{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {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 Input" 2 19 "" 0 1 255 0 0 1 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 "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warni ng" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 " " 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot " 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{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 "Normal" -1 257 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 "Maple Output" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 25 "MAPLE Worksheet Number 10" }} {PARA 256 "" 0 "" {TEXT -1 20 "Sequences and Series" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}{PARA 0 "" 0 " " {TEXT -1 135 "Students often confuse the terms \"sequence\" and \"se ries.\" \"Sequence\" refers to an ordered, usually infinitely long, l ist of numbers " }}{PARA 256 "" 0 "" {TEXT -1 3 " \{ " }{XPPEDIT 18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[2]" "6 #&%\"aG6#\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[3]" "6#&%\"aG6#\" \"$" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[4]" "6#&%\"aG6#\"\"%" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[5]" "6#&%\"aG6#\"\"&" }{TEXT -1 10 " ,... .., " }{XPPEDIT 18 0 "a[i]" "6#&%\"aG6#%\"iG" }{TEXT -1 19 " , . . . \+ . . . \} ." }}{PARA 0 "" 0 "" {TEXT -1 85 "As we saw before, the MAPL E command for generating a finite portion of a sequence is" }}{PARA 256 "" 0 "" {TEXT -1 91 "seq(expression in terms of some index, index= starting integer value..ending integer value);" }}{PARA 0 "" 0 "" {TEXT -1 96 "Use this command to generate the first 20 numbers in the \+ following sequences beginning with i=1." }}{PARA 257 "" 0 "" {TEXT -1 3 "a. " }{XPPEDIT 18 0 "\{1/i\}" "6#<#*&\"\"\"F%%\"iG!\"\"" }{TEXT -1 9 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(1/i, i=1 ..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "66\"\"\"#F#\"\"##F#\"\"$#F#\" \"%#F#\"\"&#F#\"\"'#F#\"\"(#F#\"\")#F#\"\"*#F#\"#5#F#\"#6#F#\"#7#F#\"# 8#F#\"#9#F#\"#:#F#\"#;#F#\"#<#F#\"#=#F#\"#>#F#\"#?" }}}{PARA 257 "" 0 "" {TEXT -1 4 "b. " }{XPPEDIT 18 0 "\{(1/i)^2\}" "6#<#*$*&\"\"\"F&%\" iG!\"\"\"\"#" }{TEXT -1 5 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "seq((1/i)^2, i=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "66\"\" \"#F#\"\"%#F#\"\"*#F#\"#;#F#\"#D#F#\"#O#F#\"#\\#F#\"#k#F#\"#\")#F#\"$+ \"#F#\"$@\"#F#\"$W\"#F#\"$p\"#F#\"$'>#F#\"$D##F#\"$c##F#\"$*G#F#\"$C$# F#\"$h$#F#\"$+%" }}}{PARA 257 "" 0 "" {TEXT -1 4 "c. " }{XPPEDIT 18 0 "\{(1/2)^i\}" "6#<#)*&\"\"\"F&\"\"#!\"\"%\"iG" }{TEXT -1 7 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "seq((1/2)^i, i=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "66#\"\"\"\"\"##F$\"\"%#F$\"\")#F$\"#;# F$\"#K#F$\"#k#F$\"$G\"#F$\"$c##F$\"$7&#F$\"%C5#F$\"%[?#F$\"%'4%#F$\"%# >)#F$\"&%Q;#F$\"&oF$#F$\"&Ob'#F$\"'s58#F$\"'W@E#F$\"')GC&#F$\"(w&[5" } }}{PARA 0 "" 0 "" {TEXT -1 67 "\"Series\" refers to the sum of a seque nce and the notation used is " }{XPPEDIT 18 0 "sum(a[i],i=i[0]..infin ity)" "6#-%$sumG6$&%\"aG6#%\"iG/F);&F)6#\"\"!%)infinityG" }{TEXT -1 205 ". Of course we can't add infinitely many numbers together so we \+ do what we always do in Caluclus, we add more and more finitely many o f the numbers together and hope for a limit. To this end we call \+ " }{XPPEDIT 18 0 "S[n]=sum(a[i],i=i[0]..n)" "6#/&%\"SG6#%\"nG-%$sumG6$ &%\"aG6#%\"iG/F.;&F.6#\"\"!F'" }{TEXT -1 34 " the n-th partial sum a nd define" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "sum(a[ i],i=i[0]..infinity)" "6#-%$sumG6$&%\"aG6#%\"iG/F);&F)6#\"\"!%)infinit yG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "limit(S[n],n=infinity)" "6#-%&li mitG6$&%\"SG6#%\"nG/F)%)infinityG" }{TEXT -1 3 " ." }}{PARA 0 "" 0 " " {TEXT -1 49 "The MAPLE command for generating a partial sum is" }} {PARA 256 "" 0 "" {TEXT -1 92 "sum(expression in terms of some index, \+ index=beginning integer value..ending integer value);" }}{PARA 0 "" 0 "" {TEXT -1 225 "Use this command to generate the 20-th partial sums f or a, b, and c above. Using cap S will display the notation. For exam ple to display the notation and compute the sum of the first 20 terms \+ of (a) use the command sequence " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Sum(1/i,i=1..20)=sum(1/i,i=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(%\"iG!\"\"/F);F(\"#?#\")N^$e&\")/&> b\"" }}}{PARA 0 "" 0 "" {TEXT -1 25 "Now do the same thing for" }} {PARA 0 "" 0 "" {TEXT -1 4 " b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Sum((1/i)^2, i=1..20)=sum((1/i)^2, i=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*$)%\"iG\"\"#F(!\"\"/F+;F(\"#?# \"2TEaJd(**H<\"2?2F)>v%Q3\"" }}}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 0 "" 0 "" {TEXT -1 2 "c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Sum((1/2)^i, i=1..20) = sum((1/2)^i, i=1..20) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$)#\"\"\"\"\"#%\"iG/F+;F)\"#?#\"(v&[5\"(w &[5" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The series " }{XPPEDIT 18 0 "su m(1/i,i=1..infinity)" "6#-%$sumG6$*&\"\"\"F'%\"iG!\"\"/F(;F'%)infinity G" }{TEXT -1 50 " is called the \"harmonic\" series. The series \+ " }{XPPEDIT 18 0 "sum(r^i, i=0..infinity)" "6#-%$sumG6$)%\"rG%\"iG/F(; \"\"!%)infinityG" }{TEXT -1 65 " is called the \"geometric series wit h ratio r\" and the series " }{XPPEDIT 18 0 "sum((1/i^p),i=1..infini ty)" "6#-%$sumG6$*&\"\"\"F')%\"iG%\"pG!\"\"/F);F'%)infinityG" }{TEXT -1 126 " is called a \"p-series.\" The series in (b) is a p-series \+ with p=2 and the series in (c) is a gerometric series with ratio " } {XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 4 " . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " A series " }{XPPEDIT 18 0 "sum(a[i],i=i[0]..infinity)" "6#-%$sumG6$&%\"aG6#%\" iG/F);&F)6#\"\"!%)infinityG" }{TEXT -1 44 " is called \"convergent w ith limit S\" if " }{XPPEDIT 18 0 "limit(S[n],n=infinity) =S" "6#/-% &limitG6$&%\"SG6#%\"nG/F*%)infinityGF(" }{TEXT -1 27 " and we use the notation " }{XPPEDIT 18 0 "sum(a[i],i=i[0]..infinity)=S" "6#/-%$sumG 6$&%\"aG6#%\"iG/F*;&F*6#\"\"!%)infinityG%\"SG" }{TEXT -1 4 " , " }} {PARA 0 "" 0 "" {TEXT -1 217 "otherwise it is said to \"diverge.\" Fo r each of the harmonic, p-series, and geometric series above, determin e if the series diverges or converges by computing first the n-th part ial sum then taking its' limit as n -> " }{XPPEDIT 18 0 "infinity" "6# %)infinityG" }{TEXT -1 47 " by performing the following command sequen ces:" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "Sa[n]:=sum(1/i,i=1..n) ;" "6#>&%#SaG6#%\"nG-%$sumG6$*&\"\"\"F,%\"iG!\"\"/F-;F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#SaG6#%\"nG,&-%$PsiG6#,&F'\"\"\"F-F-F-%&gamm aGF-" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "limit(Sa[n],n=infinit y);" "6#-%&limitG6$&%#SaG6#%\"nG/F)%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 39 "Does it converge or diverge? Diverges." }}{EXCHG {PARA 0 "> \+ " 0 "" {XPPEDIT 19 1 "Sb[n]:=sum(1/i^2,i=1..n);" "6#>&%#SbG6#%\"nG-%$s umG6$*&\"\"\"F,*$%\"iG\"\"#!\"\"/F.;F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#SbG6#%\"nG,&-%$PsiG6$\"\"\",&F'F,F,F,!\"\"*&#F,\"\"'F,)%#PiG \"\"#F,F," }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "limit(Sb[n],n=in finity);" "6#-%&limitG6$&%#SbG6#%\"nG/F)%)infinityG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*$)%#PiG\"\"#\"\"\"#F(\"\"'" }}}{PARA 0 "" 0 "" {TEXT 257 28 "Does it converge or diverge?" }{TEXT -1 15 " Converges \+ to " }{XPPEDIT 18 0 "Pi^2/6;" "6#*&%#PiG\"\"#\"\"'!\"\"" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "Sc[n]:=sum((1/2)^i,i=0..n);" "6#>&%#S cG6#%\"nG-%$sumG6$)*&\"\"\"F-\"\"#!\"\"%\"iG/F0;\"\"!F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%#ScG6#%\"nG,&)#\"\"\"\"\"#,&F'F+F+F+!\"#F,F+ " }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "limit(Sc[n],n=infinity); " "6#-%&limitG6$&%#ScG6#%\"nG/F)%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT 258 28 "Does it converg e or diverge?" }{TEXT -1 16 " Converges to 2" }}{PARA 0 "" 0 "" {TEXT -1 12 "Now define " }{XPPEDIT 18 0 "Sp[n]" "6#&%#SpG6#%\"nG" } {TEXT -1 69 " to be the n-th partial sum for a general p-series. In other words" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "Sp[n]:=sum(1/i ^p,i=1..n);" "6#>&%#SpG6#%\"nG-%$sumG6$*&\"\"\"F,)%\"iG%\"pG!\"\"/F.;F ,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#SpG6#%\"nG-%$sumG6$*&\"\"\" F,)%\"iG%\"pG!\"\"/F.;F,F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 221 "Experiment by substituting various values of p into this expression and computing the limit, to determine for what v alues of p does the p-series converge, and for what values does it div erge. For example to compute with " }{XPPEDIT 18 0 "p=1/3" "6#/%\"pG* &\"\"\"F&\"\"$!\"\"" }{TEXT -1 40 " perform the following command sequ ence:" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(p=1/3,Sp[n]);" " 6#-%%subsG6$/%\"pG*&\"\"\"F)\"\"$!\"\"&%#SpG6#%\"nG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$sumG6$*&\"\"\"F'*$)%\"iG#F'\"\"$F'!\"\"/F*;F'%\"nG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(%,n=infinity);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 33 "What is your conclusion? Diverges" }} {PARA 0 "" 0 "" {TEXT -1 59 "Show sufficiently many examples to suppor t your conclusion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(p =9/10, Sp[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sumG6$*&\"\"\"F'* $)%\"iG#\"\"*\"#5F'!\"\"/F*;F'%\"nG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(%, n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "subs(p=1, Sp[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sumG6$*&\"\"\"F'%\"iG!\"\"/F(;F'%\"nG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "We know that diverges." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(p=11/10, Sp[n]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$sumG6$*&\"\"\"F'*$)%\"iG#\"#6\"#5F'!\"\"/F*;F'%\"n G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit(%, n=infinity); \+ evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%ZetaG6##\"#6\"#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Y[We5!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Conjecture: The p-series converges for p>1" }}}{PARA 0 "" 0 "" {TEXT -1 170 "Similarily experiment with different values of r in the geometric series to determine for what values of r the geome tric series converges and for what values it diverges." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Gr[n] := sum(r^i, i=1..n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#GrG6#%\"nG,&*&)%\"rG,&F'\"\"\"F-F- F-,&F+F-F-!\"\"F/F-*&F+F-F.F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(r=9/10, Gr[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&)#\" \"*\"#5,&%\"nG\"\"\"F*F*!#5F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(%, n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(r=11/10, Gr[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&)#\"#6\"#5,&%\"nG\"\"\"F*F*F'F&!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(%, n=infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " subs(r=1, Gr[n]);" }}{PARA 8 "" 1 "" {TEXT -1 43 "Error, numeric excep tion: division by zero\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sum(1^i, i=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"nG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(%, n=infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(r=-1/3, Gr[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&)#!\"\"\"\"$,&%\"nG\"\"\"F*F*#!\"$\"\"%#F*F-F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(%, n=infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6##!\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(r=-11/10, Gr[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&)#!#6\"#5,&%\"nG\"\"\"F*F*#!#5\"#@#\"#6F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(%, n=infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%*undefinedG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 260 82 "What is your conclusion? The series converges for abs (r)<1, diverges otherwise." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Let's try to use MAPLE to \"prove\" our conclusion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Int(1/x^p,x)=int(1/x^p,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F()%\"xG%\"pG!\" \"F*,$*(,&F+F(F(F,F,F*F(-%$expG6#*&F+F(-%#lnG6#F*F(F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(1/x^p,x=1..n)=int(1/x^p,x=1..n) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F()%\"xG%\"pG! \"\"/F*;F(%\"nG*(,&F/F,-%$expG6#*&F+F(-%#lnG6#F/F(F(F(,&F+F(F(F,F,F2F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$)%\"xG,$%\"pG!\"\"/F(;\"\"\"%\"nG, $*&,&)F/,&F*F+F.F.F.F.F+F.,&F*F.F.F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int(1/x^p,x=1..infinity)=limit(rhs(%),n=infinity);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F()%\"xG%\"pG!\"\"/ F*;F(%)infinityG-%&limitG6$,$*&,&)%\"nG,&F+F,F(F(F(F(F,F(,&F+F(F(F,F,F ,/F7F/" }}}{PARA 258 "" 1 "" {TEXT -1 92 "Notice that this expression \+ goes to infinity if the \"effect\" of n is in the numerator, i.e. " } {XPPEDIT 18 0 "1-p>0" "6#2\"\"!,&\"\"\"F&%\"pG!\"\"" }{TEXT -1 14 ", a nd goes to " }{XPPEDIT 18 0 "1/(p-1)" "6#*&\"\"\"F$,&%\"pGF$F$!\"\"F' " }{TEXT -1 53 " when the effect of n is in the denominator, i.e. \+ " }{XPPEDIT 18 0 "1-p<0" "6#2,&\"\"\"F%%\"pG!\"\"\"\"!" }{TEXT -1 78 " . What does this have to do with a p series? Well if we graph the f unction " }{XPPEDIT 18 0 "x^(-p)" "6#)%\"xG,$%\"pG!\"\"" }{TEXT -1 523 " from x=0 to x=n we observe that the n rightboxes are always unde r the graph while for x=1..n+1 the n left boxes are always above the g raph. But the sum of the n rightboxes from 0 to n is the n-th partial \+ sum of the p-series and the sum of the n leftboxes from 1 to n+1 are t he n-th partial sum of the p-series. Thus the limit of the integral is infinite or finite if and only if the p-series is correspondingly inf inite and finite. To illustrate this observation do the appropriate l eft and right box plots and sums for " }{XPPEDIT 18 0 "x^(1/2)" "6#)% \"xG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 109 " using 10 boxes and observe t he relationship between the area under the graph and that included in \+ the boxes." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "rightbox(1/x^( 1/2),x=0..10,10, view=[0..10,0..1.5]); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6/-%'CURVESG6&7jo7$$\"3S+++v1h6o!#? $\"3FBc6aTk67!#;7$$\"33+++N@Ki8!#>$\"3?b>gxzhn&)!#<7$$\"3<+++-K[V?F1$ \"3SPFE'yIa*pF47$$\"3;+++qUkCFF1$\"3O;\"y0x?#egF47$$\"32+++P`!eS$F1$\" 3V!G$o'QP'=aF47$$\"3s*****\\Smp3%F1$\"3(eO#oXl^Y\\F47$$\"3)******>ZF\" oZF1$\"35R.4UWezXF47$$\"3K+++S&)G\\aF1$\"3gx4!)))*3QG%F47$$\"3'******z g\\/8'F1$\"3mut)o%Q\")QSF47$$\"3A+++v1h6oF1$\"3a[MjS_bJQF47$$\"3))**** *HurF\\(F1$\"3B%**\\/eWKl$F47$$\"3W******4G$R<)F1$\"35eMq#R:x\\$F47$$ \"3R+++xQ4b))F1$\"3i9.J]i\\gLF47$$\"3/+++X\\DO&*F1$\"3=/([Q@b#QKF47$$ 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"Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" }}}}{PARA 0 " " 0 "" {TEXT -1 48 "Notice it is obvious from this picture that if " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "int(1/x^(1/2),x=1..infinity)" "6#-%$i ntG6$*&\"\"\"F')%\"xG*&F'F'\"\"#!\"\"F,/F);F'%)infinityG" }{TEXT -1 16 " is finite then " }{XPPEDIT 18 0 "sum(1/i^(1/2),i=1..infinity)" "6 #-%$sumG6$*&\"\"\"F')%\"iG*&F'F'\"\"#!\"\"F,/F);F'%)infinityG" }{TEXT -1 16 " is also finite." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "I nt(1/(x^(1/2)),x = 1 .. infinity)=Limit(Int(1/x^(1/2),x=1..n),n=infini ty);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$-%%sqrtG 6#%\"xGF(!\"\"/F-;F(%)infinityG-%&LimitG6$-F%6$F'/F-;F(%\"nG/F9F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(1/x^(1/2),x=1..n)=int(1/ x^(1/2),x=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\" F(*$-%%sqrtG6#%\"xGF(!\"\"/F-;F(%\"nG,&*$-F+6#F1F(\"\"#F6F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Limit(Int(1/x^(1/2),x=1..n), n=infinity)=limit(int(1/x^(1/2),x=1..n),n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%$IntG6$*&\"\"\"F+*$-%%sqrtG6#%\"xGF+! \"\"/F0;F+%\"nG/F4%)infinityGF6" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Thus \+ we have no information about whether or not " }{XPPEDIT 18 0 "sum(1/(i ^(1/2)),i = 1 .. infinity)" "6#-%$sumG6$*&\"\"\"F')%\"iG*&F'F'\"\"#!\" \"F,/F);F'%)infinityG" }{TEXT -1 55 " is finite or infinite. Contin ue with the following:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "le ftbox(1/x^(1/2),x=1..11,10);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6/-%)POLYGONSG6$7&7$$\"\"\"\"\"!$F*F*7$F(F(7$$\"\"#F* F(7$F.F+-%&COLORG6&%$RGBG$\"\"(!\"\"$\"\"*F7F5-F$6$7&F07$F.$\"+5y1rq!# 57$$\"\"$F*F>7$FBF+F1-F$6$7&FD7$FB$\"+$p-Nx&F@7$$\"\"%F*FI7$FLF+F1-F$6 $7&FN7$FL$\"+++++]F@7$$\"\"&F*FS7$FVF+F1-F$6$7&FX7$FV$\"+cf8sWF@7$$\" \"'F*Fgn7$FjnF+F1-F$6$7&F\\o7$Fjn$\"+1H[#3%F@7$$F6F*Fao7$FdoF+F1-F$6$7 &Feo7$Fdo$\"+JZkzPF@7$$\"\")F*Fjo7$F]pF+F1-F$6$7&F_p7$F]p$\"+1R`NNF@7$ $F9F*Fdp7$FgpF+F1-F$6$7&Fhp7$Fgp$\"+LLLLLF@7$$\"#5F*F]q7$F`qF+F1-F$6$7 &Fbq7$F`q$\"+gwFiJF@7$$\"#6F*Fgq7$FjqF+F1-%'CURVESG6&7UF,7$$\"3FLL$3x& )*36!#<$\"3SF^]a#>f\\*!#=7$$\"3ammmT:(z@\"Fdr$\"3/)\\@)f06h!*Fgr7$$\"3 3+]7y%*z78Fdr$\"3)4)znZ*>xs)Fgr7$$\"3SLLe9ui29Fdr$\"3!f&4!ea8'G%)Fgr7$ $\"3ymm;z_\"4i\"Fdr$\"3i!fCTZBX&yFgr7$$\"3#pmmT&phN=Fdr$\"33s&3m-%*3Q( Fgr7$$\"3JLLe*=)H\\?Fdr$\"33K*Q2z)\\&)pFgr7$$\"3rmm\"z/3uC#Fdr$\"3ZLW] ;*40n'Fgr7$$\"3n***\\7LRDX#Fdr$\"3@Hd*eOdaQ'Fgr7$$\"3$om;zR'okEFdr$\"3 Kf%\\`O**f7'Fgr7$$\"3I++D1J:wGFdr$\"3q6VBJyz_V\\ %Fgr7$$\"3l++D\"=lj;&Fdr$\"3-=1e8=a*R%Fgr7$$\"3R++vV&RYJ%Fgr7$$\"3BML$e9Ege&Fdr$\"3QW8k5N0JUFgr7$$\"3]LLeR\"3Gy&Fdr$\"3huLo 8;WeTFgr7$$\"3emm;/T1&*fFdr$\"3B=JFnJ;%3%Fgr7$$\"3=nm\"zRQb@'Fdr$\"3l? Z9FN26SFgr7$$\"3:++v=>Y2kFdr$\"3PF:1FZa]RFgr7$$\"3Znm;zXu9mFdr$\"3u(*e %H-a\"))QFgr7$$\"34+++]y))GoFdr$\"3aK!o(G_qEQFgr7$$\"3H++]i_QQqFdr$\"3 kV%R'4TKpPFgr7$$\"3a++D\"y%3TsFdr$\"3T$p@+M#>;PFgr7$$\"3+++]P![hY(Fdr$ \"3(4N;wP_(fOFgr7$$\"3iKLL$Qx$owFdr$\"3O?.+AF<6OFgr7$$\"3Y+++v.I%)yFdr $\"3y&p\\1&3QhNFgr7$$\"3?mm\"zpe*z!)Fdr$\"3&*3j.:n*z^$Fgr7$$\"3;,++D\\ 'QH)Fdr$\"3@Y0&[AMBZ$Fgr7$$\"31KLe9S8&\\)Fdr$\"3XI`f+R&4V$Fgr7$$\"3h,+ D1#=bq)Fdr$\"3+@K\")HGC*Q$Fgr7$$\"3!QLL$3s?6*)Fdr$\"3#z(o0@\"**)\\LFgr 7$$\"3a***\\7`Wl7*Fdr$\"3\"3k.FUV,J$Fgr7$$\"3enmmm*RRL*Fdr$\"3I^([L)>; tKFgr7$$\"3$zmmTvJga*Fdr$\"35[\"p'=lfOKFgr7$$\"3]MLe9tOc(*Fdr$\"3h-lWe y^,KFgr7$$\"31,++]Qk\\**Fdr$\"3,#*3_j)p-<$Fgr7$$\"3[LL3dg6<5!#;$\"3]&H 1FKdb8$Fgr7$$\"3ymmmw(Gp.\"Fg`l$\"3?raq;rX0JFgr7$$\"3-+]7oK0e5Fg`l$\"3 p?/9=**HuIFgr7$$\"36+](=5s#y5Fg`l$\"3;,W&4?S`/$Fgr7$Fjq$\"3HOwxXM6:IFg r-%'COLOURG6&F4$\"*++++\"!\")F+F+-%*THICKNESSG6#F/-%&STYLEG6#%%LINEG-% +AXESLABELSG6$Q\"x6\"Q!F]cl-%%VIEWG6$;F(Fjq%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 1 0" "Curve 11" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "From this picture it fo llows that if " }{XPPEDIT 18 0 "int(1/(x^(1/2)),x = 1 .. infinity)" "6#-%$intG6$*&\"\"\"F')%\"xG*&F'F'\"\"#!\"\"F,/F);F'%)infinityG" } {TEXT -1 20 " is infinite then " }{XPPEDIT 18 0 "sum(1/(i^(1/2)),i = 1 .. infinity)" "6#-%$sumG6$*&\"\"\"F')%\"iG*&F'F'\"\"#!\"\"F,/F);F'% )infinityG" }{TEXT -1 114 " is also infinite. Thus the sum is infinite from above. Of course MAPLE can arrive at either conclusion directly :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Int(1/x^(1/2),x=1..infi nity)=int(1/x^(1/2),x=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$IntG6$*&\"\"\"F(*$-%%sqrtG6#%\"xGF(!\"\"/F-;F(%)infinityGF1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Sum(1/i^(1/2),i=1..infinity) =sum(1/i^(1/2),i=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ SumG6$*&\"\"\"F(*$-%%sqrtG6#%\"iGF(!\"\"/F-;F(%)infinityGF1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "90 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }