{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 "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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 115 0 230 1 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 0 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 273 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 274 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 261 275 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Norma l" -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 1 0 1 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 "Maple O utput" -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 "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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 39 "More series involving Bernoulli n umbers" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., \+ Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Maclaurin series for " }{XPPEDIT 18 0 "x/(sinh* x);" "6#*&%\"xG\"\"\"*&%%sinhGF%F$F%!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "We can find the Maclaurin series for " }{XPPEDIT 18 0 "x/(sinh*x);" "6#*&%\"xG\"\"\"*&%%sinhGF%F$F%!\"\"" }{TEXT -1 24 " by using the identity " }{XPPEDIT 18 0 "coth(x/2)-coth*x = 1/(sinh* x);" "6#/,&-%%cothG6#*&%\"xG\"\"\"\"\"#!\"\"F**&F&F*F)F*F,*&F*F**&%%si nhGF*F)F*F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "This identity can be checked \"by hand\" . . . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "coth(x/2)-coth*x \+ = ``((exp(x/2)+exp(-x/2))/(exp(x/2)-exp(-x/2)))-``((exp(x)+exp(-x))/(e xp(x)-exp(-x)));" "6#/,&-%%cothG6#*&%\"xG\"\"\"\"\"#!\"\"F**&F&F*F)F*F ,,&-%!G6#*&,&-%$expG6#*&F)F*F+F,F*-F56#,$*&F)F*F+F,F,F*F*,&-F56#*&F)F* F+F,F*-F56#,$*&F)F*F+F,F,F,F,F*-F06#*&,&-F56#F)F*-F56#,$F)F,F*F*,&-F56 #F)F*-F56#,$F)F,F,F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = ``((exp(x)+ 1)/(exp(x)-1))-``((exp(2*x)+1)/(exp(2*x)-1));" "6#/%!G,&-F$6#*&,&-%$ex pG6#%\"xG\"\"\"F.F.F.,&-F+6#F-F.F.!\"\"F2F.-F$6#*&,&-F+6#*&\"\"#F.F-F. F.F.F.F.,&-F+6#*&F:F.F-F.F.F.F2F2F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = (exp(2*x)+2*exp(x)+1-(exp(2*x)+1))/(exp(2*x)-1);" "6#/%!G*&,*-%$expG 6#*&\"\"#\"\"\"%\"xGF,F,*&F+F,-F(6#F-F,F,F,F,,&-F(6#*&F+F,F-F,F,F,F,! \"\"F,,&-F(6#*&F+F,F-F,F,F,F5F5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2* exp(x)/(exp(2*x)-1)" "6#/%!G*(\"\"#\"\"\"-%$expG6#%\"xGF',&-F)6#*&F&F' F+F'F'F'!\"\"F0" }{XPPEDIT 18 0 "`` = 2/(exp(x)-exp(-x));" "6#/%!G*&\" \"#\"\"\",&-%$expG6#%\"xGF'-F*6#,$F,!\"\"F0F0" }{XPPEDIT 18 0 " ``= 1/ (sinh*x)" "6#/%!G*&\"\"\"F&*&%%sinhGF&%\"xGF&!\"\"" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 25 " . . . or by using Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "coth(x/ 2)-coth(x);\n``=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%%c othG6#,$*&\"\"#!\"\"%\"xG\"\"\"F,F,-F%6#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&-%%sinhG6#%\"xG!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x/2" "6#*&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "coth(x/2) = 1+``;" "6#/-%%cothG6#*&%\"xG\"\"\"\"\"#!\" \",&F)F)%!GF)" }{XPPEDIT 18 0 "Sum(``(beta[2*k]/(2*k)!)*x^(2*k),k = 1 \+ .. infinity)" "6#-%$SumG6$*&-%!G6#*&&%%betaG6#*&\"\"#\"\"\"%\"kGF0F0-% *factorialG6#*&F/F0F1F0!\"\"F0)%\"xG*&F/F0F1F0F0/F1;F0%)infinityG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "-2*Pi < x;" "6#2,$*&\"\"#\"\"\"%#PiGF' !\"\"%\"xG" }{XPPEDIT 18 0 "`` < 2*Pi;" "6#2%!G*&\"\"#\"\"\"%#PiGF'" } {TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*coth*x = 1+``;" "6#/*(%\"xG\"\"\"% %cothGF&F%F&,&F&F&%!GF&" }{XPPEDIT 18 0 "Sum(``(beta[2*k]*2^(2*k)/(2*k )!)*x^(2*k),k = 1 .. infinity)" "6#-%$SumG6$*&-%!G6#*(&%%betaG6#*&\"\" #\"\"\"%\"kGF0F0)F/*&F/F0F1F0F0-%*factorialG6#*&F/F0F1F0!\"\"F0)%\"xG* &F/F0F1F0F0/F1;F0%)infinityG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-Pi < x ;" "6#2,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 "`` < Pi;" "6#2%!G%#PiG" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "x*coth(x/2)-x*coth*x = 2+2" "6#/,&*&% \"xG\"\"\"-%%cothG6#*&F&F'\"\"#!\"\"F'F'*(F&F'F)F'F&F'F-,&F,F'F,F'" } {XPPEDIT 18 0 "Sum(``(beta[2*k]/(2*k)!)*x^(2*k),k = 1 .. infinity)" "6 #-%$SumG6$*&-%!G6#*&&%%betaG6#*&\"\"#\"\"\"%\"kGF0F0-%*factorialG6#*&F /F0F1F0!\"\"F0)%\"xG*&F/F0F1F0F0/F1;F0%)infinityG" }{XPPEDIT 18 0 " `` -1-``" "6#,(%!G\"\"\"F%!\"\"F$F&" }{XPPEDIT 18 0 "Sum(``(beta[2*k]*2^( 2*k)/(2*k)!)*x^(2*k),k = 1 .. infinity)" "6#-%$SumG6$*&-%!G6#*(&%%beta G6#*&\"\"#\"\"\"%\"kGF0F0)F/*&F/F0F1F0F0-%*factorialG6#*&F/F0F1F0!\"\" F0)%\"xG*&F/F0F1F0F0/F1;F0%)infinityG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-Pi < x;" "6#2,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 "`` < Pi;" "6#2%!G% #PiG" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 6 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x/(sinh*x) = 1+``;" " 6#/*&%\"xG\"\"\"*&%%sinhGF&F%F&!\"\",&F&F&%!GF&" }{XPPEDIT 18 0 "Sum(b eta[2*k]*``((2-2^(2*k))/(2*k)!)*x^(2*k),k = 1 .. infinity)" "6#-%$SumG 6$*(&%%betaG6#*&\"\"#\"\"\"%\"kGF,F,-%!G6#*&,&F+F,)F+*&F+F,F-F,!\"\"F, -%*factorialG6#*&F+F,F-F,F5F,)%\"xG*&F+F,F-F,F,/F-;F,%)infinityG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "-Pi < x;" "6#2,$%#PiG!\"\"%\"xG" } {XPPEDIT 18 0 "`` < Pi;" "6#2%!G%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "that is, using the f act that the Bernoulli numbers " }{XPPEDIT 18 0 "beta[n]" "6#&%%betaG6 #%\"nG" }{TEXT -1 6 " with " }{TEXT 262 1 "n" }{TEXT -1 9 " odd and " }{XPPEDIT 18 0 "n<>1" "6#0%\"nG\"\"\"" }{TEXT -1 11 " are zero, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x/(sinh*x) = 1+``" "6 #/*&%\"xG\"\"\"*&%%sinhGF&F%F&!\"\",&F&F&%!GF&" }{XPPEDIT 18 0 "Sum(be ta[n]*``((2-2^n)/n!)*x^n,n = 2 .. infinity)" "6#-%$SumG6$*(&%%betaG6#% \"nG\"\"\"-%!G6#*&,&\"\"#F+)F1F*!\"\"F+-%*factorialG6#F*F3F+)%\"xGF*F+ /F*;F1%)infinityG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-Pi < x;" "6#2,$% #PiG!\"\"%\"xG" }{XPPEDIT 18 0 "`` < Pi;" "6#2%!G%#PiG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 26 "_________________ _________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "alias(beta=bernoulli):\n1+Sum(beta( n)*((2-2^n))/n!*x^n,n=2..14);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%$SumG6$**-%%betaG6#%\"nGF$,&\"\"#F$)F.F,!\"\"F$-%*fa ctorialGF+F0)%\"xGF,F$/F,;F.\"#9F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,2\"\"\"F$*&#F$\"\"'F$*$)%\"xG\"\"#F$F$!\"\"*&#\"\"(\"$g$F$*$)F*\"\"%F $F$F$*&#\"#J\"&?^\"F$*$)F*F'F$F$F,*&#\"$F\"\"'+[gF$*$)F*\"\")F$F$F$*&# \"#t\"(S9U$F$*$)F*\"#5F$F$F,*&#\"(xWT\"\"-+S=PQlF$*$)F*\"#7F$F$F$*&#\" %\">)\",+[7it$F$*$)F*\"#9F$F$F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "taylor(x/sinh(x),x,16);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"\"\"!#!\"\"\"\"'\"\"##\" \"(\"$g$\"\"%#!#J\"&?^\"F)#\"$F\"\"'+[g\"\")#!#t\"(S9U$\"#5#\"(xWT\"\" -+S=PQl\"#7#!%\">)\",+[7it$\"#9-%\"OG6#F%\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Graphical illustration" }}{PARA 0 "" 0 "" {TEXT -1 37 "The first picture shows the graph of " }{XPPEDIT 18 0 "y = x/(sinh*x);" "6#/%\"yG*&%\"xG\"\"\"*&%%sinhGF'F&F'!\"\"" }{TEXT -1 20 ", which is shown in " }{TEXT 267 5 "black" }{TEXT -1 102 ", alo ng with the graphs of some polynomials obtained by truncating the corr esponding Maclaurin series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "plot([x/sinh(x),seq(convert(taylor (x/sinh(x),x,2*i),polynom),i=2..7)],\n x=-4..4,y=-1..2,color=[black, red,coral,green,brown,magenta,blue],\n thickness=[2,1$6]);" }}{PARA 13 "" 1 "" {GLPLOT2D 401 404 404 {PLOTDATA 2 "6+-%'CURVESG6%7W7$$!\"% \"\"!$\"3]CY.8Gul9!#=7$$!3ommmmFiDQ!#<$\"3cT,mbj8p;F-7$$!35LLLo!)*Qn$F 1$\"35!4E!)*y(e'=F-7$$!3nmmmwxE.NF1$\"3\">q\"H@+!36#F-7$$!3YmmmOk]JLF1 $\"3klYN@ZN%Q#F-7$$!3_LLL[9cgJF1$\"3U>e!ye0^o#F-7$$!3smmmhN2-IF1$\"3;O 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" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 27 "___________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "alias(beta=b ernoulli):\n1+Sum(abs(beta(n))*((2^n-2))/n!*x^n,n=2..14);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%$SumG6$*&*(-%$absG6#-%%b etaG6#%\"nGF$,&)\"\"#F0F$!\"#F$F$)%\"xGF0F$F$-%*factorialGF/!\"\"/F0;F 3\"#9F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2\"\"\"F$*$)%\"xG\"\"#F$#F $\"\"'*$)F'\"\"%F$#\"\"(\"$g$*$)F'F*F$#\"#J\"&?^\"*$)F'\"\")F$#\"$F\" \"'+[g*$)F'\"#5F$#\"#t\"(S9U$*$)F'\"#7F$#\"(xWT\"\"-+S=PQl*$)F'\"#9F$# \"%\">)\",+[7it$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "By consid ering complex numbers, and using the relation " }{XPPEDIT 18 0 "sin*x = -i*sinh*i*x;" "6#/*&%$sinG\"\"\"%\"xGF&,$**%\"iGF&%%sinhGF&F*F&F'F& !\"\"" }{TEXT -1 50 ", this series can be obtained from the series for " }{XPPEDIT 18 0 "x/(sinh*x);" "6#*&%\"xG\"\"\"*&%%sinhGF%F$F%!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Graphical illustration" }}{PARA 0 "" 0 "" {TEXT -1 37 "The first picture shows the graph of " } {XPPEDIT 18 0 "y = x/(sin*x);" "6#/%\"yG*&%\"xG\"\"\"*&%$sinGF'F&F'!\" \"" }{TEXT -1 20 ", which is shown in " }{TEXT 265 5 "black" }{TEXT -1 102 ", along with the graphs of some polynomials obtained by trunca ting the corresponding Maclaurin series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "p1 := plot([x/sin(x) ,seq(convert(taylor(x/sin(x),x,2*i),polynom),i=2..7)],\n x=-4..4,y=- 1..6,color=[black,red,coral,green,brown,magenta,blue],\n thickness=[ 2,1$6],discont=true):\np2 := plot([[[-Pi,-1],[-Pi,6]],[[Pi,-1],[Pi,6]] ],color=black,\n linestyle=3):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 411 370 370 {PLOTDATA 2 "6--%'CURVESG6(7gn7$$!\"% \"\"!$!3a4OCN[R&G&!#<7$$!3E;e\"H;*G\")RF-$!3)QA&)HeG![`F-7$$!3)*p^hj*3 ]'RF-$!3RXO)RY1eS&F-7$$!3M,(3o<+n%RF-$!3_mzC[akuaF-7$$!3?7'yu-q#GRF-$! 3W)[6@#)>$[bF-7$$!3Kb6Vtu#*4RF-$!3_i.Mt,HEcF-7$$!3w2)=yd@H*QF-$!3zY*3r \"4*Hq&F-7$$!3eQ@$z&HJvQF-$!3A&zTi8#=(y&F-7$$!3saf0)4-r&QF-$!3AZjL&z5( zeF-7$$!3v]$Q1k\\*QQF-$!3k'*4_\"Rmy(fF-7$$!3x1P'pgx-#QF-$!3gwNRrRZ&3'F -7$$!3a!oR&p7$Q!QF-$!3ivk#>Jgi='F-7$$!3i^Y!*)e;`y$F-$!3]m9P>t)pI'F-7$$ !33\"37_)esmPF-$!3:&='**e')eOkF-7$$!3Ik1V(H5)[PF-$!3+M@1#p?,d'F-7$$!3d b6QF7aKPF-$!3cO@ZN$f$*p'F-7$$!3,WT^Hd>8PF-$!3y[F#[H>Q'oF-7$$!3Y_wEbv!o p$F-$!3c\"Q./ARJ,(F-7$$!3S9On7UuxOF-$!3)ohC$3s_*>(F-7$$!3)pz=V#*p3m$F- $!33Q-G)3EqP(F-7$$!3uHFcbhNUOF-$!3)Gy5rdLne(F-7$$!3hA?epmsCOF-$!3V6!f$ \\&[D!yF-7$$!3*3o:/:Kjg$F-$!3#)=jdCqVY!)F-7$$!3k`NnR-W*e$F-$!3-fdXh]>* G)F-7$$!3!QH?>.?7d$F-$!3I!\\R-f(pt&)F-7$$!3yc2wcVH_NF-$!3)>XL%zmN(*))F -7$$!3qV*)*3]>e`$F-$!3=wk\"R)))f0#*F-7$$!31hZA0i-=NF-$!3QcXN1z4q&*F-7$ $!3w&4=(*)Rk*\\$F-$!3o#Q*o2([h)**F-7$$!3Q%*)*4y0m\"[$F-$!3SrH:ih\"Q/\" !#;7$$!3Q([pEqgUY$F-$!3#Hk)3D))[#4\"Fit7$$!3\"*p:g26%\\W$F-$!3#zD!*\\. 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\"\"-%$SumG6$*&-%!G6#*(&%%betaG6#%\"nGF))\"\"#F7F)*&-%*factorialG6#F7F )F7F)F+F))F*F7F)/F7;F9%)infinityG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-P i < x;" "6#2,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 "`` < Pi;" "6#2%!G%#PiG " }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 273 23 "_ ______________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "alias(beta=bernoulli):\n Sum(beta(n)*2^n/(n*n!)*x^n,n=2..14);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,-%%betaG6#%\"nG\"\"\")\"\"#F*F+F*!\"\"-%*fac torialGF)F.)%\"xGF*F+/F*;F-\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0* &#\"\"\"\"\"'F&*$)%\"xG\"\"#F&F&F&*&#F&\"$!=F&*$)F*\"\"%F&F&!\"\"*&#F& \"%NGF&*$)F*F'F&F&F&*&#F&\"&+y$F&*$)F*\"\")F&F&F2*&#F&\"'vxYF&*$)F*\"# 5F&F&F&*&#\"$\"p\"+]s2JQF&*$)F*\"#7F&F&F2*&#F+\"*vDqF\"F&*$)F*\"#9F&F& F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "taylor(ln(sinh(x)/x),x,16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG#\"\"\"\"\"'\"\"##!\"\"\"$!=\"\"%#F&\"%NGF'#F*\" &+y$\"\")#F&\"'vxY\"#5#!$\"p\"+]s2JQ\"#7#F(\"*vDqF\"\"#9-%\"OG6#F&\"#; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Graphical i llustration" }}{PARA 0 "" 0 "" {TEXT -1 37 "The first picture shows th e graph of " }{XPPEDIT 18 0 "y = ln(sinh*x/x);" "6#/%\"yG-%#lnG6#*(%%s inhG\"\"\"%\"xGF*F+!\"\"" }{TEXT -1 20 ", which is shown in " }{TEXT 269 5 "black" }{TEXT -1 102 ", along with the graphs of some polynomia ls obtained by truncating the corresponding Maclaurin series." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "plot([ln(sinh(x)/x),seq(convert(taylor(ln(sinh(x)/x),x,2*i),polyn om),\n i=2..7)],x=-5..5,y=-1..3,color=[black,red,coral,green,brown,m agenta,blue],\n thickness=[2,1$6]);" }}{PARA 13 "" 1 "" {GLPLOT2D 316 358 358 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$!\"&\"\"!$\"3y$eXg]ptp#!#< 7$$!3YLLLe%G?y%F-$\"3OH&fxCYR_#F-7$$!3OmmT&esBf%F-$\"3iBf_PF-$\"3%o:(*pgZkt\"F-7$$!3K++vo1 YZNF-$\"3]([c^z^se\"F-7$$!3;LL3-OJNLF-$\"33(=CaIJjV\"F-7$$!3p***\\P*o% Q7$F-$\"33DjU1xp*G\"F-7$$!3Kmmm\"RFj!HF-$\"3/L%)=JgHV6F-7$$!33LL$e4OZr #F-$\"3t\"y>\\6*\\=5F-7$$!3u*****\\n\\!*\\#F-$\"3o_^'o4$=K))!#=7$$!3%) *****\\ixCG#F-$\"3ymz6@+/OvFdo7$$!3#******\\KqP2#F-$\"3u_K%)RMD`jFdo7$ $!39LL3-TC%)=F-$\"3Zb'*e1d7U`Fdo7$$!3[mmm\"4z)e;F-$\"3LTm)ygIoA%Fdo7$$ !3Mmmmm`'zY\"F-$\"3i'*yFv\"))RO$Fdo7$$!3#****\\(=t)eC\"F-$\"3w\"3i<4G] Y#Fdo7$$!3!ommmh5$\\5F-$\"3?\"f./A)3s!p!41tS\\'!#>7$$!3ndmmT&Q(RTFd o$\"3*o+B'y#)4SGFbr7$$!3%\\mmTg=><#Fdo$\"3!f\"HdK_s\\y!#?7$$!3vDMLLe*e $\\F]s$\"3e*f\\?)z]gS!#B7$$\"3!=nm\"zRQb@Fdo$\"3/xM$e@X3t(F]s7$$\"3_,+ ](=>Y2%Fdo$\"3GhbR@G$>v#Fbr7$$\"3summ\"zXu9'Fdo$\"3cZ9j*yB5A'Fbr7$$\"3 #4+++]y))G)Fdo$\"3&[%GW4_&*>6Fdo7$$\"3H++]i_QQ5F-$\"3jw>$3ewlt\"Fdo7$$ \"3b++D\"y%3T7F-$\"3))[ED\\?\"pW#Fdo7$$\"3+++]P![hY\"F-$\"33c-e!3hhN$F do7$$\"3iKLL$Qx$o;F-$\"3WB.(Hi\"zrUFdo7$$\"3Y+++v.I%)=F-$\"3&zW.jQ;CM& Fdo7$$\"3?mm\"zpe*z?F-$\"3_@XzE`J(Q'Fdo7$$\"3;,++D\\'QH#F-$\"3y%4$=va^ -wFdo7$$\"3%HL$e9S8&\\#F-$\"3^!\\=t+t\"3))Fdo7$$\"3s++D1#=bq#F-$\"3o.N AV0g75F-7$$\"3\"HLL$3s?6HF-$\"3%ykU[MFl9\"F-7$$\"3a***\\7`Wl7$F-$\"3Ee 'H`eU:H\"F-7$$\"3enmmm*RRL$F-$\"3d8X74hON9F-7$$\"3%zmmTvJga$F-$\"3Kabh &QBie\"F-7$$\"3]MLe9tOcPF-$\"3)zo6&[:AR$F-7$F>$\"3U&oq?pZ.*GF-7$FC$\"3na#Q\"*zS8g#F-7$FH$\"3qmn!)f 5*pM#F-7$FM$\"3y*pag'GT(4#F-7$FR$\"3m\\.//G0a=F-7$FW$\"3F?>$fB.ki\"F-7 $Ffn$\"3(yKDz\")*y29F-7$F[o$\"3XMnLy')HG7F-7$F`o$\"3Ynioz[(3/\"F-7$Ffo $\"3YKF/\"=SGo)Fdo7$F[p$\"3y35vM*Qv;(Fdo7$F`p$\"3$H/$RgIHP:f$Fdo7$F_q$\"3ssn.&oeqe#Fdo7$Fdq$\"3A McT]z3N=Fdo7$Fiq$\"3KSsQ=]Ce6Fdo7$F^r$\"3Kg4z9+_ylFbr7$Fdr$\"3e(R$*))> Ri&GFbr7$Fir$\"3e*y3_q]?'yF]s7$F_s$\"3Kyjez7^gSFcs7$Fes$\"3Xfsui,!Gu(F ]s7$Fjs$\"3?iQ_?p3nFFbr7$F_t$\"3w7F-7$Ffw$\"3AE` (\\B@DT\"F-7$F[x$\"3eY:)4X8#H;F-7$F`x$\"3!**)GA]f__=F-7$Fex$\"3'f0$fLN s&4#F-7$Fjx$\"3m,]M+fr^BF-7$F_y$\"3w/Q(pvZ**f#F-7$Fdy$\"3_yd%zTj(**GF- 7$Fiy$\"33mrlk#z<=$F-7$F^z$\"3]_lmS*zo\\$F-7$Fcz$\"3Ou@`!R.C\"QF-7$Fhz Ff[l-F[[l6&F][l$\"*++++\"!\")$F*F*F[el-F_[l6#\"\"\"-F$6%7eo7$F($\"3aPW WWWWWpFdo7$$!3Gmm;HU,\"*[F-$\"3c#yX])>kx!)Fdo7$F/$\"3C*3pE'z)41*Fdo7$$ !3!***\\(=_+so%F-$\"3mwmHe#y6!)*Fdo7$F4$\"3wH?T;F'R/\"F-7$$!3O**\\7`'G d[%F-$\"3)=O8Y@kU5\"F-7$F9$\"3sX`MG!*4`6F-7$$!3#HLLL)QtrUF-$\"39]ES'4- 9>\"F-7$F>$\"3\"z,fetD&>7F-7$$!35+]7GCadSF-$\"3L5,h$[)4Q7F-7$FC$\"37/g [3g%zC\"F-7$$!3ALL$eMu6!RF-$\"3/,eTdJt\\7F-7$$!3w***\\7)ok^QF-$\"3_1>f '3\\)\\7F-7$$!3vmmm;%>@!QF-$\"3W$Ht$)ec$[7F-7$FH$\"3])=WGl<`C\"F-7$$!3 /nmT5j-]OF-$\"3:9+l6-PM7F-7$FM$\"3lTj\"*>Ze<7F-7$FR$\"3)HP1SW]l;\"F-7$ FW$\"3)f&>wQdO(4\"F-7$Ffn$\"3O#[MOQ:9,\"F-7$F[o$\"3G1)fp\\^bE*Fdo7$F`o $\"3\"=()Qlw2>C)Fdo7$Ffo$\"3'y7_)3f+vrFdo7$F[p$\"3mwv7Zm1ShFdo7$F`p$\" 3]H:9\\'Fbr7$Fdr$\"3223<%*H#*RGFbr7$$!3Ihm\"HdGe:$Fdo$\"3UP*o\"*GlVl\"Fbr 7$Fir$\"3'R`]pL)o\\yF]s7$$!3g***\\7yQ16\"Fdo$\"3Srk>b)=]0#F]s7$F_s$\"3 33g.#)z]gSFcs7$$\"3womT5D,`5Fdo$\"3?W]ZZePZ=F]s7$Fes$\"3L6_(=(*43t(F]s 7$$\"3mOLL$e,]6$Fdo$\"3*\\dE95v>h\"Fbr7$Fjs$\"3g$Rs/QtiFbr7$Fdt$\"3`3H6wq')=6Fdo7$Fit$\"3*=u1*4Q[K3Cr)zZ85F-7$F[x$\"3S1(4s:Y$)4\"F-7 $F`x$\"3]XI(4Zbh;\"F-7$Fex$\"3!H!>u2@J<7F-7$$\"3A,]PM&*>^OF-$\"3@2j%fW HXB\"F-7$Fjx$\"3!4!z:lBgX7F-7$$\"39,vV[ko/QF-$\"3)e9i]Zs%[7F-7$$\"3yn; 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Feao$\"3TyJ4z'>+\"=F\\^r7$F\\^q$\"3)fdS9&[]1?F\\^r7$Fgfm$\"3]=-p$3VKA# F\\^r7$F]bo$\"3-Zj(***y=iCF\\^r7$Fbbo$\"3=c#[dDoas#F\\^r7$Fgbo$\"3nJ#* zPJS:IF\\^r7$F\\[lF_\\s-F[el6$F]el#\"#5F_elFb[lF`elFfel" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Maclaurin series for " }{XPPEDIT 18 0 "ln (sin*x/x);" "6#-%#lnG6#*(%$sinG\"\"\"%\"xGF(F)!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "taylor(ln(sin(x)/x),x,16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3 %\"xG#!\"\"\"\"'\"\"##F&\"$!=\"\"%#F&\"%NGF'#F&\"&+y$\"\")#F&\"'vxY\"# 5#!$\"p\"+]s2JQ\"#7#!\"#\"*vDqF\"\"#9-%\"OG6#\"\"\"\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin series for " }{XPPEDIT 18 0 "ln(sin*x/x)" "6#-%#lnG6#*(%$sinG\"\"\"%\"xGF(F)!\"\"" }{TEXT -1 55 " can be expressed in terms of the Bernoulli numbers as " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(sin*x/x)=-``" "6#/-%#lnG 6#*(%$sinG\"\"\"%\"xGF)F*!\"\",$%!GF+" }{XPPEDIT 18 0 "Sum(abs(beta[n] )*``(2^n/(n!*n))*x^n,n = 2 .. infinity)" "6#-%$SumG6$*(-%$absG6#&%%bet aG6#%\"nG\"\"\"-%!G6#*&)\"\"#F-F.*&-%*factorialG6#F-F.F-F.!\"\"F.)%\"x GF-F./F-;F4%)infinityG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-Pi < x;" "6 #2,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 "`` < Pi;" "6#2%!G%#PiG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 274 26 "__________ ________________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "alias(beta=bernoulli):\n-Sum (abs(beta(n))*2^n/(n*n!)*x^n,n=2..14);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$SumG6$*&*(-%$absG6#-%%betaG6#%\"nG\"\"\")\"\"#F/F0 )%\"xGF/F0F0*&F/F0-%*factorialGF.F0!\"\"/F/;F2\"#9F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)%\"xG\"\"#\"\"\"#!\"\"\"\"'*$)F&\"\"%F(#F*\"$!= *$)F&F+F(#F*\"%NG*$)F&\"\")F(#F*\"&+y$*$)F&\"#5F(#F*\"'vxY*$)F&\"#7F(# !$\"p\"+]s2JQ*$)F&\"#9F(#!\"#\"*vDqF\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "By considering complex numbers, and using the relation " }{XPPEDIT 18 0 "sin*x = -i*sinh(i*x);" "6#/*&%$s inG\"\"\"%\"xGF&,$*&%\"iGF&-%%sinhG6#*&F*F&F'F&F&!\"\"" }{TEXT -1 50 " , this series can be obtained from the series for " }{XPPEDIT 18 0 "ln (sinh*x/x);" "6#-%#lnG6#*(%%sinhG\"\"\"%\"xGF(F)!\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Graphical illustrati on" }}{PARA 0 "" 0 "" {TEXT -1 37 "The first picture shows the graph o f " }{XPPEDIT 18 0 "y = ln(sin*x/x);" "6#/%\"yG-%#lnG6#*(%$sinG\"\"\"% \"xGF*F+!\"\"" }{TEXT -1 20 ", which is shown in " }{TEXT 271 5 "black " }{TEXT -1 102 ", along with the graphs of some polynomials obtained \+ by truncating the corresponding Maclaurin series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 282 "p1 := plot( [ln(sin(x)/x),seq(convert(taylor(ln(sin(x)/x),x,2*i),polynom),i=2..7)] ,\n x=-4.5..4.5,y=-5..1,color=[black,red,coral,green,brown,magenta,b lue],\n thickness=[2,1$6]):\np2 := plot([[[-Pi,-5],[-Pi,1]],[[Pi,-5] ,[Pi,1]]],color=black,\n linestyle=3):\nplots[display]([p1,p2]);" }} {PARA 13 "" 1 "" {GLPLOT2D 330 258 258 {PLOTDATA 2 "6--%'CURVESG6%7fo7 $$!32+++w[,RJ!#<$!32@%)*\\xPZ5(F*7$$!3$*******R#[I8$F*$!3$Ra))z;'\\/fF *7$$!3z*****Rg\"3FJF*$!3!y&)Q\"\\,)HP&F*7$$!3-+++n\\6@JF*$!3A?m7M'zm-& F*7$$!3'******>L[^6$F*$!3_q8A^r5pZF*7$$!3>+++&p\"=4JF*$!3G'z$ySLzjXF*7 $$!30+++f]@.JF*$!3q)R6\"Rc'HR%F*7$$!3#)*****>U[s4$F*$!3k%*[Y7XhYUF*7$$ !3#*******[^J&3$F*$!31&G_'4@k/SF*7$$!35+++x=QtIF*$!3,!*>(z#oq3QF*7$$!3 =+++/'[91$F*$!3F!>%H)e))Qk$F*7$$!3#******>L:&\\IF*$!3i\"y7C#y,IF*$!3+tW_?T$*pIF*7$$!3/+ ++PA*z(HF*$!32$p'*=ZUg!HF*7$$!3=+++LA?aHF*$!3)*G5Xk!\\Ow#F*7$$!3y***** zA7/$HF*$!3S[+]h%Hwj#F*7$$!3%)*****HAAm!HF*$!3\\**)zQ51X_#F*7$$!3-+++9 A/fGF*$!3c#*)[oDExK#F*7$$!38+++/AY6GF*$!3iU*ov!H?g@F*7$$!3$******zUyNr #F*$!3[@S%yS)ex=F*7$$!3=+++_Yp:EF*$!3g$HoF002l\"F*7$$!37+++p&y%HDF*$!3 %)RA7$3'4#[\"F*7$$!31+++'[iKW#F*$!3f,^hv'Q]L\"F*7$$!39+++(f.iM#F*$!3?& ))pzQ![*=\"F*7$$!3A+++3Z9\\AF*$!3_*4Q!yZzg5F*7$$!37+++i)HU0#F*$!3h`]cA 27;%)!#=7$$!3#******>H$Rm=F*$!3)*)=Uf'4[$o'Fis7$$!3#******>p>ep\"F*$!3 owN,X3-g`Fis7$$!3#******>=\"*H\\\"F*$!3.:739\\6QSFis7$$!3'*******H)o6K \"F*$!3%HF)G:$G+5$Fis7$$!32+++(e)H@6F*$!3gL;j6')3\">#Fis7$$!3P+++]bzV% *Fis$!3fH3cWGHL:Fis7$$!3Q+++qLr-vFis$!3)f$\\;i-Rk&*!#>7$$!3E+++5TMacFi s$!3Y-.aS.d'Q&F]v7$$!3@+++!pkds$Fis$!376x/fTNCBF]v7$$!39+++Snsa>Fis$!3 Mp#*HIFis$!3Y Bx[OPbziF]w7$$\"3#*******ps:nOFis$!3#yc(3$Gu9D#F]v7$$\"3[******47qKbFi s$!3U6)=8M*)[:&F]v7$$\"3=+++]1**fuFis$!3;%Ht&\\!\\OX*F]v7$$\"3W+++gtYX $*Fis$!3ia15\\?_+:Fis7$$\"3&******HIwp6\"F*$!3Y!frRbMM<#Fis7$$\"36+++M K`>8F*$!3_)o+9&[#=4$Fis7$$\"3.+++X'R:]\"F*$!3'f%*GKv6*)3%Fis7$$\"3'*** ***zLqep\"F*$!3_Lk$p:$Qg`Fis7$$\"3-+++GG'>(=F*$!3EX))*pj2/t'Fis7$$\"3/ +++K%yW1#F*$!3]&Q;&HBO?&)Fis7$$\"3#)*****Hh?cC#F*$!3A'y1rs'Rc5F*7$$\"3 %*******)\\$HSBF*$!3o]:q]\"*>\"=\"F*7$$\"3:+++'Qm\\V#F*$!3%>'=)Q&HF*$!3)\\E7lxgIF*7$$ \"33+++(H1W-$F*$!3w%pP!\\W*HD$F*7$$\"3'******Him#[IF*$!3_Q&*Hm8o([$F*7 $$\"3?+++(y'>gIF*$!3#fk3\"oW,GOF*7$$\"3#*******\\p7sIF*$!33^1xPm4!z$F* 7$$\"33+++8r0%3$F*$!3%RMLN=]@)RF*7$$\"39+++vs)f4$F*$!3Ke&e#HJ==UF*7$$ \"38+++eB&>5$F*$!3k(*3uCF?gVF*7$$\"3&*******Qu\"z5$F*$!3!*zX%H1N^_%F*7 $$\"3&)*****4_#)Q6$F*$!36%Q_.Yb>s%F*7$$\"36+++-w%)>JF*$!3K38oC?Bm\\F*7 $$\"3$******Ho7e7$F*$!331'foj\\()G&F*7$$\"3$)*****\\wx<8$F*$!3g[o2x;Zl dF*7$$\"34+++YGuPJF*$!3U=vN\"GQKq'F*7$%*undefinedGFdbl-%'COLOURG6&%$RG BG\"\"!FiblFibl-%*THICKNESSG6#\"\"#-F$6%7S7$$!3++++++++XF*$!3+++++++vL F*7$$!3%)****\\7c#QI%F*$!3%H$=!Q[_r3$F*7$$!3=+](oKNJ8%F*$!3U&oV]gMr%GF *7$$!3s***\\([iIZg32@F*7$$!3++](ovKtP$F*$!3;\"[V%eF1,>F*7$$!35+]( =g9F>$F*$!3u1V]@W!*)p\"F*7$$!3)***\\(=C#y,IF*$!3X$os7x#y,:F*7$$!3;+]P/ AY6GF*$!3%f&[5@mQ<8F*7$$!3x****\\_Yp:EF*$!3$e^>Dv4.9\"F*7$$!34++D'[iKW #F*$!3s`v_%H>#\\**Fis7$$!3u****\\2Z9\\AF*$!3Dl(ea_'3J%)Fis7$$!3=++]i)H U0#F*$!3:;YkY05LqFis7$$!3u****\\#H$Rm=F*$!3I;$[r`1d!eFis7$$!39+](=p>ep \"F*$!3#pO=*yt+$z%Fis7$$!3'*****\\#=\"*H\\\"F*$!3\\8z.&yP]r$Fis7$Fjt$! 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O&F]t7$F^^m$!3nb@PT0OInF]t7$Fc^m$!3&)F]t7$Fh^m$!3w.nO!**oh 0\"F.7$F]_m$!3PDYLVSZ?8F.7$Fb_m$!3%3V8&**pi_;F.7$Fg_m$!3]@<<%pM`7#F.7$ F\\`m$!3I\"3\"\\PFH)y#F.7$Fa`m$!3YXPBV#GL&QF.7$Ff`m$!3+0s7$Q4rk&F.7$F[ am$!3=d%\\K#3:O&)F.7$F`am$!3'[$*zBTp:Z\"F_[o7$F]in$!37KC9Vy]<>F_[o7$Fe am$!3Em%R!*=T]_#F_[o7$Fh]n$!3)3)eR4n7=MF_[o7$Fjam$!3))\\Rm+lWhYF_[o7$F a^q$!37/jF_[o7$Fi^q$!3w!R%oeueStF_[o7$F _bm$!3WV5&RV'e_&)F_[o7$$\"3MCcE-G*)GVF.$!3M(=ls;BiG*F_[o7$F_fo$!3EBOX( [z$35F[[s7$$\"3Gvo/t0yxVF.$!3K!p=sSq]4\"F[[s7$Fh^n$!3#=AG+yy#*=\"F[[s7 $Fg\\r$!3aa!GQ0N;H\"F[[s7$Fgfo$!3'GJF\\%*>GS\"F[[s7$F_]r$!3kD%H0@oN_\" F[[s7$FdbmFijr-Fgbm6$Fibm#\"#5F[cmF^clF\\cmFbcm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "d/dx" "6#*&%\" dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ln(cosh*x)] = tan h*x;" "6#/7#-%#lnG6#*&%%coshG\"\"\"%\"xGF**&%%tanhGF*F+F*" }{TEXT -1 27 ", the Maclaurin series for " }{XPPEDIT 18 0 "ln(cosh*x)" "6#-%#lnG 6#*&%%coshG\"\"\"%\"xGF(" }{TEXT -1 57 " can be obtained by integratin g the Maclaurin series for " }{XPPEDIT 18 0 "tanh*x" "6#*&%%tanhG\"\" \"%\"xGF%" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 275 1 " " }{XPPEDIT 18 0 "tanh*x = Sum(beta[n]*``(2^n*(2^n-1)/n!)*x^(n -1),n = 2 .. infinity);" "6#/*&%%tanhG\"\"\"%\"xGF&-%$SumG6$*(&%%betaG 6#%\"nGF&-%!G6#*()\"\"#F/F&,&)F5F/F&F&!\"\"F&-%*factorialG6#F/F8F&)F', &F/F&F&F8F&/F/;F5%)infinityG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-Pi/2 \+ < x;" "6#2,$*&%#PiG\"\"\"\"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` < Pi/2; " "6#2%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "Check that the first few terms of the Maclaurin series ex pansion for " }{XPPEDIT 18 0 "ln(cosh*x)" "6#-%#lnG6#*&%%coshG\"\"\"% \"xGF(" }{TEXT -1 30 " coincide with those given by:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(cosh*x) = Sum(beta[n]*``(2^n*(2 ^n-1)/(n!*n))*x^n,n = 2 .. infinity);" "6#/-%#lnG6#*&%%coshG\"\"\"%\"x GF)-%$SumG6$*(&%%betaG6#%\"nGF)-%!G6#*()\"\"#F2F),&)F8F2F)F)!\"\"F)*&- %*factorialG6#F2F)F2F)F;F))F*F2F)/F2;F8%)infinityG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-Pi/2 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 69 "Check that the first \+ few terms of the Maclaurin series expansion for " }{XPPEDIT 18 0 "ln(c os*x)" "6#-%#lnG6#*&%$cosG\"\"\"%\"xGF(" }{TEXT -1 30 " coincide with \+ those given by:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln (cos*x) = -`` " "6#/-%#lnG6#*&%$cosG\"\"\"%\"xGF),$%!G!\"\"" } {XPPEDIT 18 0 "Sum(abs(beta[n])*``(2^n*(1-2^n)/(n!*n))*x^n,n = 2 .. in finity);" "6#-%$SumG6$*(-%$absG6#&%%betaG6#%\"nG\"\"\"-%!G6#*()\"\"#F- F.,&F.F.)F4F-!\"\"F.*&-%*factorialG6#F-F.F-F.F7F.)%\"xGF-F./F-;F4%)inf inityG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-Pi/2 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 33 "_________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG \"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ln(tanh*x/x)] = cot h*x-1/x-tan*x;" "6#/7#-%#lnG6#*(%%tanhG\"\"\"%\"xGF*F+!\"\",(*&%%cothG F*F+F*F**&F*F*F+F,F,*&%$tanGF*F+F*F," }{TEXT -1 27 ", the Maclaurin se ries for " }{XPPEDIT 18 0 "ln(tanh*x/x);" "6#-%#lnG6#*(%%tanhG\"\"\"% \"xGF(F)!\"\"" }{TEXT -1 57 " can be obtained by integrating the Macla urin series for " }{XPPEDIT 18 0 "coth*x-1/x-tanh*x;" "6#,(*&%%cothG\" \"\"%\"xGF&F&*&F&F&F'!\"\"F)*&%%tanhGF&F'F&F)" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 70 "Check that the first few terms of the Mac laurin series expansion for l" }{XPPEDIT 18 0 "n(tanh*x/x);" "6#-%\"nG 6#*(%%tanhG\"\"\"%\"xGF(F)!\"\"" }{TEXT -1 30 " coincide with those gi ven by:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(tanh*x/ x) = Sum(beta[n]*``(2^n*(2-2^n)/(n!*n))*x^n,n = 2 .. infinity);" "6#/- %#lnG6#*(%%tanhG\"\"\"%\"xGF)F*!\"\"-%$SumG6$*(&%%betaG6#%\"nGF)-%!G6# *()\"\"#F3F),&F9F))F9F3F+F)*&-%*factorialG6#F3F)F3F)F+F))F*F3F)/F3;F9% )infinityG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-Pi/2 < x;" "6#2,$*&%#Pi G\"\"\"\"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` < Pi/2;" "6#2%!G*&%#PiG\" \"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 65 "Construct an animation to illustrate the \+ interval of convergence." }}{PARA 0 "" 0 "" {TEXT -1 33 "_____________ ____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 70 "Check that the first \+ few terms of the Maclaurin series expansion for l" }{XPPEDIT 18 0 "n(t an*x/x);" "6#-%\"nG6#*(%$tanG\"\"\"%\"xGF(F)!\"\"" }{TEXT -1 30 " coin cide with those given by:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "ln(tan*x/x) = Sum(abs(beta[n])*``(2^n*(2^n-2)/(n!*n))*x ^n,n = 2 .. infinity);" "6#/-%#lnG6#*(%$tanG\"\"\"%\"xGF)F*!\"\"-%$Sum G6$*(-%$absG6#&%%betaG6#%\"nGF)-%!G6#*()\"\"#F6F),&)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 33 "_______________________________ __" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 }