{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 "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 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 "Ti mes" 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 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 Outpu t" -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 "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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "Multiplication of power series" } }{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 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 56 "From multiplying polynomials to multiplying power serie s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 260 27 "multiplying two polynomials" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x)=a[0]+a[1]*x+a[2] *x^2+ `. . . `+a[n]*x^n" "6#/-%\"pG6#%\"xG,,&%\"aG6#\"\"!\"\"\"*&&F*6# F-F-F'F-F-*&&F*6#\"\"#F-*$F'F4F-F-%'.~.~.~GF-*&&F*6#%\"nGF-)F'F:F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "q(x)=b[0]+b[1]*x+b[2]*x^2+ `. . . `+b [m]*x^m" "6#/-%\"qG6#%\"xG,,&%\"bG6#\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6 #\"\"#F-*$F'F4F-F-%'.~.~.~GF-*&&F*6#%\"mGF-)F'F:F-F-" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 19 "the coefficient of " }{XPPEDIT 18 0 "x^ k" "6#)%\"xG%\"kG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "Sum(a[i]*b[j],i+j = k);" "6#-%$SumG6$*&&%\"aG6#%\"i G\"\"\"&%\"bG6#%\"jGF+/,&F*F+F/F+%\"kG" }{TEXT -1 16 " ------- (i). \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 134 "The following commands perform polynomial multiplication in a way which i llustrates this formula for the coefficients of the product.." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "n := 5:\nm := 6:\npx := Sum(a[j]*x^j,j=0..n);\nqx := Sum(b[j]*x^j ,j=0..m);\nrx := px*qx;\nrx := expand(value(rx)):\nsx := 0:\nfor i fro m 1 to n+m do sx := sx+coeff(rx,x^i)*x^i end do:\nsx;\nn := 'n': m := \+ 'm':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pxG-%$SumG6$*&&%\"aG6#%\"jG \"\"\")%\"xGF,F-/F,;\"\"!\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q xG-%$SumG6$*&&%\"bG6#%\"jG\"\"\")%\"xGF,F-/F,;\"\"!\"\"'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#rxG*&-%$SumG6$*&&%\"aG6#%\"jG\"\"\")%\"xGF-F. /F-;\"\"!\"\"&F.-F'6$*&&%\"bGF,F.F/F./F-;F3\"\"'F." }}{PARA 12 "" 1 " " {XPPMATH 20 "6#,8*&,&*&&%\"aG6#\"\"!\"\"\"&%\"bG6#F+F+F+*&&F(F.F+&F- F)F+F+F+%\"xGF+F+*&,(*&F'F+&F-6#\"\"#F+F+*&&F(F7F+F1F+F+*&F0F+F,F+F+F+ )F2F8F+F+*&,**&F0F+F6F+F+*&F:F+F,F+F+*&F'F+&F-6#\"\"$F+F+*&&F(FCF+F1F+ F+F+)F2FDF+F+*&,,*&F0F+FBF+F+*&F'F+&F-6#\"\"%F+F+*&&F(FMF+F1F+F+*&F:F+ F6F+F+*&FFF+F,F+F+F+)F2FNF+F+*&,.*&F0F+FLF+F+*&F:F+FBF+F+*&FFF+F6F+F+* &FPF+F,F+F+*&F'F+&F-6#\"\"&F+F+*&&F(FfnF+F1F+F+F+)F2FgnF+F+*&,.*&F0F+F enF+F+*&FPF+F6F+F+*&FinF+F,F+F+*&F'F+&F-6#\"\"'F+F+*&FFF+FBF+F+*&F:F+F LF+F+F+)F2FcoF+F+*&,,*&FFF+FLF+F+*&FPF+FBF+F+*&FinF+F6F+F+*&F0F+FaoF+F +*&F:F+FenF+F+F+)F2\"\"(F+F+*&,**&FFF+FenF+F+*&FinF+FBF+F+*&F:F+FaoF+F +*&FPF+FLF+F+F+)F2\"\")F+F+*&,(*&FPF+FenF+F+*&FinF+FLF+F+*&FFF+FaoF+F+ F+)F2\"\"*F+F+*&,&*&FPF+FaoF+F+*&FinF+FenF+F+F+)F2\"#5F+F+*(FinF+)F2\" #6F+FaoF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 30 "Multiplication of power series" }{TEXT -1 87 " is performed by using the same formul a (i) for the coefficients in the product series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " If" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=a[0]+a[1]*x+a[2]*x^2+a[3]*x^3+`. . . `+a[i]*x^i+` . . . `" "6#/-%\"fG6#%\"xG,0&%\"aG6#\"\"!\"\"\"*&&F*6# F-F-F'F-F-*&&F*6#\"\"#F-*$F'F4F-F-*&&F*6#\"\"$F-*$F'F9F-F-%'.~.~.~GF-* &&F*6#%\"iGF-)F'F?F-F-%(~.~.~.~GF-" }{XPPEDIT 18 0 "`` = Sum(a[i]*x^i, i = 0 .. infinity);" "6#/%!G-%$SumG6$*&&%\"aG6#%\"iG\"\"\")%\"xGF,F-/F ,;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "abs(x) " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 46 "Examples of the multiplication of power series" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 27 "We investigate the product " } {XPPEDIT 18 0 "exp(x)*``(1/(1-x)) = exp(x)/(1-x);" "6#/*&-%$expG6#%\"x G\"\"\"-%!G6#*&F)F),&F)F)F(!\"\"F/F)*&-F&6#F(F),&F)F)F(F/F/" }{TEXT -1 39 " in terms of a product of power series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The Maclaurin series expa nsion of " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 3 " is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(x) = 1+x+x^2 /2!+x^3/3!+x^4/4!+` . . . `;" "6#/-%$expG6#%\"xG,.\"\"\"F)F'F)*&F'\"\" #-%*factorialG6#F+!\"\"F)*&F'\"\"$-F-6#F1F/F)*&F'\"\"%-F-6#F5F/F)%(~.~ .~.~GF)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(a[n]*x^n,n = 0 .. infinity);" "6#/%!G-%$SumG6$*&&%\"aG6 #%\"nG\"\"\")%\"xGF,F-/F,;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a[n] = 1/n!;" "6#/&%\"aG 6#%\"nG*&\"\"\"F)-%*factorialG6#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "a := n \+ -> 1/n!;\nSum(a(n)*x^n,n=0..10);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&\"\"\"F--%*f actorialG6#9$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$* &)%\"xG%\"nG\"\"\"-%*factorialG6#F)!\"\"/F);\"\"!\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8\"\"\"F$%\"xGF$*&#F$\"\"#F$)F%F(F$F$*&#F$\"\"'F$ )F%\"\"$F$F$*&#F$\"#CF$)F%\"\"%F$F$*&#F$\"$?\"F$)F%\"\"&F$F$*&#F$\"$?( F$)F%F,F$F$*&#F$\"%S]F$)F%\"\"(F$F$*&#F$\"&?.%F$)F%\"\")F$F$*&#F$\"'!) GOF$)F%\"\"*F$F$*&#F$\"(+)GOF$)F%\"#5F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "taylor(exp(x),x,11 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"xG\"\"\"\"\"!F%F%#F%\"\"#F( #F%\"\"'\"\"$#F%\"#C\"\"%#F%\"$?\"\"\"&#F%\"$?(F*#F%\"%S]\"\"(#F%\"&?. %\"\")#F%\"'!)GO\"\"*#F%\"(+)GO\"#5-%\"OG6#F%\"#6" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The Maclaurin series expansion of " }{XPPEDIT 18 0 "1/(1-x)" "6#*&\"\"\"F$,&F$F$%\"xG!\"\"F'" }{TEXT -1 4 " is" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/(1-x) = 1+x+x^2+x^ 3+x^4+` . . . `;" "6#/*&\"\"\"F%,&F%F%%\"xG!\"\"F(,.F%F%F'F%*$F'\"\"#F %*$F'\"\"$F%*$F'\"\"%F%%(~.~.~.~GF%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(b[n]*x^n,n = 0 .. infinity) ;" "6#/%!G-%$SumG6$*&&%\"bG6#%\"nG\"\"\")%\"xGF,F-/F,;\"\"!%)infinityG " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "b[n] = 1;" "6#/&%\"bG6#%\"nG\"\"\"" }{TEXT -1 10 " for all " } {TEXT 263 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "b := n -> 1;\nSum(b(n)*x^n,n =0..10);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$)%\"xG%\"nG/F(;\"\"!\"#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,8\"\"\"F$%\"xGF$*$)F%\"\"#F$F$*$)F%\" \"$F$F$*$)F%\"\"%F$F$*$)F%\"\"&F$F$*$)F%\"\"'F$F$*$)F%\"\"(F$F$*$)F%\" \")F$F$*$)F%\"\"*F$F$*$)F%\"#5F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "taylor(1/(1-x),x,11);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"xG\"\"\"\"\"!F%F%F%\"\"#F%\"\"$F% \"\"%F%\"\"&F%\"\"'F%\"\"(F%\"\")F%\"\"*F%\"#5-%\"OG6#F%\"#6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "exp(x)*[1 /(1-x)] = exp(x)/(1-x);" "6#/*&-%$expG6#%\"xG\"\"\"7#*&F)F),&F)F)F(!\" \"F-F)*&-F&6#F(F),&F)F)F(F-F-" }{TEXT -1 4 ", so" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(x)/(1-x) = Sum(c[n]*x^n,n = 0 .. i nfinity);" "6#/*&-%$expG6#%\"xG\"\"\",&F)F)F(!\"\"F+-%$SumG6$*&&%\"cG6 #%\"nGF))F(F3F)/F3;\"\"!%)infinityG" }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[n]=Sum(a[i]*b[n-i],i=0..n)" "6 #/&%\"cG6#%\"nG-%$SumG6$*&&%\"aG6#%\"iG\"\"\"&%\"bG6#,&F'F0F/!\"\"F0/F /;\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "c := n -> Sum(a(i)*b(n-i),i=0..n); \nadd(c(n)*x^n,n=0..10);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"cGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$SumG6$*&-%\"aG6#%\"iG\"\" \"-%\"bG6#,&9$F4F3!\"\"F4/F3;\"\"!F9F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8-%$SumG6$*&\"\"\"F(-%*factorialG6#%\"iG!\"\"/F,;\"\"! F0F(*&-F%6$F'/F,;F0F(F(%\"xGF(F(*&-F%6$F'/F,;F0\"\"#F()F6F\"$?(F$)F' \"\"'F$F$*&#\"$&o\"$_#F$)F'\"\"(F$F$*&#\"','4\"\"&?.%F$)F'F.F$F$*&#\"& T')*\"&)GOF$)F'\"\"*F$F$*&#\"(,T')*\"(+)GOF$)F'\"#5F$F$" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "taylo r(exp(x)/(1-x),x,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"xG\"\"\" \"\"!\"\"#F%#\"\"&F'F'#\"\")\"\"$F,#\"#l\"#C\"\"%#\"$j\"\"#gF)#\"%d>\" $?(\"\"'#\"$&o\"$_#\"\"(#\"','4\"\"&?.%F+#\"&T')*\"&)GO\"\"*#\"(,T')* \"(+)GO\"#5-%\"OG6#F%\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 5 "No tes" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "1. The radius of \+ convergence of the power series for " }{XPPEDIT 18 0 "exp(x)/(1-x)" "6 #*&-%$expG6#%\"xG\"\"\",&F(F(F'!\"\"F*" }{TEXT -1 6 " is 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "2. The multipl ication of these power series can be achieved using procedures from th e Maple package " }{TEXT 0 9 "powseries" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "with(p owseries):\nF := 'F': G := 'G':\npowcreate(F(n)=1/n!):\ntpsform(F,x,16 );\npowcreate(G(n)=1):\ntpsform(G,x,16);\nP := multiply(F,G):\ntpsform (P,x,11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+E%\"xG\"\"\"\"\"!F%F%#F% \"\"#F(#F%\"\"'\"\"$#F%\"#C\"\"%#F%\"$?\"\"\"&#F%\"$?(F*#F%\"%S]\"\"(# F%\"&?.%\"\")#F%\"'!)GO\"\"*#F%\"(+)GO\"#5#F%\")+o\"*R\"#6#F%\"*+;+z% \"#7#F%\"++3-Fi\"#8#F%\",+7Hyr)\"#9#F%\".+!oVn28\"#:-%\"OG6#F%\"#;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+E%\"xG\"\"\"\"\"!F%F%F%\"\"#F%\"\"$F% \"\"%F%\"\"&F%\"\"'F%\"\"(F%\"\")F%\"\"*F%\"#5F%\"#6F%\"#7F%\"#8F%\"#9 F%\"#:-%\"OG6#F%\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"xG\"\"\" \"\"!\"\"#F%#\"\"&F'F'#\"\")\"\"$F,#\"#l\"#C\"\"%#\"$j\"\"#gF)#\"%d>\" $?(\"\"'#\"$&o\"$_#\"\"(#\"','4\"\"&?.%F+#\"&T')*\"&)GO\"\"*#\"(,T')* \"(+)GO\"#5-%\"OG6#F%\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 29 "We investigate the product " } {XPPEDIT 18 0 "sin*x*cos*x = sin*2*x/2;" "6#/**%$sinG\"\"\"%\"xGF&%$co sGF&F'F&**F%F&\"\"#F&F'F&F*!\"\"" }{TEXT -1 39 " in terms of a product of power series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The Maclaurin series expansion of " }{XPPEDIT 18 0 "sin*x )" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "sin*x = x+x^3/3!-x^5/5!+x^7/7!-x^9/9!+ ` . . . `;" "6#/*&%$sinG\"\"\"%\"xGF&,.F'F&*&F'\"\"$-%*factorialG6#F*! \"\"F&*&F'\"\"&-F,6#F0F.F.*&F'\"\"(-F,6#F4F.F&*&F'\"\"*-F,6#F8F.F.%(~. ~.~.~GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Sum(a[n]*x^n,n = 0 .. infinity);" "6#/%!G-%$SumG6$ *&&%\"aG6#%\"nG\"\"\")%\"xGF,F-/F,;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a[n]=sin(n*Pi/2) /n!" "6#/&%\"aG6#%\"nG*&-%$sinG6#*(F'\"\"\"%#PiGF-\"\"#!\"\"F--%*facto rialG6#F'F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "a := n -> sin(n*Pi/2)/n!;\nSum(a(n) *x^n,n=0..13);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6 #%\"nG6\"6$%)operatorG%&arrowGF(*&-%$sinG6#,$*&9$\"\"\"%#PiGF3#F3\"\"# F3-%*factorialG6#F2!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$S umG6$*&*&-%$sinG6#,$*&%\"nG\"\"\"%#PiGF.#F.\"\"#F.)%\"xGF-F.F.-%*facto rialG6#F-!\"\"/F-;\"\"!\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0%\"xG \"\"\"*$)F$\"\"$F%#!\"\"\"\"'*$)F$\"\"&F%#F%\"$?\"*$)F$\"\"(F%#F*\"%S] *$)F$\"\"*F%#F%\"'!)GO*$)F$\"#6F%#F*\")+o\"*R*$)F$\"#8F%#F%\"++3-Fi" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "taylor(sin(x),x,14);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG \"\"\"F%#!\"\"\"\"'\"\"$#F%\"$?\"\"\"&#F'\"%S]\"\"(#F%\"'!)GO\"\"*#F' \")+o\"*R\"#6#F%\"++3-Fi\"#8-%\"OG6#F%\"#9" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 34 "The Maclaurin series expansion of " }{XPPEDIT 18 0 "cos *x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "cos*x = 1-x^2/2+x^4/4!-x^6/6!+` . . . \+ `;" "6#/*&%$cosG\"\"\"%\"xGF&,,F&F&*&F'\"\"#F*!\"\"F+*&F'\"\"%-%*facto rialG6#F-F+F&*&F'\"\"'-F/6#F2F+F+%(~.~.~.~GF&" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(b[n]*x^n,n = 0 .. infinity);" "6#/%!G-%$SumG6$*&&%\"bG6#%\"nG\"\"\")%\"xGF,F-/F,; \"\"!%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "b[n] = cos(n*Pi/2)/n!;" "6#/&%\"bG6#%\"nG*&-%$cosG6#*(F'\"\"\"%#PiG F-\"\"#!\"\"F--%*factorialG6#F'F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "b := n -> co s(n*Pi/2)/n!;\nSum(b(n)*x^n,n=0..12);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&-%$cosG6#,$* &9$\"\"\"%#PiGF3#F3\"\"#F3-%*factorialG6#F2!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&*&-%$cosG6#,$*&%\"nG\"\"\"%#PiGF.#F.\" \"#F.)%\"xGF-F.F.-%*factorialG6#F-!\"\"/F-;\"\"!\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$*$)%\"xG\"\"#F$#!\"\"F(*$)F'\"\"%F$#F$\"#C *$)F'\"\"'F$#F*\"$?(*$)F'\"\")F$#F$\"&?.%*$)F'\"#5F$#F*\"(+)GO*$)F'\"# 7F$#F$\"*+;+z%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "taylor(cos(x),x,13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"\"\"!#!\"\"\"\"#F)#F%\"#C\"\"%#F(\"$?(\"\" '#F%\"&?.%\"\")#F(\"(+)GO\"#5#F%\"*+;+z%\"#7-%\"OG6#F%\"#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "sin*x*cos*x = sin *2*x/2;" "6#/**%$sinG\"\"\"%\"xGF&%$cosGF&F'F&**F%F&\"\"#F&F'F&F*!\"\" " }{TEXT -1 4 ", so" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "sin*2*x = Sum(c[n]*x^n,n = 0 .. infinity);" "6#/*(%$sinG\"\"\"\"\"# F&%\"xGF&-%$SumG6$*&&%\"cG6#%\"nGF&)F(F0F&/F0;\"\"!%)infinityG" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c [n]=Sum(a[i]*b[n-i],i=0..n)" "6#/&%\"cG6#%\"nG-%$SumG6$*&&%\"aG6#%\"iG \"\"\"&%\"bG6#,&F'F0F/!\"\"F0/F/;\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "c := \+ n -> Sum(a(i)*b(n-i),i=0..n);\nSum(c(n)*x^n,n=0..13);\nvalue(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGf*6#%\"nG6\"6$%)operatorG%&arrow GF(-%$SumG6$*&-%\"aG6#%\"iG\"\"\"-%\"bG6#,&9$F4F3!\"\"F4/F3;\"\"!F9F(F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-F$6$*&*&-%$sinG6#,$ *&%\"iG\"\"\"%#PiGF1#F1\"\"#F1-%$cosG6#,$*&,&%\"nGF1F0!\"\"F1F2F1F3F1F 1*&-%*factorialG6#F0F1-F?6#F:F1F " 0 "" {MPLTEXT 1 0 24 "taylor(sin(2*x)/2,x,14);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"F%#!\"#\"\"$F(#\"\"#\"#: \"\"&#!\"%\"$:$\"\"(#F*\"%NG\"\"*#F.\"'Df:\"#6#\"\"%\"(v53'\"#8-%\"OG6 #F%F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 5 "Notes" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 58 "1. The radius of convergence of all t hree power series is " }{XPPEDIT 18 0 "R=infinity" "6#/%\"RG%)infinity G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "2. We have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "sin*2*x/2 = Sum(sin(n*Pi/2)*2^(n-1)*x^n/n!,n = 2 .. inf inity);" "6#/**%$sinG\"\"\"\"\"#F&%\"xGF&F'!\"\"-%$SumG6$**-F%6#*(%\"n GF&%#PiGF&F'F)F&)F',&F1F&F&F)F&)F(F1F&-%*factorialG6#F1F)/F1;F'%)infin ityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Sum(sin(n*Pi/2)*2^(n-1)/n!*x^n,n=0..13); \nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&*(-%$sinG6#, $*&%\"nG\"\"\"%#PiGF.#F.\"\"#F.)F1,&F-F.!\"\"F.F.)%\"xGF-F.F.-%*factor ialG6#F-F4/F-;\"\"!\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0%\"xG\"\" \"*$)F$\"\"$F%#!\"#F(*$)F$\"\"&F%#\"\"#\"#:*$)F$\"\"(F%#!\"%\"$:$*$)F$ \"\"*F%#F/\"%NG*$)F$\"#6F%#F5\"'Df:*$)F$\"#8F%#\"\"%\"(v53'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "3. The multipl ication of these power series can be acheived using procedures from th e Maple package " }{TEXT 0 9 "powseries" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "with(p owseries):\nF := 'F': G := 'G':\npowcreate(F(n)=sin(n*Pi/2)/n!):\ntpsf orm(F,x,14);\npowcreate(G(n)=cos(n*Pi/2)/n!):\ntpsform(G,x,14);\nP := \+ multiply(F,G):\ntpsform(P,x,14);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3 %\"xG\"\"\"F%#!\"\"\"\"'\"\"$#F%\"$?\"\"\"&#F'\"%S]\"\"(#F%\"'!)GO\"\" *#F'\")+o\"*R\"#6#F%\"++3-Fi\"#8-%\"OG6#F%\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"\"\"!#!\"\"\"\"#F)#F%\"#C\"\"%#F(\"$?(\"\" '#F%\"&?.%\"\")#F(\"(+)GO\"#5#F%\"*+;+z%\"#7-%\"OG6#F%\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"F%#!\"#\"\"$F(#\"\"#\"#:\"\"&#! \"%\"$:$\"\"(#F*\"%NG\"\"*#F.\"'Df:\"#6#\"\"%\"(v53'\"#8-%\"OG6#F%\"#9 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 27 "We investigate the product " }{XPPEDIT 18 0 "exp(x)*exp(x ) = exp(2*x);" "6#/*&-%$expG6#%\"xG\"\"\"-F&6#F(F)-F&6#*&\"\"#F)F(F)" }{TEXT -1 39 " in terms of a product of power series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The Maclaurin series e xpansion of " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(x) = 1+x+ x^2/2!+x^3/3!+x^4/4!+` . . . `;" "6#/-%$expG6#%\"xG,.\"\"\"F)F'F)*&F' \"\"#-%*factorialG6#F+!\"\"F)*&F'\"\"$-F-6#F1F/F)*&F'\"\"%-F-6#F5F/F)% (~.~.~.~GF)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Sum(a[n]*x^n,n = 0 .. infinity);" "6#/%!G-%$SumG6$ *&&%\"aG6#%\"nG\"\"\")%\"xGF,F-/F,;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a[n] = 1/n!;" "6 #/&%\"aG6#%\"nG*&\"\"\"F)-%*factorialG6#F'!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "a := n -> 1/n!;\nSum(a(n)*x^n,n=0..10);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&\"\"\"F --%*factorialG6#9$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Su mG6$*&)%\"xG%\"nG\"\"\"-%*factorialG6#F)!\"\"/F);\"\"!\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8\"\"\"F$%\"xGF$*$)F%\"\"#F$#F$F(*$)F%\"\"$ F$#F$\"\"'*$)F%\"\"%F$#F$\"#C*$)F%\"\"&F$#F$\"$?\"*$)F%F.F$#F$\"$?(*$) F%\"\"(F$#F$\"%S]*$)F%\"\")F$#F$\"&?.%*$)F%\"\"*F$#F$\"'!)GO*$)F%\"#5F $#F$\"(+)GO" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "taylor(exp(x),x,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#F%\"\"'\"\"$#F%\"#C\"\"%#F%\"$? \"\"\"&#F%\"$?(F*#F%\"%S]\"\"(#F%\"&?.%\"\")#F%\"'!)GO\"\"*#F%\"(+)GO \"#5-%\"OG6#F%\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "exp(x)*exp(x) = exp(2*x); " "6#/*&-%$expG6#%\"xG\"\"\"-F&6#F(F)-F&6#*&\"\"#F)F(F)" }{TEXT -1 4 " , so" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(2*x) = S um(c[n]*x^n,n = 0 .. infinity);" "6#/-%$expG6#*&\"\"#\"\"\"%\"xGF)-%$S umG6$*&&%\"cG6#%\"nGF))F*F2F)/F2;\"\"!%)infinityG" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[n] = Sum(a[i]*a[ n-i],i = 0 .. n);" "6#/&%\"cG6#%\"nG-%$SumG6$*&&%\"aG6#%\"iG\"\"\"&F-6 #,&F'F0F/!\"\"F0/F/;\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "c := n -> Sum(a(i) *a(n-i),i=0..n);\nSum(c(n)*x^n,n=0..10);\nvalue(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"cGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$SumG6$*&- %\"aG6#%\"iG\"\"\"-F16#,&9$F4F3!\"\"F4/F3;\"\"!F8F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$SumG6$*&-F$6$*&\"\"\"F**&-%*factorialG6#%\"iG F*-F-6#,&%\"nGF*F/!\"\"F*F4/F/;\"\"!F3F*)%\"xGF3F*/F3;F7\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8\"\"\"F$%\"xG\"\"#*$)F%F&F$F&*$)F%\"\"$F$# \"\"%F+*$)F%F-F$#F&F+*$)F%\"\"&F$#F-\"#:*$)F%\"\"'F$#F-\"#X*$)F%\"\"(F $#\"\")\"$:$*$)F%F?F$#F&F@*$)F%\"\"*F$#F-\"%NG*$)F%\"#5F$#F-\"&vT\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "taylor(exp(2*x),x,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"x G\"\"\"\"\"!\"\"#F%F'F'#\"\"%\"\"$F*#F'F*F)#F)\"#:\"\"&#F)\"#X\"\"'#\" \")\"$:$\"\"(#F'F4F3#F)\"%NG\"\"*#F)\"&vT\"\"#5-%\"OG6#F%\"#6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 5 "Notes" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 53 "1. The radius of convergence of the power series for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "exp(2*x)" "6#-%$expG6#*&\"\"#\"\"\"%\"xGF(" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "R=infinity" "6#/%\"RG%)infinityG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "2. We have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "exp(2*x) = Sum(2^n*x^n/n!,n = 0 .. infinity);" "6#/-%$e xpG6#*&\"\"#\"\"\"%\"xGF)-%$SumG6$*()F(%\"nGF))F*F0F)-%*factorialG6#F0 !\"\"/F0;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Sum(2^n/n!*x^n,n=0..1 0);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&*&)\"\"#% \"nG\"\"\")%\"xGF*F+F+-%*factorialG6#F*!\"\"/F*;\"\"!\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8\"\"\"F$%\"xG\"\"#*$)F%F&F$F&*$)F%\"\"$F$#\" \"%F+*$)F%F-F$#F&F+*$)F%\"\"&F$#F-\"#:*$)F%\"\"'F$#F-\"#X*$)F%\"\"(F$# \"\")\"$:$*$)F%F?F$#F&F@*$)F%\"\"*F$#F-\"%NG*$)F%\"#5F$#F-\"&vT\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "3. The m ultiplication of these power series can be acheived using procedures f rom the Maple package " }{TEXT 0 9 "powseries" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "with(powseries):\nF := 'F':\npowcreate(F(n)=1/n!):\ntpsform(F,x,1 1);\nP := multiply(F,F):\ntpsform(P,x,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#F%\"\"'\"\"$#F%\"#C\" \"%#F%\"$?\"\"\"&#F%\"$?(F*#F%\"%S]\"\"(#F%\"&?.%\"\")#F%\"'!)GO\"\"*# F%\"(+)GO\"#5-%\"OG6#F%\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"xG \"\"\"\"\"!\"\"#F%F'F'#\"\"%\"\"$F*#F'F*F)#F)\"#:\"\"&#F)\"#X\"\"'#\" \")\"$:$\"\"(#F'F4F3#F)\"%NG\"\"*#F)\"&vT\"\"#5-%\"OG6#F%\"#6" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 27 "We investigate the product " }{XPPEDIT 18 0 "sin*x*s ec*x = tan*x;" "6#/**%$sinG\"\"\"%\"xGF&%$secGF&F'F&*&%$tanGF&F'F&" } {TEXT -1 39 " in terms of a product of power series." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The Maclaurin series ex pansion of " }{XPPEDIT 18 0 "sin*x)" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "sin*x = x+x^3/3!-x^5/5!+x^7/7!-x^9/9!+` . . . `;" "6#/*&%$sinG\"\"\"%\"xGF&,. F'F&*&F'\"\"$-%*factorialG6#F*!\"\"F&*&F'\"\"&-F,6#F0F.F.*&F'\"\"(-F,6 #F4F.F&*&F'\"\"*-F,6#F8F.F.%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(a[n]*x^n,n = 0 .. infin ity);" "6#/%!G-%$SumG6$*&&%\"aG6#%\"nG\"\"\")%\"xGF,F-/F,;\"\"!%)infin ityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "a[n]=sin(n*Pi/2)/n!" "6#/&%\"aG6#%\"nG*&-%$sinG6#*(F'\" \"\"%#PiGF-\"\"#!\"\"F--%*factorialG6#F'F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "a := n -> sin(n*Pi/2)/n!;\nSum(a(n)*x^n,n=0..15);\nvalue(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&-%$si nG6#,$*&9$\"\"\"%#PiGF3#F3\"\"#F3-%*factorialG6#F2!\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&*&-%$sinG6#,$*&%\"nG\"\"\"% #PiGF.#F.\"\"#F.)%\"xGF-F.F.-%*factorialG6#F-!\"\"/F-;\"\"!\"#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,2%\"xG\"\"\"*$)F$\"\"$F%#!\"\"\"\"'*$ )F$\"\"&F%#F%\"$?\"*$)F$\"\"(F%#F*\"%S]*$)F$\"\"*F%#F%\"'!)GO*$)F$\"#6 F%#F*\")+o\"*R*$)F$\"#8F%#F%\"++3-Fi*$)F$\"#:F%#F*\".+!oVn28" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "taylor(sin(x),x,16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\" \"\"F%#!\"\"\"\"'\"\"$#F%\"$?\"\"\"&#F'\"%S]\"\"(#F%\"'!)GO\"\"*#F'\") +o\"*R\"#6#F%\"++3-Fi\"#8#F'\".+!oVn28\"#:-%\"OG6#F%\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The Maclaurin series expansion of " } {XPPEDIT 18 0 "sec*x" "6#*&%$secG\"\"\"%\"xGF%" }{TEXT -1 3 " is" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "sec*x = 1+x^2/2+5*x^ 4/24+61*x^6/720+277*x^8/8064+50521*x^10/3628800+540553*x^12/95800320+1 99360981*x^14/87178291200+` . . . `;" "6#/*&%$secG\"\"\"%\"xGF&,4F&F&* &F'\"\"#F*!\"\"F&*(\"\"&F&*$F'\"\"%F&\"#CF+F&*(\"#hF&*$F'\"\"'F&\"$?(F +F&*(\"$x#F&*$F'\"\")F&\"%k!)F+F&*(\"&@0&F&*$F'\"#5F&\"(+)GOF+F&*(\"'` 0aF&*$F'\"#7F&\")?.!e*F+F&*(\"*\")4O*>F&*$F'\"#9F&\",+7Hyr)F+F&%(~.~.~ .~GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(b[n]*x^n,n = 0 .. infinity);" "6#/%!G-%$SumG6$*&&%\"bG6 #%\"nG\"\"\")%\"xGF,F-/F,;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "b[n] = abs(epsilon[n])/n !;" "6#/&%\"bG6#%\"nG*&-%$absG6#&%(epsilonG6#F'\"\"\"-%*factorialG6#F' !\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "epsilon[n]" "6#&%(epsilonG 6#%\"nG" }{TEXT -1 26 " is the n th Euler number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "alias(epsilo n=euler):\nb := n -> abs(epsilon(n))/n!;\nSum(b(n)*x^n,n=0..15);\nvalu e(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGf*6#%\"nG6\"6$%)operato rG%&arrowGF(*&-%$absG6#-%(epsilonG6#9$\"\"\"-%*factorialGF2!\"\"F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&*&-%$absG6#-%(epsilonG6 #%\"nG\"\"\")%\"xGF.F/F/-%*factorialGF-!\"\"/F.;\"\"!\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2\"\"\"F$*$)%\"xG\"\"#F$#F$F(*$)F'\"\"%F$#\"\" &\"#C*$)F'\"\"'F$#\"#h\"$?(*$)F'\"\")F$#\"$x#\"%k!)*$)F'\"#5F$#\"&@0& \"(+)GO*$)F'\"#7F$#\"'`0a\")?.!e**$)F'\"#9F$#\"*\")4O*>\",+7Hyr)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "taylor(sec(x),x,14);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\" \"\"\"\"!#F%\"\"#F(#\"\"&\"#C\"\"%#\"#h\"$?(\"\"'#\"$x#\"%k!)\"\")#\"& @0&\"(+)GO\"#5#\"'`0a\")?.!e*\"#7-%\"OG6#F%\"#9" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "sin*x*sec*x = tan*x;" "6#/** %$sinG\"\"\"%\"xGF&%$secGF&F'F&*&%$tanGF&F'F&" }{TEXT -1 4 ", so" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "tan*x = Sum(c[n]*x^n ,n = 0 .. infinity);" "6#/*&%$tanG\"\"\"%\"xGF&-%$SumG6$*&&%\"cG6#%\"n GF&)F'F/F&/F/;\"\"!%)infinityG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[n]=Sum(a[i]*b[n-i],i=0..n)" "6#/ &%\"cG6#%\"nG-%$SumG6$*&&%\"aG6#%\"iG\"\"\"&%\"bG6#,&F'F0F/!\"\"F0/F/; \"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "c := n -> Sum(a(i)*b(n-i),i=0..n); \nSum(c(n)*x^n,n=0..15);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"cGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$SumG6$*&-%\"aG6#%\"iG\"\" \"-%\"bG6#,&9$F4F3!\"\"F4/F3;\"\"!F9F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-F$6$*&*&-%$sinG6#,$*&%\"iG\"\"\"%#PiGF1#F1 \"\"#F1-%$absG6#-%(epsilonG6#,&%\"nGF1F0!\"\"F1F1*&-%*factorialG6#F0F1 -F@F:F1F=/F0;\"\"!F " 0 "" {MPLTEXT 1 0 20 "taylor(tan(x ),x,16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"F%#F%\"\"$F'# \"\"#\"#:\"\"&#\"#<\"$:$\"\"(#\"#i\"%NG\"\"*#\"%#Q\"\"'Df:\"#6#\"&W=# \"(v53'\"#8#\"'p&H*\"*vG^Q'F*-%\"OG6#F%\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 5 "Notes" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 24 "1. The power series for " }{XPPEDIT 18 0 "sec*x)" "6#*&%$secG\"\"\"% \"xGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan*x)" "6#*&%$tanG\"\"\"% \"xGF%" }{TEXT -1 33 " each have radius of convergence " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "2. We have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "tan*x = Sum(abs(beta[n])*` `(2^n*(2^n-1)/n!)*x^(n-1),n = 2 .. infinity);" "6#/*&%$tanG\"\"\"%\"xG F&-%$SumG6$*(-%$absG6#&%%betaG6#%\"nGF&-%!G6#*()\"\"#F2F&,&)F8F2F&F&! \"\"F&-%*factorialG6#F2F;F&)F',&F2F&F&F;F&/F2;F8%)infinityG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "beta[n] " "6#&%%betaG6#%\"nG" }{TEXT -1 8 " is the " }{TEXT 264 1 "n" }{TEXT -1 21 " th Bernoulli number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "alias(beta=bernoulli):\nSum( abs(beta(n))*2^n*(2^n-1)/n!*x^(n-1),n=2..16);\nvalue(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$SumG6$*,-%$absG6#-%%betaG6#%\"nG\"\"\")\"\"#F -F.,&F/F.F.!\"\"F.-%*factorialGF,F2)%\"xG,&F-F.F.F2F./F-;F0\"#;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,2%\"xG\"\"\"*&#F%\"\"$F%*$)F$F(F%F%F% *&#\"\"#\"#:F%*$)F$\"\"&F%F%F%*&#\"#<\"$:$F%*$)F$\"\"(F%F%F%*&#\"#i\"% NGF%*$)F$\"\"*F%F%F%*&#\"%#Q\"\"'Df:F%*$)F$\"#6F%F%F%*&#\"&W=#\"(v53'F %*$)F$\"#8F%F%F%*&#\"'p&H*\"*vG^Q'F%*$)F$F.F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "3. The multiplication of these power series can be achieved using procedures from the Maple pa ckage " }{TEXT 0 9 "powseries" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "with(powser ies):\nF := 'F': G := 'G':\npowcreate(F(n)=sin(n*Pi/2)/n!):\ntpsform(F ,x,16);\npowcreate(G(n)=abs(euler(n))/n!):\ntpsform(G,x,16);\nP := mul tiply(F,G):\ntpsform(P,x,16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\" xG\"\"\"F%#!\"\"\"\"'\"\"$#F%\"$?\"\"\"&#F'\"%S]\"\"(#F%\"'!)GO\"\"*#F '\")+o\"*R\"#6#F%\"++3-Fi\"#8#F'\".+!oVn28\"#:-%\"OG6#F%\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"\"\"!#F%\"\"#F(#\"\"&\"#C\"\"%# \"#h\"$?(\"\"'#\"$x#\"%k!)\"\")#\"&@0&\"(+)GO\"#5#\"'`0a\")?.!e*\"#7# \"*\")4O*>\",+7Hyr)\"#9-%\"OG6#F%\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"F%#F%\"\"$F'#\"\"#\"#:\"\"&#\"#<\"$:$\"\"(#\"#i\"%NG\" \"*#\"%#Q\"\"'Df:\"#6#\"&W=#\"(v53'\"#8#\"'p&H*\"*vG^Q'F*-%\"OG6#F%\"# ;" }}}{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 "" }}{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 }