34n>B84a?7Fgt7$$\"#KF)$\"3g2()G^Gy!y'!# E7$$\"#LF)$\"3!>G\\Se,rw$F\\v7$$\"#MF)$\"3/72O8U$G4#F\\v7$$\"#NF)$\"3S %G*3ucoi6F\\v7$$\"#OF)$\"3+\"e^g&[Ofk!#F7$$\"#PF)$\"3[m(R6.O&)e$Faw7$$ \"#QF)$\"3U\")4TG6j$*>Faw7$$\"#RF)$\"3&>@<\"\\Gd26Faw7$$\"#SF)$\"3CAc4 &\\#=`h!#G7$$\"#TF)$\"3)zc(\\>ZV=MFfx7$$\"#UF)$\"3;r3s*RI\"**=Ffx7$$\" #VF)$\"3%GP*GWC2b5Ffx7$$\"#WF)$\"3%[S&QzN^he!#H7$$\"#XF)$\"3A\"*=*HV'R cKF[z7$$\"#YF)$\"3lR*R%H\"4\"4=F[z7$$\"#ZF)$\"37LmYF1105F[z7$$\"#[F)$ \"3M2Cf_,n$e&!#I7$$\"#\\F)$\"3tZCmt*Q?5$F`[l7$$\"#]F)$\"3rE!o`)\\NB " 0 " " {MPLTEXT 1 0 48 "plot([[n,s(n)] $n=0..50], x=0..50, style=point);" } }{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7U7 $$\"\"!F)$\"\"\"F)7$F*$\"3ebbbbbbb:!#<7$$\"\"#F)$\"3g(>k3`(>k=F/7$$\" \"$F)$\"3RaX#\\Hlc.#F/7$$\"\"%F)$\"3hup%Q;D48#F/7$$\"\"&F)$\"3Q>$[KJZQ =#F/7$$\"\"'F)$\"3`x,O2&[K@#F/7$$\"\"(F)$\"3i()*)3/DeHAF/7$$\"\")F)$\" 3%R5;n%plQAF/7$$\"\"*F)$\"3&)o6t.$)pVAF/7$$\"#5F)$\"3SF%Gw0*\\YAF/7$$ \"#6F)$\"3/$z:?.b![AF/7$$\"#7F)$\"3/'*4Bi%>*[AF/7$$\"#8F)$\"3WU%R7q*R \\AF/7$$\"#9F)$\"3)*y&)o+lm\\AF/7$$\"#:F)$\"3+mZgAZ\")\\AF/7$$\"#;F)$ \"3!*eP6oq*)\\AF/7$$\"#F)$\"3q:'*\\]B)*\\AF/7$$\"#?F)$\"3o3?s%>!**\\AF/7$$\"#@F)$\"3 EQLi_X**\\AF/7$$\"#AF)$\"3OK'zO(p**\\AF/7$$\"#BF)$\"3s14r=$)**\\AF/7$$ \"#CF)$\"3e.0&f1***\\AF/7$$\"#DF)$\"3E8O3\"[***\\AF/7$$\"#EF)$\"3)*=Jr 6(***\\AF/7$$\"#FF)$\"3#4iS)R)***\\AF/7$$\"#GF)$\"3[nD-6****\\AF/7$$\" #HF)$\"3+$4o0&****\\AF/7$$\"#IF)$\"3oHy`s****\\AF/7$$\"#JF)$\"3QQKu%)* ***\\AF/7$$\"#KF)$\"3U@S_\"*****\\AF/7$$\"#LF)$\"3;B6H&*****\\AF/7$$\" #MF)$\"31dRQ(*****\\AF/7$$\"#NF)$\"3-Vma)*****\\AF/7$$\"#OF)$\"3mzD>** ****\\AF/7$$\"#PF)$\"3/L9b******\\AF/7$$\"#QF)$\"3C'z](******\\AF/7$$ \"#RF)$\"3U`:')******\\AF/7$$\"#SF)$\"33&3B*******\\AF/7$$\"#TF)$\"3Ip s&*******\\AF/7$$\"#UF)$\"3ygi(*******\\AF/7$$\"#VF)$\"3)=\"o)******* \\AF/7$$\"#WF)$\"3)Hn#********\\AF/7$$\"#XF)$\"3[Hf********\\AF/7$$\"# YF)$\"3qQx********\\AF/7$$\"#ZF)$\"3oV()********\\AF/7$$\"#[F)$\"3!>I* ********\\AF/7$$\"#\\F)$\"3)=h*********\\AF/7$$\"#]F)$\"3i%y********* \\AF/-%'COLOURG6&%$RGBG$FZ!\"\"F(F(-%&STYLEG6#%&POINTG-%+AXESLABELSG6$ Q\"x6\"Q!6\"-%%VIEWG6$;F(F[[l%(DEFAULTG" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The sequences of terms appears to converges to 0, and you can check with " }{TEXT 259 0 "" }{TEXT -1 0 "" }{TEXT 260 5 "limit" }{TEXT -1 24 " that this is correct. " }{TEXT 261 50 "This does not m ean that the series has a sum of 0." }{TEXT -1 57 " The sum of the se ries is the limit of the partial sums:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(s(n), n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"*\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Thi s means we can write " }{XPPEDIT 18 0 "sum((5/9)^n,n = 0 .. infinity) \+ = 9/4;" "6#/-%$sumG6$)*&\"\"&\"\"\"\"\"*!\"\"%\"nG/F-;\"\"!%)infinityG *&F+F*\"\"%F," }{TEXT -1 23 " . Not surprisingly, " }{TEXT 262 5 "Ma ple" }{TEXT -1 45 " could have given us this answer in one step." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sum((5/9)^n, n=0..infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"*\"\"%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Quest ion 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 260 "For each of the followin g series, define the sequence of terms and the sequence of partial sum s, then find the sum of the series by taking the limit of the latter, \+ if it exists. (Some of the series may diverge.) For each example, ch eck your result by asking " }{TEXT 263 5 "Maple" }{TEXT -1 19 " to sum the series." }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "sum( 5/((-2)^n),n = 1 .. infinity);" "6#-%$sumG6$*&\"\"&\"\"\"),$\"\"#!\"\" %\"nGF,/F-;F(%)infinityG" }{TEXT -1 20 " (b) " } {XPPEDIT 18 0 "sum(1/(n*(n+1)),n = 1 .. infinity);" "6#-%$sumG6$*&\"\" \"F'*&%\"nGF',&F)F'F'F'F'!\"\"/F);F'%)infinityG" }{TEXT -1 19 " \+ (c) " }{XPPEDIT 18 0 "sum(1/n,n = 1 .. infinity);" "6#-%$sumG6 $*&\"\"\"F'%\"nG!\"\"/F(;F'%)infinityG" }{TEXT -1 26 " \+ (d) " }{XPPEDIT 18 0 "sum(1/(n^2),n = 1 .. infinity);" "6#-%$sum G6$*&\"\"\"F'*$%\"nG\"\"#!\"\"/F);F'%)infinityG" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 18 "(Hint for (b): if " }{TEXT 264 5 "Maple" }{TEXT -1 76 " has trouble computing the partial sums of this one, use partial fractions.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 265 10 "Solutions." }{TEXT -1 27 " Here is the first series:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a := proc (n) 5/(-2)^n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "s := p roc(n) sum(a(j), j=1..n) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(s(n), n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\" &\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Series (a) converges, a nd its sum is -5/3. We can now do the others just by re-defining " } {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 46 "; since all our series star t the summation at " }{XPPEDIT 18 0 "n = 1;" "6#/%\"nG\"\"\"" }{TEXT -1 21 ", the expression for " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 13 " in terms of " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 29 " will b e the same every time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "a := proc(n) 1/(n*(n+1)) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(s(n), n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Series (b) converges, with \+ sum 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a := proc(n) 1/n \+ end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(s(n), n=infin ity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The answer " }{XPPEDIT 18 0 "infinity;" "6#%)in finityG" }{TEXT -1 32 " shows that series (c) diverges." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a := proc(n) 1/n^2 end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(s(n), n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#PiG\"\"#\"\"\"#F(\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Series (d) converges, with sum " } {XPPEDIT 18 0 "Pi^2;" "6#*$%#PiG\"\"#" }{TEXT -1 34 "/6 . (Strange bu t true: what has " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 70 " have \+ to do with summing reciprocals of squares of positive integers?)" }}}} }}{MARK "1 12 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }