{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 266 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "Red Emphasis" -1 267 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 268 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 257 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 257 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 257 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 257 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 257 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 257 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 257 282 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 267 284 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 267 285 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey \+ Emphasis" -1 286 "Times" 1 12 96 52 84 1 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 "Heading 1" -1 3 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "M aple 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 Output" -1 12 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 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 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 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "The Euler-Maclaurin summation for mula" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Ca nada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "load procedures for series" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 286 8 "series.m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 5 "EMsum" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can \+ be read into a Maple session by a command similar to the one that foll ows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read \"K:\\\\Maple/procdrs/series.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "The Eule r-Maclaurin summation formula for infinite series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "a[n] = g(n);" "6#/&%\"aG6#%\"nG-%\"gG6#F'" }{TEXT -1 21 ", where the function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" } {TEXT -1 16 " is defined for " }{XPPEDIT 18 0 "x>0" "6#2\"\"!%\"xG" } {TEXT -1 109 ", then we can approximate the tail of an infinite series by an integral together with other correction terms." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity) = Sum( a[i],i = 1 .. n-1)" "6#/-%$SumG6$&%\"aG6#%\"nG/F*;\"\"\"%)infinityG-F% 6$&F(6#%\"iG/F3;F-,&F*F-F-!\"\"" }{XPPEDIT 18 0 "``+a[n]/2+ Int(g(x),x = n .. infinity)-``" "6#,*%!G\"\"\"*&&%\"aG6#%\"nGF%\"\"#!\"\"F%-%$In tG6$-%\"gG6#%\"xG/F3;F*%)infinityGF%F$F," }{XPPEDIT 18 0 "Sum(``(beta[ 2*k]/(2*k)!)*`@@`(g,2*k-1)*``(n),k = 1 .. infinity);" "6#-%$SumG6$*(-% !G6#*&&%%betaG6#*&\"\"#\"\"\"%\"kGF0F0-%*factorialG6#*&F/F0F1F0!\"\"F0 -%#@@G6$%\"gG,&*&F/F0F1F0F0F0F6F0-F(6#%\"nGF0/F1;F0%)infinityG" } {TEXT -1 13 " ------- (i)," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 284 38 "______________________________________" }{TEXT -1 15 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " where " }{XPPEDIT 18 0 " `@@`(g,2*k- 1)*``(x)" "6#*&-%#@@G6$%\"gG,&*&\"\"#\"\"\"%\"kGF+F+F+!\"\"F+-%!G6#%\" xGF+" }{TEXT -1 7 " is the" }{XPPEDIT 18 0 "``(2*k-1);" "6#-%!G6#,&*& \"\"#\"\"\"%\"kGF)F)F)!\"\"" }{TEXT -1 18 " st derivative of " } {XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta[2*k];" "6#&%%betaG6#*&\"\"#\"\"\"%\"kGF(" }{TEXT -1 59 " is a Bernouilli number, defined by the generating function" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "t/(exp(t)-1) = Sum(beta[k]*t^k /k!,k = 0 .. infinity);" "6#/*&%\"tG\"\"\",&-%$expG6#F%F&F&!\"\"F+-%$S umG6$*(&%%betaG6#%\"kGF&)F%F3F&-%*factorialG6#F3F+/F3;\"\"!%)infinityG " }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT 285 11 "___________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "series(t/(exp(t)-1),t,22):\nconvert(%,polynom); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,:\"\"\"F$*&#F$\"\"#F$%\"tGF$!\"\" *&#F$\"#7F$*$)F(F'F$F$F$*&#F$\"$?(F$*$)F(\"\"%F$F$F)*&#F$\"&S-$F$*$)F( \"\"'F$F$F$*&#F$\"(+'47F$*$)F(\"\")F$F$F)*&#F$\")g,!z%F$*$)F(\"#5F$F$F $*&#\"$\"p\".+!oVn28F$*$)F(F,F$F$F)*&#F$\",+'\\UsuF$*$)F(\"#9F$F$F$*&# \"% " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Note that the Bernoulli numbers with odd index gr eater than 1 are all 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 9 "bernoulli" }{TEXT -1 33 " generates the Bernoulli numbers." }}{PARA 0 "" 0 "" {TEXT -1 61 "The following code constructs the same polynomial as before. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Sum(bernoulli(k)*t^k/k!,k=0..20);\n``=value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$SumG6$*(-%*bernoulliG6#%\"kG\"\"\")%\"tGF*F+-%*fac torialGF)!\"\"/F*;\"\"!\"#?" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%!G,: \"\"\"F&*&#F&\"\"#F&%\"tGF&!\"\"*&#F&\"#7F&*$)F*F)F&F&F&*&#F&\"$?(F&*$ )F*\"\"%F&F&F+*&#F&\"&S-$F&*$)F*\"\"'F&F&F&*&#F&\"(+'47F&*$)F*\"\")F&F &F+*&#F&\")g,!z%F&*$)F*\"#5F&F&F&*&#\"$\"p\".+!oVn28F&*$)F*F.F&F&F+*&# F&\",+'\\UsuF&*$)F*\"#9F&F&F&*&#\"% " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 90 "The series on the right of (i) \+ is an asymptotic series, and as a result may not converge. " }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 266 17 "improper integral" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = n .. infinity)" "6#-%$IntG6$-%\" fG6#%\"xG/F);%\"nG%)infinityG" }{TEXT -1 16 " is defined by " } {XPPEDIT 18 0 "Limit(Int(f(x),x = n .. m),m=infinity)" "6#-%&LimitG6$- %$IntG6$-%\"fG6#%\"xG/F,;%\"nG%\"mG/F0%)infinityG" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Computing " }{XPPEDIT 18 0 "Sum(1/n^2,n=1..infinity)" " 6#-%$SumG6$*&\"\"\"F'*$%\"nG\"\"#!