{VERSION 3 0 "SGI MIPS UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "with(MS): " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 184 "This example looks at the Bernoul li numbers. But for this example, modify the equation, so that it can be demonstrated how common factors of polynomials can be taken out. \+ So examine " }{XPPEDIT 18 0 "(x^2+x)/(x*exp(x)-x+exp(x)-1) = sum(b[i]* x^i/i!,i = 0 .. infinity)/sum(d[j]*x^j/j!,j = 0 .. infinity);" "6#/*&, &*\$%\"xG\"\"#\"\"\"F'F)F),**&F'F)-%\$expG6#F'F)F)F'!\"\"-F-6#F'F)\"\"\" F/F/*&-%\$sumG6\$*(&%\"bG6#%\"iGF))F'F;F)-%*factorialG6#F;F//F;;\"\"!%)i nfinityGF)-F56\$*(&%\"dG6#%\"jGF))F'FJF)-F>6#FJF//FJ;FBFCF/" }{TEXT -1 36 ". Now multisection this by 4 at 2." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "So the bottom can be " }{XPPEDIT 18 0 "product((x*om ega[4]^i+1)*(exp(x*omega[4]^i)-1),i = 0 .. 3) = (product(x*omega[4]^i- 1,i = 0 .. 3)*product(exp(x*omega[4]^i)-1,i = 0 .. 3));" "6#/-%(produc tG6\$*&,&*&%\"xG\"\"\")&%&omegaG6#\"\"%%\"iGF+F+\"\"\"F+F+,&-%\$expG6#*& F*F+)&F.6#\"\"%F1F+F+\"\"\"!\"\"F+/F1;\"\"!\"\"\$*&-F%6\$,&*&F*F+)&F.6# \"\"%F1F+F+\"\"\"F=/F1;F@\"\"\$F+-F%6\$,&-F56#*&F*F+)&F.6#\"\"%F1F+F+\" \"\"F=/F1;F@\"\"\$F+" }{TEXT -1 176 ". So there is a polynomial that c an be factored out. After this simply work out the normal linear rec urrence relation for the bottom. This could have done automatically b y:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "`bottom/ms/factor`((x +1)*(exp(x)-1),f,x,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/-%\"fG6#%\" xG,&-F%6#,&F'\"\"\"!\")F,\"\"%-F%6#,&F'F,!\"%F,!\"\$F%F'7+/-F%6#\"\"!F8 /-F%6#F,F8/-F%6#\"\"#F8/-F%6#\"\"\$F8/-F%6#F.!#C/-F%6#\"\"&F8/-F%6#\"\" 'F8/-F%6#\"\"(F8/-F%6#\"\")\"#c,&*\$)F'F.\"\"\"!\"\"F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Where the last value is the polynomial th at is pulled out." }}{PARA 0 "" 0 "" {TEXT -1 89 "The top can be simil arly manipulated so as to get the common polynomial to be pulled out. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`top/ms/factor`(x^2+x, \+ (x+1)*(exp(x)-1),f,x,4,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6'/-%\"fG6 #%\"xG,&-F%6#,&F'\"\"\"!\"#F,\"\"%-F%6#,&F'F,!\"\"F,!\"\$F%F'7B/-F%6#\" \"!F8/-F%6#F,F8/-F%6#\"\"#F8/-F%6#\"\"\$F8/-F%6#F.F8/-F%6#\"\"&!#5/-F%6 #\"\"'F8/-F%6#\"\"(F8/-F%6#\"\")F8/-F%6#\"\"*\"#I/-F%6#\"#5F8/-F%6#\"# 6F8/-F%6#\"#7F8/-F%6#\"#8!\$I\"/-F%6#\"#9F8/-F%6#\"#:F8/-F%6#\"#;F8/-F% 6#\"#<\"\$5&/-F%6#\"#=F8/-F%6#\"#>F8/-F%6#\"#?F8/-F%6#\"#@!%]?/-F%6#\"# AF8/-F%6#\"#BF8/-F%6#\"#CF8/-F%6#\"#D\"%!>)/-F%6#\"#EF8/-F%6#\"#FF8/-F %6#\"#GF8/-F%6#\"#H!&qF\$/-F%6#FfnF8/-F%6#\"#JF8*,,&F'F,%\"IGF2F,F1F,,& F'F,FetF,F,F'F,,&F'F,F,F,F," }}}}{MARK "0 0 0" 10 }{VIEWOPTS 1 1 0 1 1 1803 }