\"\"/F);F'%)infinityG" }{TEXT -1 44 " using the Euler-Maclaurin summation formula" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "In the case of the series " }{XPPEDIT 18 0 "Sum(1/(n^2),n = 1 .. infinity)" "6#-%$Su mG6$*&\"\"\"F'*$%\"nG\"\"#!\"\"/F);F'%)infinityG" }{TEXT -1 7 ", let \+ " }{XPPEDIT 18 0 "g(x) = 1/(x^2);" "6#/-%\"gG6#%\"xG*&\"\"\"F)*$F'\"\" #!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/x^2,x=n..m) = -1/x" " 6#/-%$IntG6$*&\"\"\"F(*$%\"xG\"\"#!\"\"/F*;%\"nG%\"mG,$*&F(F(F*F,F," } {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([m, ``],[``, ``],[n, ``]) = \+ 1/n-1/m;" "6#/-%*PIECEWISEG6%7$%\"mG%!G7$F)F)7$%\"nGF),&*&\"\"\"F/F,! \"\"F/*&F/F/F(F0F0" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "so \+ " }{XPPEDIT 18 0 "Int(g(x),x = n .. infinity) = Limit(Int(1/(x^2),x = \+ n .. m),m = infinity);" "6#/-%$IntG6$-%\"gG6#%\"xG/F*;%\"nG%)infinityG -%&LimitG6$-F%6$*&\"\"\"F5*$F*\"\"#!\"\"/F*;F-%\"mG/F;F." }{XPPEDIT 18 0 "`` = Limit(``(1/n-1/m),m = infinity);" "6#/%!G-%&LimitG6$-F$6#,& *&\"\"\"F,%\"nG!\"\"F,*&F,F,%\"mGF.F./F0%)infinityG" }{XPPEDIT 18 0 "` `=1/n" "6#/%!G*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "g* `'`(x)=-2/x^3, `@@`(g,2)*``(x)=3!/x^4, `@@`(g,3)*``(x)=-4!/x^5,` . . . `" "6&/*&%\"gG\"\"\"-%\"'G6#%\"xGF&,$*&\"\"#F&*$F*\"\"$!\"\"F0/*&-%#@ @G6$F%F-F&-%!G6#F*F&*&-%*factorialG6#F/F&*$F*\"\"%F0/*&-F46$F%F/F&-F76 #F*F&,$*&-F;6#F>F&*$F*\"\"&F0F0%(~.~.~.~G" }{TEXT -1 18 ",and, in gene ral, " }{XPPEDIT 18 0 "`@@`(g,2*k-1)*``(x)=-(2*k)!/(x^(2*k+1))" "6#/*& -%#@@G6$%\"gG,&*&\"\"#\"\"\"%\"kGF,F,F,!\"\"F,-%!G6#%\"xGF,,$*&-%*fact orialG6#*&F+F,F-F,F,)F2,&*&F+F,F-F,F,F,F,F.F." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(``(beta[2*k]/(2*k)!)*`@@`(g,2*k-1)*``(n),k = 1 . . infinity) = -Sum(beta[2*k]/(n^(2*k+1)),k = 1 .. infinity);" "6#/-%$S umG6$*(-%!G6#*&&%%betaG6#*&\"\"#\"\"\"%\"kGF1F1-%*factorialG6#*&F0F1F2 F1!\"\"F1-%#@@G6$%\"gG,&*&F0F1F2F1F1F1F7F1-F)6#%\"nGF1/F2;F1%)infinity G,$-F%6$*&&F-6#*&F0F1F2F1F1)F@,&*&F0F1F2F1F1F1F1F7/F2;F1FCF7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The Euler-Maclaurin summation formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity) = Sum(a[i],i = 1 \+ .. n-1)" "6#/-%$SumG6$&%\"aG6#%\"nG/F*;\"\"\"%)infinityG-F%6$&F(6#%\"i G/F3;F-,&F*F-F-!\"\"" }{XPPEDIT 18 0 "``+a[n]/2+ Int(g(x),x = n .. inf inity)-``" "6#,*%!G\"\"\"*&&%\"aG6#%\"nGF%\"\"#!\"\"F%-%$IntG6$-%\"gG6 #%\"xG/F3;F*%)infinityGF%F$F," }{XPPEDIT 18 0 "Sum(``(beta[2*k]/(2*k)! )*`@@`(g,2*k-1)*``(n),k = 1 .. infinity);" "6#-%$SumG6$*(-%!G6#*&&%%be taG6#*&\"\"#\"\"\"%\"kGF0F0-%*factorialG6#*&F/F0F1F0!\"\"F0-%#@@G6$%\" gG,&*&F/F0F1F0F0F0F6F0-F(6#%\"nGF0/F1;F0%)infinityG" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 8 "becomes:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n^2),n = 1 .. infinity) = Sum(1/(i^2),i = 1 .. n-1)+1/(2*n^2)+1/n+Sum(beta[2*k]/(n^(2*k+1)),k = 1 .. infinity);" "6#/-%$SumG6$*&\"\"\"F(*$%\"nG\"\"#!\"\"/F*;F(%)infinityG,*-F%6$*&F(F( *$%\"iGF+F,/F5;F(,&F*F(F(F,F(*&F(F(*&F+F(*$F*F+F(F,F(*&F(F(F*F,F(-F%6$ *&&%%betaG6#*&F+F(%\"kGF(F()F*,&*&F+F(FDF(F(F(F(F,/FD;F(F/F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "In the following computation we take " }{XPPEDIT 18 0 "n = 20" "6# /%\"nG\"#?" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 6 "Step I" }{TEXT 269 2 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "Compute the partial sum " }{XPPEDIT 18 0 "S[1] = Sum(1/( i^2),i = 1 .. n-1);" "6#/&%\"SG6#\"\"\"-%$SumG6$*&F'F'*$%\"iG\"\"#!\" \"/F-;F',&%\"nGF'F'F/" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "n := 20;\nS1 := evalf(add(1/i^2,i=1..n-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#S1G$\"+WKm$f\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 272 7 "Step II" }{TEXT 271 2 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "Add the 1st correction term " }{XPPEDIT 18 0 "1/(2*n^2)" "6#*&\"\"\"F$*&\"\"#F$*$%\"nGF&F$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "term := evalf(1/(2*n^2));\nS2 := S1 + term;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%termG$\"++++]7!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S2G$\"+WK \"\\f\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 8 "Step III" }{TEXT 273 2 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "Add \+ the 2nd correction term " }{XPPEDIT 18 0 "Int(1/(x^2),x = n .. infini ty) = 1/n;" "6#/-%$IntG6$*&\"\"\"F(*$%\"xG\"\"#!\"\"/F*;%\"nG%)infinit yG*&F(F(F/F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "term := evalf(1/n);\nS3 := S 2 + term;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%termG$\"+++++]!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S3G$\"+WK\"\\k\"!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 7 "Step IV" }{TEXT 275 2 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "Add a partial sum of the asy mptotic series." }}{PARA 0 "" 0 "" {TEXT -1 92 "Trial and error can be used to see how many terms are necessary to obtain 10 digit accuracy. " }}{PARA 0 "" 0 "" {TEXT -1 36 "In fact just 2 terms are sufficient. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "term := evalf(add(bernoulli(2*k)/n^(2*k+1),k=1..2));\nS4 := S3 + term;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%termG$\"+n;H#3#!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S4G$\"+nS$\\k\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(P i^2/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+oS$\\k\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Computing " }{XPPEDIT 18 0 "z eta(3) = Sum(1/(n^3),n = 1 .. infinity);" "6#/-%%zetaG6#\"\"$-%$SumG6$ *&\"\"\"F,*$%\"nGF'!\"\"/F.;F,%)infinityG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 101 " In this example, we let Maple do all the work of finding the integral \+ and derivatives in the formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity) = Sum(a[i],i = 1 .. n-1)" " 6#/-%$SumG6$&%\"aG6#%\"nG/F*;\"\"\"%)infinityG-F%6$&F(6#%\"iG/F3;F-,&F *F-F-!\"\"" }{XPPEDIT 18 0 "``+a[n]/2+ Int(g(x),x = n .. infinity)-`` " "6#,*%!G\"\"\"*&&%\"aG6#%\"nGF%\"\"#!\"\"F%-%$IntG6$-%\"gG6#%\"xG/F3 ;F*%)infinityGF%F$F," }{XPPEDIT 18 0 "Sum(``(beta[2*k]/(2*k)!)*`@@`(g, 2*k-1)*``(n),k = 1 .. infinity);" "6#-%$SumG6$*(-%!G6#*&&%%betaG6#*&\" \"#\"\"\"%\"kGF0F0-%*factorialG6#*&F/F0F1F0!\"\"F0-%#@@G6$%\"gG,&*&F/F 0F1F0F0F0F6F0-F(6#%\"nGF0/F1;F0%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a[n]=1/n^3" "6#/&%\"aG6#% \"nG*&\"\"\"F)*$F'\"\"$!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "g(x ) = 1/(x^3);" "6#/-%\"gG6#%\"xG*&\"\"\"F)*$F'\"\"$!\"\"" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 8 "We take " }{XPPEDIT 18 0 "n = 20" "6# /%\"nG\"#?" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 6 "Step I" }{TEXT 277 2 " " }}{PARA 0 "" 0 "" {TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Int(g(x),x = 20 .. infinity) = \+ Int(1/(x^3),x = 20 .. infinity);" "6#/-%$IntG6$-%\"gG6#%\"xG/F*;\"#?%) infinityG-F%6$*&\"\"\"F2*$F*\"\"$!\"\"/F*;F-F." }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "g := x -> 1/x^3;\nn := 'n':\nInt(g(x),x=20..infinity);\nint_correc t := value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$ %)operatorG%&arrowGF(*&\"\"\"F-*$)9$\"\"$F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$)%\"xG\"\"$F'!\"\"/F*;\"#?%)i nfinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,int_correctG#\"\"\"\"$+ )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 7 "Ste p II" }{TEXT 279 2 " " }}{PARA 0 "" 0 "" {TEXT -1 9 " Compute " } {XPPEDIT 18 0 "Sum(``(beta[2*k]/(2*k)!)*`@@`(g,2*k-1)*``(20),k = 1 .. \+ 5);" "6#-%$SumG6$*(-%!G6#*&&%%betaG6#*&\"\"#\"\"\"%\"kGF0F0-%*factoria lG6#*&F/F0F1F0!\"\"F0-%#@@G6$%\"gG,&*&F/F0F1F0F0F0F6F0-F(6#\"#?F0/F1;F 0\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "k := 'k':\ng := x -> 1/x^3:\nSum(b ernoulli(2*k)*(D@@(2*k-1))(g)(20)/(2*k)!,k=1..5);\nbernoulli_correct : = evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(-%*bernoulli G6#,$*&\"\"#\"\"\"%\"kGF-F-F----%#@@G6$%\"DG,&*&F,F-F.F-F-F-!\"\"6#%\" gG6#\"#?F--%*factorialGF)F7/F.;F-\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2bernoulli_correctG$!+d6?h:!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 8 "Step III" }{TEXT 281 2 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "Add the initial partial sum, the term " } {XPPEDIT 18 0 "g(n)/2;" "6#*&-%\"gG6#%\"nG\"\"\"\"\"#!\"\"" }{TEXT -1 32 ", and the other two corrections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "evalf(add(g(i),i=1..19)+g( 20)/2+int_correct-bernoulli_correct);\nevalf(Zeta(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.p0-7!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+.p0-7!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "An Eu ler-Maclaurin summation procedure: " }{TEXT 0 5 "EMsum" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "EMsum: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 17 "Calling S equence:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " EMsum( " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 2 ", \+ " }{TEXT 261 1 "n" }{TEXT -1 29 "=m..infinity) \n EMvalue(Sum(" } {XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 2 ", " }{TEXT 283 1 " n" }{TEXT -1 14 "=m..infinity))" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT 262 4 " - " }{TEXT -1 51 " an expression in volving a single variable, say n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 23 7 "m - " }{TEXT -1 29 "the initial index of the sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "EMsum" }{TEXT -1 43 " attempts to \+ evaluate the sum of a series " }{XPPEDIT 18 0 "Sum(a[n],n = m .. infin ity);" "6#-%$SumG6$&%\"aG6#%\"nG/F);%\"mG%)infinityG" }{TEXT -1 52 " \+ by applying the Euler-Maclaurin summation formula:" }}{PARA 257 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(a[n],n = m .. infinity) = Sum(a[ i],i = m .. n-1)+a[n]/2+Int(f(x),x = n .. infinity)-Sum(B[2*k]*[f^[2*k -1]*`(`*n*`)`/(2*k)!],k = 1 .. infinity);" "6#/-%$SumG6$&%\"aG6#%\"nG/ F*;%\"mG%)infinityG,*-F%6$&F(6#%\"iG/F4;F-,&F*\"\"\"F8!\"\"F8*&&F(6#F* F8\"\"#F9F8-%$IntG6$-%\"fG6#%\"xG/FD;F*F.F8-F%6$*&&%\"BG6#*&F=F8%\"kGF 8F87#*,)FB7#,&*&F=F8FNF8F8F8F9F8%\"(GF8F*F8%\")GF8-%*factorialG6#*&F=F 8FNF8F9F8/FN;F8F.F9" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 73 "in which the infinite asymptotic series is approximated by a finite sum. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 8 "Opti ons:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 121 "numterms=p\nThe number of terms to be used in the initial partial sum minus 1, that i s, n in the above formula when m = 1." }}{PARA 0 "" 0 "" {TEXT -1 35 " The default is \"numterms=2*Digits\"." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 113 "bernoulli=q\nThe maximum number of t erms to be used in the asymptotic series which involves the Bernoulli \+ numbers." }}{PARA 0 "" 0 "" {TEXT -1 34 "The default is \"bernoulli=Di gits\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "info=true/false\nThe option info=true gives the value of the succ essive approximations as the three corrections are added to the initia l partial sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 16 "How to activate:" } {TEXT 256 1 "\n" }{TEXT -1 155 "To make the procedure active, open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection." }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 16 ": The procedure " }{TEXT 0 5 "EMsum" }{TEXT -1 54 " may be invoked by means of the pair of commands: \n \+ " }{TEXT 0 22 "Sum(??,n=m..infinity);" }{TEXT -1 4 "\n " }{TEXT 0 11 "EMvalue(%);" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "EMsum: impleme tation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4597 "EMvalue := proc(ff)\n local ok,an,rng;\n ok := \+ false;\n if type(ff,'function') and op(0,ff)='Sum' and nops(ff)=2 th en\n an := op(1,ff);\n rng := op(2,ff);\n if type(an,al gebraic) and type(rng,name=integer..infinity) then\n ok := tru e;\n end if;\n end if; \n if ok then\n EMsum(an,rng,arg s[2..nargs])\n else\n error \"the 1st argument must have the fo rm: Sum(a(n),n=m..infinity)\"\n end if;\nend proc: # EMvalue\n\nEMsu m := proc(an::algebraic,rng::name=integer..infinity)\n\nlocal Options, n,start,prntflg,j,term,fn,temp,val,defint,fact,\n sum,saveDigits, h,eps,ntrms,maxb,fx,x,R,valR,termOK;\n \n n := op(1,rng);\n if n ot type(indets(an,name) minus \{n\},set(realcons)) then\n erro r \"the 1st argument, %1, is invalid .. it should be an expression whi ch depends only on the variable %2\",an,n;\n end if;\n start := op (1,op(2,rng));\n \n # Get the options \"numterms\",\"maxbernoulli \" and \"info\".\n # Set the default values to start with.\n ntrms := Digits*2;\n maxb := ntrms;\n prntflg := false;\n if nargs > \+ 2 then\n Options:=[args[3..nargs]];\n if not type(Options,li st(equation)) then\n error \"each optional argument must be an equation\"\n end if;\n if hasoption(Options,'numterms','ntr ms','Options') then\n if not type(ntrms,posint) then\n \+ error \"\\\"numterms\\\" must be a positive integer\"\n en d if;\n end if;\n if hasoption(Options,'maxbernoulli','maxb' ,'Options') then\n if not type(maxb,posint) then\n \+ error \"\\\"maxbernoulli\\\" must be a positive integer\"\n en d if;\n end if;\n if hasoption(Options,'info','prntflg','Opt ions') then\n if prntflg<>true then prntflg := false end if;\n end if;\n if nops(Options)>0 then\n error \"%1 is n ot a valid option for %2\",op(1,Options),procname;\n end if;\n \+ end if;\n\n # Increase precision for the computation\n saveDigits \+ := Digits;\n Digits := Digits + min(max(trunc(Digits/3),5),10);\n \+ eps := Float(1,-(saveDigits+min(trunc(saveDigits/10),3)));\n\n sum : = 0;\n for j from start to ntrms-1 do\n term := traperror(evalf (subs(n=j,an)));\n if term=lasterror or not type(term,numeric) th en\n error \"a non-numeric value occurred\"\n end if;\n \+ sum := sum + term;\n if abs(term) < eps*abs(sum) then\n \+ if prntflg then\n print(`initial sum of `||j||` terms co nverged`);\n end if;\n Digits := saveDigits;\n \+ return evalf(sum);\n end if;\n end do;\n\n if prntflg then\n j := j - 1;\n print(`partial sum of `||j||` terms ... `, sum)\n end if;\n term := traperror(evalf(subs(n=ntrms,an)*0.5));\n if term=lasterror or not type(term,numeric) then\n error \"a n on-numeric value occurred\"\n end if;\n sum := sum + term;\n\n f n := an;\n if prntflg then\n j := j + 1;\n print(`add half of term `||j||` ... `,sum);\n print(``);\n fx := subs(n=x ,fn);\n print(`integral correction ... `,Int(fx,x=ntrms..infini ty));\n end if;\n termOK := false;\n val := int(fn,n=ntrms..infi nity);\n \n if indets(val,'specfunc(anything,int)')=\{\} then\n \+ if prntflg then\n valR := int(fn,n=ntrms..R);\n if \+ indets(valR,'specfunc(anything,int)')=\{\} then\n print(`= \+ `,Limit(valR,R=infinity));\n end if;\n end if;\n te rm := traperror(evalf(val));\n termOK := term<>lasterror and type (term,numeric);\n end if;\n\n if not termOK then \n if hasf un(fn,factorial) then\n fn := convert(fn,GAMMA);\n end if ;\n term := traperror(evalf(Int(fn,n=ntrms..infinity)));\n t ermOK := term<>lasterror and type(term,numeric);\n end if;\n\n if \+ not termOK then\n error \"failed to calculate integral correction \"\n end if;\n \n if prntflg then\n print(`= `,term);\n print(``);\n end if;\n\n sum := sum + term;\n if prntflg t hen print(`add integral correction ... `,sum) end if;\n fn := diff (fn,n);\n fact := 2;\n for j from 1 to maxb do\n temp := trap error(evalf(subs(n=ntrms,fn)));\n if lasterror=term or not type(t emp,numeric) then\n error \"a non-numeric value occurred for a derivative\"\n end if;\n term := evalf(bernoulli(2*j)*temp/ fact);\n sum := sum - term;\n if abs(term) <= eps*abs(sum) t hen break end if;\n fact := fact*(2*j+1)*(2*j+2);\n fn := di ff(fn,n$2);\n end do;\n if j=maxb+1 then\n WARNING(\"reached \+ maximum number of 'Bernoulli' terms\");\n end if;\n if prntflg the n print(`add asymptotic correction ... `,sum) end if;\n Digits := \+ saveDigits;\n evalf(sum);\nend proc: # EMsum" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are gi ven in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 5 "EMsum" }{TEXT -1 40 ": examples - series with positive term s " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 1" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(1/(n^2),n = 1 .. infinity) = Pi^2/6;" "6#/-%$SumG6$ *&\"\"\"F(*$%\"nG\"\"#!\"\"/F*;F(%)infinityG*&%#PiGF+\"\"'F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "n := 'n':\nEMsum(1/n^2,n=1..infinity,info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%@partial~sum~of~19~terms~~...~~~G$\"0 .8RCjOf\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;add~half~of~term~20~ ...~~~G$\"0.8RC8\\f\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%;integral~correction~...~~~G-%$IntG6$ *&\"\"\"F(*$)%\"xG\"\"#F(!\"\"/F+;\"#?%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G-%&LimitG6$,$*(\"#?!\"\",&%\"RG\"\"\"F)F*F-F,F*F- /F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G$\"0+++++++&!# ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?add~integral~correction~...~~~G$\"0.8RC8\\k\"!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%Aadd~asymptotic~correction~...~~~G$\"0A[o1M\\k \"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nS$\\k\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eval f(evalf(Pi^2/6,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nS$\\k\"! \"*" }}}{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 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Zeta(3) = Sum(1/(n^ 3),n = 1 .. infinity);" "6#/-%%ZetaG6#\"\"$-%$SumG6$*&\"\"\"F,*$%\"nGF '!\"\"/F.;F,%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "n := 'n':\nevalf(EMsum( 1/n^3,n=1..infinity),50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S0M#H' )\\w!**\\9^h\"Q(*R&G%ffJ!p0-7!#\\" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(Zeta(3),50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S0M#H')\\w!**\\9^h\"Q(*R&G%ffJ!p0-7!#\\" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Zeta(1.1) = Sum(1/(n^1.1),n = 1 .. infinity);" "6#/-%%ZetaG6#-%&FloatG6$\"#6!\"\"-%$SumG6$*&\"\" \"F0)%\"nG-F(6$F*F+F+/F2;F0%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "n := 'n ':\nevalf(EMsum(1/n^1.1,n=1..infinity),50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S]<#GX[*RI_N " 0 "" {MPLTEXT 1 0 20 "evalf(Zeta (1.1),50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S]<#GX[*RI_N " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example \+ 4" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(ln(n)/(n^2), n = 1 .. infinity);" "6#-%$SumG6$*&-%#lnG6#%\"nG\"\"\"*$F*\"\"#!\"\"/F *;F+%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "n := 'n':\nevalf(EMsum(ln(n) /n^2,n=1..infinity,info=true),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %@partial~sum~of~39~terms~~...~~~G$\";6;+)!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;add~half~of~term~40~...~~~G$\";]dYEh8DPW'zJ?) !#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;integral~correction~...~~~G-%$IntG6$*&-%#lnG6#%\"xG\" \"\"F+!\"#/F+;\"#S%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~ G-%&LimitG6$,$*&#\"\"\"\"#SF**&,,*&F+F*-%#lnG6#%\"RGF*!\"\"F+F3*(\"\"$ F*-F06#\"\"#F*F2F*F**&-F06#\"\"&F*F2F*F*F2F*F*F2F3F*F*/F2%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G$\";R68d2%[GN')>A<\"!#E" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%?add~integral~correction~...~~~G$\";*)of$)o(*4!z]*Rv$*!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Aadd~asymptotic~correction~...~~~G$\";4uDq` P%eJa#[v$*!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5q`P%eJa#[v$*!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalf(sum(ln(n)/n^2,n=1..infinity),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5q`P%eJa#[v$*!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(1/(2^n),n = 1 .. infinity) = 1;" "6#/-%$SumG6$*&\" \"\"F()\"\"#%\"nG!\"\"/F+;F(%)infinityGF(" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "n := \+ 'n':\nevalf(EMsum(1/2^n,n=1..infinity,info=true),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@partial~sum~of~39~terms~~...~~~G$\";:akf55=)**** *******!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;add~half~of~term~40~.. .~~~G$\";hSt%zvN')***********!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;integral~correction~...~~~G-%$I ntG6$*&\"\"\"F()\"\"#%\"xG!\"\"/F+;\"#S%)infinityG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%$=~~G-%&LimitG6$,$*&#\"\"\"\".wxi6&*4\"F**(,&\".wxi6 &*4\"!\"\")\"\"#%\"RGF*F*-%#lnG6#F1F/F0F/F*F*/F2%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G$\";w#)zTG!fif\\B@J\"!#P" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?add~int egral~correction~...~~~G$\";?.LW\")y%*************!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Aadd~asymptotic~correction~...~~~G$\";()********* ***************!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5+++++++++5!# >" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(2^n/n!,n = 1 .. infinity);" "6#-%$SumG6$*&)\"\"#%\"nG\"\"\"-%*factorialG6#F)!\"\"/F); F*%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "n := 'n':\nEMsum(2^n/n!,n=1. .infinity,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Binitial~sum ~of~18~terms~convergedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*4c!*Q' !\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Sum(2^n/n!,n=1..infinity);\nvalue(%);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)\"\"#%\"nG\"\"\"-%*factori alG6#F)!\"\"/F);F*%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$ expG6#\"\"#\"\"\"F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*4c!*Q '!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 7" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(10^n/n!,n = 1 . . infinity);" "6#-%$SumG6$*&)\"#5%\"nG\"\"\"-%*factorialG6#F)!\"\"/F); F*%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "n := 'n':\nEMsum(10^n/n!,n=1 ..infinity,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@partial~su m~of~19~terms~~...~~~G$\"0@V\\)y$\\>#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;add~half~of~term~20~...~~~G$\"0([vVI*p>#!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;integral~cor rection~...~~~G-%$IntG6$*&)\"#5%\"xG\"\"\"-%*factorialG6#F*!\"\"/F*;\" #?%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G$\"0&z,4'=\"4`! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?add~integral~correction~...~~~G$\"0*QOi@I-A!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Aadd~asymptotic~correction~...~~~G$\" 0s![zla-A!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zla-A!\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "n := 'n':\nEMsum(10^n/n!,n=1..infinity,numterms=50,info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Binitial~sum~of~38~terms~convergedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zla-A!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evalf(sum(10^n/n!,n=1..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zla-A!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" } {TEXT 0 5 "EMsum" }{TEXT -1 44 ": examples - series with some negative terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(-cos(n*Pi)/n,n = 1 .. infinity) = 1-1/2+1/3-1/ 4+1/5-1/6+` . . . `;" "6#/-%$SumG6$,$*&-%$cosG6#*&%\"nG\"\"\"%#PiGF.F. F-!\"\"F0/F-;F.%)infinityG,0F.F.*&F.F.\"\"#F0F0*&F.F.\"\"$F0F.*&F.F.\" \"%F0F0*&F.F.\"\"&F0F.*&F.F.\"\"'F0F0%(~.~.~.~GF." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 59 "This is the alternating harmonic series w hich converges to " }{XPPEDIT 18 0 "ln(2)" "6#-%#lnG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 153 "The signs of the terms are de termined by the numerator, where the numerator is chosen in such a way that Maple can integrate the corresponding function " }{XPPEDIT 18 0 "g(x) = -cos(x*Pi)/x;" "6#/-%\"gG6#%\"xG,$*&-%$cosG6#*&F'\"\"\"%#PiG F.F.F'!\"\"F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "n := 'n':\nevalf(EMsum(-cos( n*Pi)/n,n=1..infinity,info=true),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@partial~sum~of~39~terms~~...~~~G$\";n\")G.vSp#z\"Q.eq!#E" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%;add~half~of~term~40~...~~~G$\";n\")G .vSp#z\"Q.Lp!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;integral~correction~...~~~G-%$IntG6$,$*&-%$cosG6 #*&%\"xG\"\"\"%#PiGF.F.F-!\"\"F0/F-;\"#S%)infinityG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%$=~~G-%&LimitG6$,&-%#CiG6#*&%\"RG\"\"\"%#PiGF-!\"\"- F)6#,$*&\"#SF-F.F-F-F-/F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%$=~~G$!;W[#zX2'\\LF4 " 0 "" {MPLTEXT 1 0 59 "evalf(sum(-cos(n*Pi)/n,n=1.. infinity),20);\nevalf(ln(2),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"5U4`%*f0=ZJp!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5U4`%*f0=ZJp!# ?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Ano ther way to sum the alternating harmonic series" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Sum((-1)^(n-1)/n,n = 1 .. infinity) = 1-1/2+1/3-1/4+1/5-1/6;" "6#/-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF*F*F+F*F -F+/F-;F*%)infinityG,.F*F**&F*F*\"\"#F+F+*&F*F*\"\"$F+F**&F*F*\"\"%F+F +*&F*F*\"\"&F+F**&F*F*\"\"'F+F+" }{TEXT -1 10 " + . . . ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "is to group the te rms together in pairs" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/(2*n-1)-1/(2*n)=1/((2*n-1)*(2*n))" "6#/,&*&\"\"\"F&,&*&\"\"#F& %\"nGF&F&F&!\"\"F+F&*&F&F&*&F)F&F*F&F+F+*&F&F&*&,&*&F)F&F*F&F&F&F+F&*& F)F&F*F&F&F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 13 "and calcu late" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/((2*n-1 )*(2*n)),n=1..infinity)" "6#-%$SumG6$*&\"\"\"F'*&,&*&\"\"#F'%\"nGF'F'F '!\"\"F'*&F+F'F,F'F'F-/F,;F'%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "evalf(E Msum(1/(2*n*(2*n-1)),n=1..infinity,info=true),20);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%@partial~sum~of~39~terms~~...~~~G$\";*R-KGFR5;7!yno! #E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;add~half~of~term~40~...~~~G$\" ;S;Mu0**3&3Er&oo!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;integral~correction~...~~~G-%$IntG6$,$*&\" \"\"F)*(\"\"#F)%\"xGF),&*&F+F)F,F)F)F)!\"\"F)F/F)/F,;\"#S%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G-%&LimitG6$,,*&#\"\"\"\"\"#F*- %#lnG6#,&*&F+F*%\"RGF*F*F*!\"\"F*F**&#F*F+F*-F-6#F1F*F2*&#F*F+F*-F-6# \"#zF*F2*&#\"\"$F+F*-F-6#F+F*F**&F)F*-F-6#\"\"&F*F*/F1%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G$\"90r$[!f+V.6R*G'!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?add ~integral~correction~...~~~G$\";X(y\"zk*>&)= " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 9" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(-cos(n*Pi)/sqrt(n),n = 1 .. infinity) = Sum((-1)^n/sqrt(n),n = 1 .. infinity);" "6#/-%$SumG6$,$*&-%$cosG6#*&%\"nG\"\"\"%#PiGF.F.-% %sqrtG6#F-!\"\"F3/F-;F.%)infinityG-F%6$*&),$F.F3F-F.-F16#F-F3/F-;F.F6 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 188 "This is another alte rnating series where signs of the terms are determined by the numerato r, and the numerator is chosen in such a way that Maple can integrate \+ the corresponding function " }{XPPEDIT 18 0 "g(x) = -cos(x*Pi)/sqrt(x );" "6#/-%\"gG6#%\"xG,$*&-%$cosG6#*&F'\"\"\"%#PiGF.F.-%%sqrtG6#F'!\"\" F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 70 "n := 'n':\nevalf(EMsum(-cos(n*Pi)/sqrt(n),n= 1..infinity,info=true),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@parti al~sum~of~39~terms~~...~~~G$\";Sz\\5b)o**R%f\\Wo!#E" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%;add~half~of~term~40~...~~~G$\";12]!=Pfd*Gl#R0'!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;integral~correction~...~~~G-%$IntG6$,$*&-%$cosG6#*&%\"xG\"\"\"%#P iGF.F.F-#!\"\"\"\"#F1/F-;\"#S%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G-%&LimitG6$,&*&\"\"##\"\"\"F)-%)FresnelCG6#*&F)F*%\"RGF*F+ !\"\"*&-F-6#,$*(F)F+\"#5F*F)F*F+F+F)F*F+/F0%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G$!8qaRXH2,lg?+#!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?add~integral~c orrection~...~~~G$\";O_5NU'[2$oW#>0'!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Aadd~asymptotic~correction~...~~~G$\";6ssCq.j@Mk)*[g!#E" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5Dq.j@Mk)*[g!#?" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "We can also find the su m of this series by grouping the terms together in pairs and calculati ng" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(1/sqrt(2*n -1)-1/sqrt(2*n),n = 1 .. infinity);" "6#-%$SumG6$,&*&\"\"\"F(-%%sqrtG6 #,&*&\"\"#F(%\"nGF(F(F(!\"\"F0F(*&F(F(-F*6#*&F.F(F/F(F0F0/F/;F(%)infin ityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "evalf(EMsum(1/sqrt(2*n-1)-1/sqrt(2*n),n=1 ..infinity,info=true),20);;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@parti al~sum~of~39~terms~~...~~~G$\";W$p[*o/W\\qBm%[&!#E" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%;add~half~of~term~40~...~~~G$\";S2Az@pCeE$*=)[&!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;integral~correction~...~~~G-%$IntG6$,&*&\"\"\"F)*$,&*&\"\"#F)%\"x GF)F)F)!\"\"#F)F-F/F)*(F-F/F-#F)F-F.#F/F-F//F.;\"#S%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G-%&LimitG6$,**$,&*&\"\"#\"\"\"% \"RGF,F,F,!\"\"#F,F+F,*&F+F/F-F/F.*$\"#zF/F.*(F+F,\"#5F/F+F/F,/F-%)inf inityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G$\"9FDXb$*pNo#\\xg&!#D " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?add~integral~correction~...~~~G$\";5guLdo\"=MDk*[g!#E" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Aadd~asymptotic~correction~...~~~G$\" ;llsCq.j@Mk)*[g!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5Dq.j@Mk)*[g! #?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 10 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let \+ " }{XPPEDIT 18 0 "sigma[n]" "6#&%&sigmaG6#%\"nG" }{TEXT -1 47 " be the sequence of \"signs\" defined as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma := n -> 1-2* (((n+1) mod 3) mod 2);\nseq(sigma(n),n=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-* &\"\"#F--%$modG6$-F16$,&9$F-F-F-\"\"$F/F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "66\"\"\"F#!\"\"F#F#F$F#F#F$F#F#F$F#F#F$F#F#F$F#F#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Consider the series: " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "Sum(sigma(n)/(n*sqrt(n)),n = 1 .. infinity) = 1+1/(2*sqrt(2))-1/(3*sqrt(3))+1/(4*sqrt(4))+1/(5*sqrt(5))-1/(6*sqrt (6))+1/(7*sqrt(7))+1/(8*sqrt(8))-1/(9*sqrt(9))+` . . . `;" "6#/-%$SumG 6$*&-%&sigmaG6#%\"nG\"\"\"*&F+F,-%%sqrtG6#F+F,!\"\"/F+;F,%)infinityG,6 F,F,*&F,F,*&\"\"#F,-F/6#F8F,F1F,*&F,F,*&\"\"$F,-F/6#F=F,F1F1*&F,F,*&\" \"%F,-F/6#FBF,F1F,*&F,F,*&\"\"&F,-F/6#FGF,F1F,*&F,F,*&\"\"'F,-F/6#FLF, F1F1*&F,F,*&\"\"(F,-F/6#FQF,F1F,*&F,F,*&\"\")F,-F/6#FVF,F1F,*&F,F,*&\" \"*F,-F/6#FenF,F1F1%(~.~.~.~GF," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 67 "This series converges absolutely to a sum which is no gre ater than " }{XPPEDIT 18 0 "Sum(1/(n*sqrt(n)),n = 1 .. infinity);" "6# -%$SumG6$*&\"\"\"F'*&%\"nGF'-%%sqrtG6#F)F'!\"\"/F);F'%)infinityG" } {TEXT -1 1 " " }{TEXT 265 1 "~" }{TEXT -1 13 " 2.612375349." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sigma" "6#%&sigmaG" }{TEXT -1 53 " can be extended to a continuous function as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "si gma := x ->1/3-4/3*cos(2*Pi/3*x);\nseq(sigma(n),n=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&# \"\"\"\"\"$F.*&#\"\"%F/F.-%$cosG6#,$*&#\"\"#F/F.*&%#PiGF.9$F.F.F.F.!\" \"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"F#!\"\"F#F#F$F#F#F$F #" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 " No w Maple can differentiate and integrate the corresponding function " } {XPPEDIT 18 0 "g(x) = sigma(x)/(x*sqrt(x));" "6#/-%\"gG6#%\"xG*&-%&sig maG6#F'\"\"\"*&F'F,-%%sqrtG6#F'F,!\"\"" }{TEXT -1 30 ", so we can use \+ the procedure " }{TEXT 0 5 "EMsum" }{TEXT -1 36 " to calculate the sum of the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalf(EMsum( sigma(n)/(n*sqrt(n)),n=1..infinity),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5tc:-;3;(og\"!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "We can also find the sum of this series by groupi ng together 3 terms at a time and calculating" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(1/((3*n-2)^(3/2))+1/((3*n-1)^(3/2) )-1/((3*n)^(3/2)),n=1..infinity)" "6#-%$SumG6$,(*&\"\"\"F(),&*&\"\"$F( %\"nGF(F(\"\"#!\"\"*&F,F(F.F/F/F(*&F(F(),&*&F,F(F-F(F(F(F/*&F,F(F.F/F/ F(*&F(F()*&F,F(F-F(*&F,F(F.F/F/F//F-;F(%)infinityG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "a := n -> 1/((3*n-2)^(3/2))+1/((3*n-1)^(3/2))-1/((3*n)^(3/2)):\nS um(a(n),n=1..infinity);\nevalf(%,20);\nevalf(EMsum(a(n),n=1..infinity) ,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,(*&\"\"\"F(*$),&*& \"\"$F(%\"nGF(F(\"\"#!\"\"#F-F/F(F0F(*&F(F(*$),&*&F-F(F.F(F(F(F0#F-F/F (F0F(*(\"\"*F0F-#F(F/F.#!\"$F/F0/F.;F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5tc:-;3;(og\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"5tc:-;3;(og\"!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 11" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Su m(sqrt(2)/(n^2),n = 1 .. infinity);" "6#-%$SumG6$*&-%%sqrtG6#\"\"#\"\" \"*$%\"nGF*!\"\"/F-;F+%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin ((2*n+1)*Pi/4)= 1-1/4-1/9+1/16+1/25-1/36-1/49+1/64+` . . . `" "6#/-%$s inG6#*(,&*&\"\"#\"\"\"%\"nGF+F+F+F+F+%#PiGF+\"\"%!\"\",4F+F+*&F+F+F.F/ F/*&F+F+\"\"*F/F/*&F+F+\"#;F/F+*&F+F+\"#DF/F+*&F+F+\"#OF/F/*&F+F+\"#\\ F/F/*&F+F+\"#kF/F+%(~.~.~.~GF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 67 "This series converges absolutely to a sum which is no gre ater than " }{XPPEDIT 18 0 "Sum(1/(n^2),n = 1 .. infinity)=Pi^2/6" "6# /-%$SumG6$*&\"\"\"F(*$%\"nG\"\"#!\"\"/F*;F(%)infinityG*&%#PiGF+\"\"'F, " }{TEXT -1 1 " " }{TEXT 263 1 "~" }{TEXT -1 13 " 1.644934067." }} {PARA 0 "" 0 "" {TEXT -1 153 "The signs of the terms are determined by the numerator, where the numerator is chosen in such a way that Maple can integrate the corresponding function " }{XPPEDIT 18 0 "g(x) = sq rt(2)/(x^2);" "6#/-%\"gG6#%\"xG*&-%%sqrtG6#\"\"#\"\"\"*$F'F,!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sin((2*x+1)*Pi/4)" "6#-%$sinG6#*(,&*&\" \"#\"\"\"%\"xGF*F*F*F*F*%#PiGF*\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "n := \+ 'n':\nseq(sqrt(2)*sin((2*n+1)*Pi/4),n=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "66\"\"\"!\"\"F$F#F#F$F$F#F#F$F$F#F#F$F$F#F#F$F$F#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "evalf(EMsum(sqrt(2)*sin((2*n+1)*Pi/4)/n^2,n=1..infinity,info=true) ,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@partial~sum~of~39~terms~~.. .~~~G$\";rSz.P?_7D$)3(4(!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;add~h alf~of~term~40~...~~~G$\";rSz.P?_7DL@+r!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;integral~corre ction~...~~~G-%$IntG6$*(\"\"##\"\"\"F(-%$sinG6#,$*(\"\"%!\"\",&*&F(F*% \"xGF*F*F*F*F*%#PiGF*F*F*F4!\"#/F4;\"#S%)infinityG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%$=~~G-%&LimitG6$,$*&#\"\"\"\"#SF**&,.*(F+F*\"\"##F*F /-%$sinG6#,&*&F0F**&%\"RGF*%#PiGF*F*F**&#F*\"\"%F*F8F*F*F*F***\"#?F*-% #SiG6#,$*(F/!\"\"F7F*F8F*F*F*F7F*F8F*F***F=F*-%#CiGF@F*F7F*F8F*FCF7FC* *F=F*-FF6#,$*&F=F*F8F*F*F*F8F*F7F*F***F=F*-F?FIF*F8F*F7F*FCF*F7FCF*FC/ F7%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$=~~G$\"7U]dj/$)>h7* 4%!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?add~integral~correction~...~~~G$\";\"\\W&R$30X7X7V5(! #E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Aadd~asymptotic~correction~...~ ~~G$\";*3b&\\52>@e$)[.r!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5]52> @e$)[.r!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "We can also find the sum of this series by grouping together 4 \+ terms at a time and calculating" }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Sum(1/((4*n-3)^2)-1/((4*n-2)^2)-1/((4*n-1)^2)+1/((4*n)^ 2),n = 1 .. infinity);" "6#-%$SumG6$,**&\"\"\"F(*$,&*&\"\"%F(%\"nGF(F( \"\"$!\"\"\"\"#F/F(*&F(F(*$,&*&F,F(F-F(F(F0F/F0F/F/*&F(F(*$,&*&F,F(F-F (F(F(F/F0F/F/*&F(F(*$*&F,F(F-F(F0F/F(/F-;F(%)infinityG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "evalf(EMsum(1/(4*n-3)^2-1/(4*n-2)^2-1/(4*n-1)^2+1/(4*n)^2,\n \+ n=1..infinity),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5]52>@e$)[. r!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 12 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let \+ " }{XPPEDIT 18 0 "sigma[n]" "6#&%&sigmaG6#%\"nG" }{TEXT -1 47 " be the sequence of \"signs\" defined as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sigma := n -> ((3* n mod 5) mod 2)*2-1;\nseq(sigma(n),n=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,&-%$modG6$ -F.6$,$9$\"\"$\"\"&\"\"#F6\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "66\"\"\"F#!\"\"F$F$F#F#F$F$F$F#F#F$F$F$F#F#F$F$F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the series:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "Sum(sigma(n)/(n^3),n = 1 .. infinity) = 1+1 /8-1/27-1/64-1/125+1/216+1/343-1/512-1/729-1/1000+` . . . `;" "6#/-%$S umG6$*&-%&sigmaG6#%\"nG\"\"\"*$F+\"\"$!\"\"/F+;F,%)infinityG,8F,F,*&F, F,\"\")F/F,*&F,F,\"#FF/F/*&F,F,\"#kF/F/*&F,F,\"$D\"F/F/*&F,F,\"$;#F/F, *&F,F,\"$V$F/F,*&F,F,\"$7&F/F/*&F,F,\"$H(F/F/*&F,F,\"%+5F/F/%(~.~.~.~G F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 67 "This series converg es absolutely to a sum which is no greater than " }{XPPEDIT 18 0 "Sum( 1/(n^3),n = 1 .. infinity);" "6#-%$SumG6$*&\"\"\"F'*$%\"nG\"\"$!\"\"/F );F'%)infinityG" }{TEXT -1 1 " " }{TEXT 264 1 "~" }{TEXT -1 13 " 1.202 056903." }}{PARA 0 "" 0 "" {TEXT -1 14 "We can extend " }{XPPEDIT 18 0 "sigma" "6#%&sigmaG" }{TEXT -1 37 " to a continuous function as foll ows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "sigma := x ->2/5*(sqrt(5)-1)*cos(4*Pi/5*(x+1))-\n \+ 2/5*(1+sqrt(5))*cos(2*Pi/5*(x+1))-1/5;\nseq(normal(sigma(n)),n=1..10); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6#%\"xG6\"6$%)operatorG %&arrowGF(,(*&#\"\"#\"\"&\"\"\"*&,&-%%sqrtG6#F0F1F1!\"\"F1-%$cosG6#,$* &#\"\"%F0F1*&%#PiGF1,&9$F1F1F1F1F1F1F1F1F1*&#F/F0F1*&,&F1F1F4F1F1-F96# ,$*&F.F1F?F1F1F1F1F7#F1F0F7F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,, ,*&#\"\"#\"\"&\"\"\"*&-%$cosG6#,$*(F&F(F'!\"\"%#PiGF(F(F(F'#F(F&F(F(*& #F&F'F(F*F(F/*&F%F(*&-F+6#,$*&F'F/F0F(F(F(F'F1F(F(*&F%F(F6F(F(#F(F'F/F #,,*&#F&F'F(F5F(F/*&F%F(F6F(F(*&#F&F'F(F)F(F/*&#F&F'F(F*F(F/#F(F'F/F/F " 0 "" {MPLTEXT 1 0 44 "evalf(EMsum(sigma(n)/ n^3,n=1..infinity),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5Z.?Xu9/ on5!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "We can also find the sum of this series by grouping together 5 ter ms at a time and calculating" }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Sum(1/((5*n-4)^3)+1/((5*n-3)^3)-1/((5*n-2)^3)-1/((5*n-1 )^3)-1/((5*n)^3),n = 1 .. infinity);" "6#-%$SumG6$,,*&\"\"\"F(*$,&*&\" \"&F(%\"nGF(F(\"\"%!\"\"\"\"$F/F(*&F(F(*$,&*&F,F(F-F(F(F0F/F0F/F(*&F(F (*$,&*&F,F(F-F(F(\"\"#F/F0F/F/*&F(F(*$,&*&F,F(F-F(F(F(F/F0F/F/*&F(F(*$ *&F,F(F-F(F0F/F//F-;F(%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "evalf(EMsum (1/((5*n-4)^3)+1/((5*n-3)^3)-1/((5*n-2)^3)-\n 1/((5*n-1)^3)-1/(5*n)^3 ,n=1..infinity),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5Z.?Xu9/on5 !#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T asks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 18 "Use the proce dure " }{TEXT 0 5 "EMsum" }{TEXT -1 36 " to calculate the sum of the s eries " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n*ln (n)^2),n = 2 .. infinity);" "6#-%$SumG6$*&\"\"\"F'*&%\"nGF'*$-%#lnG6#F )\"\"#F'!\"\"/F);F.%)infinityG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "correct to 20 digits." }}{PARA 0 "" 0 "" {TEXT -1 40 "___ _____________________________________" }}{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 40 "____________________ ____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 18 "Use the procedure " }{TEXT 0 5 "EMsum" }{TEXT -1 36 " to calculate the sum of the series " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n*ln(n)^(3/2)),n = 2 .. infinity);" "6#-%$SumG6$*&\"\"\"F '*&%\"nGF')-%#lnG6#F)*&\"\"$F'\"\"#!\"\"F'F1/F);F0%)infinityG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "correct to 20 digits." }} {PARA 0 "" 0 "" {TEXT -1 40 "________________________________________ " }}{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 40 "________________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }