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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 25 "Manipulating power series" }}
{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}
{PARA 0 "" 0 "" {TEXT -1 18 "Version:  4.2.2008" }}{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 48 "Introductory example of addition of power series" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 5 "Since" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1+x
) = Sum((-1)^(n+1)*``(x^n/n),n = 1 .. infinity);" "6#/-%#lnG6#,&\"\"\"
F(%\"xGF(-%$SumG6$*&),$F(!\"\",&%\"nGF(F(F(F(-%!G6#*&)F)F2F(F2F0F(/F2;
F(%)infinityG" }{TEXT -1 4 "  = " }{XPPEDIT 18 0 "x-x^2/2+x^3/3-x^4/4+
x^5/5-x^6/6+x^7/7+` . . . `;" "6#,2%\"xG\"\"\"*&F$\"\"#F'!\"\"F(*&F$\"
\"$F*F(F%*&F$\"\"%F,F(F(*&F$\"\"&F.F(F%*&F$\"\"'F0F(F(*&F$\"\"(F2F(F%%
(~.~.~.~GF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1<x" "6#2,$\"\"\"!\"\"%
\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 2 ", " }}{PARA 
0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "ln(1-x) = Sum(-x^n/n,n = 1 .. infinity);" "6#/-%#lnG6#,
&\"\"\"F(%\"xG!\"\"-%$SumG6$,$*&)F)%\"nGF(F1F*F*/F1;F(%)infinityG" }
{TEXT -1 4 "  = " }{XPPEDIT 18 0 "-x-x^2/2-x^3/3-x^4/4-x^5/5-x^6/6-x^7
/7-` . . . `;" "6#,2%\"xG!\"\"*&F$\"\"#F'F%F%*&F$\"\"$F)F%F%*&F$\"\"%F
+F%F%*&F$\"\"&F-F%F%*&F$\"\"'F/F%F%*&F$\"\"(F1F%F%%(~.~.~.~GF%" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!\"\"%\"xG" }
{XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" 
{TEXT -1 8 "we have " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " ln(1+x)+ln(1
-x) = ``" "6#/,&-%#lnG6#,&\"\"\"F)%\"xGF)F)-F&6#,&F)F)F*!\"\"F)%!G" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``(x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/6+x^7/
7+` . . . `)+``(-x-x^2/2-x^3/3-x^4/4-x^5/5-x^6/6-x^7/7-` . . . `)" "6#
,&-%!G6#,2%\"xG\"\"\"*&F(\"\"#F+!\"\"F,*&F(\"\"$F.F,F)*&F(\"\"%F0F,F,*
&F(\"\"&F2F,F)*&F(\"\"'F4F,F,*&F(\"\"(F6F,F)%(~.~.~.~GF)F)-F%6#,2F(F,*
&F(F+F+F,F,*&F(F.F.F,F,*&F(F0F0F,F,*&F(F2F2F,F,*&F(F4F4F,F,*&F(F6F6F,F
,F7F,F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "-x^2 - x^4/2 - x^6/3 - x^8/4 - \+
x^10/5 - ` . . . `" "6#,.*$%\"xG\"\"#!\"\"*&F%\"\"%F&F'F'*&F%\"\"'\"\"
$F'F'*&F%\"\")F)F'F'*&F%\"#5\"\"&F'F'%(~.~.~.~GF'" }{TEXT -1 3 " , " }
{XPPEDIT 18 0 "-1<x" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<1" "6#
2%!G\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 6 "where " }{TEXT 261 56 "corresponding terms of th
e two series are added together" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 94 "Note that the sum of the two power series obtained in thi
s way gives the Maclaurin series for " }{XPPEDIT 18 0 "ln(1-x^2)" "6#-
%#lnG6#,&\"\"\"F'*$%\"xG\"\"#!\"\"" }{TEXT -1 6 ", and " }}{PARA 256 "
" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "ln(1+x) + ln(1-x) = ln((1+x)*(1
-x))" "6#/,&-%#lnG6#,&\"\"\"F)%\"xGF)F)-F&6#,&F)F)F*!\"\"F)-F&6#*&,&F)
F)F*F)F),&F)F)F*F.F)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "ln(1-x^2)" "6#
-%#lnG6#,&\"\"\"F'*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "The addition of power \+
series by adding corresponding terms can be regarded as an obvious ext
ension of the addition of polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "taylor(ln(1+x),x,11);\np
x := convert(%,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"xG\"
\"\"F%#!\"\"\"\"#F(#F%\"\"$F*#F'\"\"%F,#F%\"\"&F.#F'\"\"'F0#F%\"\"(F2#
F'\"\")F4#F%\"\"*F6#F'\"#5F8-%\"OG6#F%\"#6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#pxG,6%\"xG\"\"\"*$)F&\"\"#F'#!\"\"F**$)F&\"\"$F'#F'F
/*$)F&\"\"%F'#F,F3*$)F&\"\"&F'#F'F7*$)F&\"\"'F'#F,F;*$)F&\"\"(F'#F'F?*
$)F&\"\")F'#F,FC*$)F&\"\"*F'#F'FG*$)F&\"#5F'#F,FK" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "taylor(ln(1-
x),x,11);\nqx := convert(%,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#+9%\"xG!\"\"\"\"\"#F%\"\"#F(#F%\"\"$F*#F%\"\"%F,#F%\"\"&F.#F%\"\"'F0
#F%\"\"(F2#F%\"\")F4#F%\"\"*F6#F%\"#5F8-%\"OG6#F&\"#6" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#qxG,6%\"xG!\"\"*$)F&\"\"#\"\"\"#F'F**$)F&\"\"$F
+#F'F/*$)F&\"\"%F+#F'F3*$)F&\"\"&F+#F'F7*$)F&\"\"'F+#F'F;*$)F&\"\"(F+#
F'F?*$)F&\"\")F+#F'FC*$)F&\"\"*F+#F'FG*$)F&\"#5F+#F'FK" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "px + qx;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\"\"#\"\"\"!\"\"*$)F&\"
\"%F(#F)F'*$)F&\"\"'F(#F)\"\"$*$)F&\"\")F(#F)F,*$)F&\"#5F(#F)\"\"&" }}
}{PARA 0 "" 0 "" {TEXT -1 70 "Compare this Taylor polynomial with the \+
order 11 Maclaurin series for " }{XPPEDIT 18 0 "ln(1-x^2)" "6#-%#lnG6#
,&\"\"\"F'*$%\"xG\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "taylor(ln(1-x^2),x
,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG!\"\"\"\"##F%F&\"\"%#F
%\"\"$\"\"'#F%F(\"\")#F%\"\"&\"#5-%\"OG6#\"\"\"\"#6" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Formal addition of power series" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 18 "Given power series" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "f(x) = Sum(a[n]*x^n,n = 0 .. infinity);" "6#/-%\"fG6#%
\"xG-%$SumG6$*&&%\"aG6#%\"nG\"\"\")F'F/F0/F/;\"\"!%)infinityG" }{TEXT 
-1 3 " = " }{XPPEDIT 18 0 "a[0]+a[1]*x+a[2]*x^2+a[3]*x^3+` . . . `" "6
#,,&%\"aG6#\"\"!\"\"\"*&&F%6#F(F(%\"xGF(F(*&&F%6#\"\"#F(*$F,F0F(F(*&&F
%6#\"\"$F(*$F,F5F(F(%(~.~.~.~GF(" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "ab
s(x) < r[1];" "6#2-%$absG6#%\"xG&%\"rG6#\"\"\"" }{TEXT -1 2 "  " }}
{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "g(x) = Sum(b[n]*x^n,n = 0 .. infinity);" "6#/-%\"gG6#%
\"xG-%$SumG6$*&&%\"bG6#%\"nG\"\"\")F'F/F0/F/;\"\"!%)infinityG" }{TEXT 
-1 2 "= " }{XPPEDIT 18 0 "b[0]+b[1]*x+b[2]*x^2+b[3]*x^3+` . . . `" "6#
,,&%\"bG6#\"\"!\"\"\"*&&F%6#F(F(%\"xGF(F(*&&F%6#\"\"#F(*$F,F0F(F(*&&F%
6#\"\"$F(*$F,F5F(F(%(~.~.~.~GF(" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "abs
(x) < r[2]" "6#2-%$absG6#%\"xG&%\"rG6#\"\"#" }{TEXT -1 2 "  " }}{PARA 
0 "" 0 "" {TEXT -1 18 "we can define the " }{TEXT 261 16 "sum power se
ries" }{TEXT -1 59 " by adding the corresponding terms of the two powe
r series." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x)+g(x) = Sum(a[n]*x^n,n = 0 .. infi
nity)+Sum(b[n]*x^n,n = 0 .. infinity);" "6#/,&-%\"fG6#%\"xG\"\"\"-%\"g
G6#F(F),&-%$SumG6$*&&%\"aG6#%\"nGF))F(F5F)/F5;\"\"!%)infinityGF)-F/6$*
&&%\"bG6#F5F))F(F5F)/F5;F9F:F)" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "abs(
x)<min(r[1],r[2])" "6#2-%$absG6#%\"xG-%$minG6$&%\"rG6#\"\"\"&F,6#\"\"#
" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " =  " }{XPPEDIT 18 
0 "``(a[0]+a[1]*x+a[2]*x^2+a[3]*x^3+` . . . `)+``(b[0]+b[1]*x+b[2]*x^2
+b[3]*x^3+` . . . `);" "6#,&-%!G6#,,&%\"aG6#\"\"!\"\"\"*&&F)6#F,F,%\"x
GF,F,*&&F)6#\"\"#F,*$F0F4F,F,*&&F)6#\"\"$F,*$F0F9F,F,%(~.~.~.~GF,F,-F%
6#,,&%\"bG6#F+F,*&&F@6#F,F,F0F,F,*&&F@6#F4F,*$F0F4F,F,*&&F@6#F9F,*$F0F
9F,F,F;F,F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }
{XPPEDIT 18 0 "a[0]+b[0]+(a[1]+b[1])*x+(a[2]+b[2])*x^2+(a[3]+b[3])*x^3
+` . . . `;" "6#,.&%\"aG6#\"\"!\"\"\"&%\"bG6#F'F(*&,&&F%6#F(F(&F*6#F(F
(F(%\"xGF(F(*&,&&F%6#\"\"#F(&F*6#F7F(F(*$F2F7F(F(*&,&&F%6#\"\"$F(&F*6#
F?F(F(*$F2F?F(F(%(~.~.~.~GF(" }{TEXT -1 3 "   " }}{PARA 256 "" 0 "" 
{TEXT -1 4 "  = " }{XPPEDIT 18 0 "c[0]+c[1]*x+c[2]*x^2+c[3]*x^3+` . . \+
. `" "6#,,&%\"cG6#\"\"!\"\"\"*&&F%6#F(F(%\"xGF(F(*&&F%6#\"\"#F(*$F,F0F
(F(*&&F%6#\"\"$F(*$F,F5F(F(%(~.~.~.~GF(" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[n] = a[n]+b[n];" "6#/&%\"cG
6#%\"nG,&&%\"aG6#F'\"\"\"&%\"bG6#F'F," }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 47 "For example, the Maclaurin series expansion of " }
{XPPEDIT 18 0 "sin(x)+1/(1-x);" "6#,&-%$sinG6#%\"xG\"\"\"*&F(F(,&F(F(F
'!\"\"F+F(" }{TEXT -1 85 " can be obtained by adding corresponding ter
ms of the Maclaurin series expansions of " }{XPPEDIT 18 0 "sin(x)" "6#
-%$sinG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "1/(1-x)" "6#*&\"\"
\"F$,&F$F$%\"xG!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "taylor(sin(x),x,15);\np
x := convert(%,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"
\"\"F%#!\"\"\"\"'\"\"$#F%\"$?\"\"\"&#F'\"%S]\"\"(#F%\"'!)GO\"\"*#F'\")
+o\"*R\"#6#F%\"++3-Fi\"#8-%\"OG6#F%\"#:" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#pxG,0%\"xG\"\"\"*&#F'\"\"'F'*$)F&\"\"$F'F'!\"\"*&#F'\"$?\"F'*
$)F&\"\"&F'F'F'*&#F'\"%S]F'*$)F&\"\"(F'F'F.*&#F'\"'!)GOF'*$)F&\"\"*F'F
'F'*&#F'\")+o\"*RF'*$)F&\"#6F'F'F.*&#F'\"++3-FiF'*$)F&\"#8F'F'F'" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "taylor(1/(1-x),x,15);\nqx := convert(%,polynom);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#+C%\"xG\"\"\"\"\"!F%F%F%\"\"#F%\"\"$F%\"\"%F%\"\"&F%
\"\"'F%\"\"(F%\"\")F%\"\"*F%\"#5F%\"#6F%\"#7F%\"#8F%\"#9-%\"OG6#F%\"#:
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#qxG,@\"\"\"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&*$)F'\"#6F&F&*$)F'\"#7F&F&
*$)F'\"#8F&F&*$)F'\"#9F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 25 ": Applying the procedure " 
}{TEXT 0 6 "taylor" }{TEXT -1 142 " to a polynomial, with the order pa
rameter set to a value greater than the degree of the polynomial, does
 not affect the polynomial except to " }{TEXT 261 47 "arrange the term
s in order of increasing degree" }{TEXT -1 192 ". This seems to be dif
ficult to achieve in any other way, because Maple usually tends to sor
t the terms of a polynomial in order of decreasing degree, starting wi
th the term of highest degree." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "taylor(px+qx,x,15);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#+A%\"xG\"\"\"\"\"!\"\"#F%F%F'#\"\"&\"
\"'\"\"$F%\"\"%#\"$@\"\"$?\"F)F%F*#\"%R]\"%S]\"\"(F%\"\")#\"'\")GO\"'!
)GO\"\"*F%\"#5#\")*z;*R\")+o\"*R\"#6F%\"#7#\"+,3-Fi\"++3-Fi\"#8F%\"#9
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Comp
are this Taylor polynomial with the order 15 Maclaurin series for " }
{XPPEDIT 18 0 "sin(x)+1/(1-x);" "6#,&-%$sinG6#%\"xG\"\"\"*&F(F(,&F(F(F
'!\"\"F+F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "taylor(sin(x)+1/(1-x),x,15);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#+C%\"xG\"\"\"\"\"!\"\"#F%F%F'#\"\"&\"
\"'\"\"$F%\"\"%#\"$@\"\"$?\"F)F%F*#\"%R]\"%S]\"\"(F%\"\")#\"'\")GO\"'!
)GO\"\"*F%\"#5#\")*z;*R\")+o\"*R\"#6F%\"#7#\"+,3-Fi\"++3-Fi\"#8F%\"#9-
%\"OG6#F%\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Multi
plying a power series by polynomial" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 35 "Consider the Maclaurin series for  " }{XPPEDIT 18 0 "f(x) = 1/(
1-x);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)F'!\"\"F+" }{TEXT -1 1 "." }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "1/(1-x) = 1+x+x^2+x^
3+x^4+x^5+x^6+x^7+x^8+x^9 + ` . . . `" "6#/*&\"\"\"F%,&F%F%%\"xG!\"\"F
(,8F%F%F'F%*$F'\"\"#F%*$F'\"\"$F%*$F'\"\"%F%*$F'\"\"&F%*$F'\"\"'F%*$F'
\"\"(F%*$F'\"\")F%*$F'\"\"*F%%(~.~.~.~GF%" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "-1<x" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"
\"" }{TEXT -1 14 "  ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 34 "To mu
ltiply this powers series by " }{XPPEDIT 18 0 "1+x" "6#,&\"\"\"F$%\"xG
F$" }{TEXT -1 32 ", we first form the series for  " }{XPPEDIT 18 0 "x/
(1-x)" "6#*&%\"xG\"\"\",&F%F%F$!\"\"F'" }{TEXT -1 40 " by multiplying \+
all the terms of (i) by " }{TEXT 263 1 "x" }{TEXT -1 2 ": " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x/(1-x) = x+x^2+x^3+x^4+x^5
+x^6+x^7+x^8+x^9+x^10+` . . . `" "6#/*&%\"xG\"\"\",&F&F&F%!\"\"F(,8F%F
&*$F%\"\"#F&*$F%\"\"$F&*$F%\"\"%F&*$F%\"\"&F&*$F%\"\"'F&*$F%\"\"(F&*$F
%\"\")F&*$F%\"\"*F&*$F%\"#5F&%(~.~.~.~GF&" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "-1<x" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<1" "6#
2%!G\"\"\"" }{TEXT -1 15 " -------  (ii)." }}{PARA 0 "" 0 "" {TEXT -1 
15 "The series for " }{XPPEDIT 18 0 "(1+x)/(1-x) = 1/(1-x)+x/(1-x)" "6
#/*&,&\"\"\"F&%\"xGF&F&,&F&F&F'!\"\"F),&*&F&F&,&F&F&F'F)F)F&*&F'F&,&F&
F&F'F)F)F&" }{TEXT -1 63 " is obtained by adding the series in (i) and
 (ii) to get . . . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "(1+x)/(1-x) = 1+2*x+2*x^2+2*x^3+2*x^4+2*x^5+2*x^6+2*x^7+2*x^8+2*x^9
+`. . . `" "6#/*&,&\"\"\"F&%\"xGF&F&,&F&F&F'!\"\"F),8F&F&*&\"\"#F&F'F&
F&*&F,F&*$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&*&F,F&*$F'\"\")F&F&*&F,F&*$
F'\"\"*F&F&%'.~.~.~GF&" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "-1<x" "6#2,$
\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 17 " \+
-------  (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "taylor((1+x)/(1-x),x,10);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#+9%\"xG\"\"\"\"\"!\"\"#F%F'F'F'\"\"$F'\"\"%F'\"\"&F'\"
\"'F'\"\"(F'\"\")F'\"\"*-%\"OG6#F%\"#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 68 "Another way to construct this series is to make use of th
e identity " }{XPPEDIT 18 0 "(1+x)/(1-x) = 2/(1-x)-1" "6#/*&,&\"\"\"F&
%\"xGF&F&,&F&F&F'!\"\"F),&*&\"\"#F&,&F&F&F'F)F)F&F&F)" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 50 "We just need to subtract 1 from the p
ower series: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "2/(1-x) = 2+2*x+2*x^2+2*x^3+2*x^4+2*x^5
+2*x^6+2*x^7+2*x^8+2*x^9+`. . . `;" "6#/*&\"\"#\"\"\",&F&F&%\"xG!\"\"F
),8F%F&*&F%F&F(F&F&*&F%F&*$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&F&*&F%F&*$F(
\"\")F&F&*&F%F&*$F(\"\"*F&F&%'.~.~.~GF&" }{TEXT -1 3 ",  " }{XPPEDIT 
18 0 "-1<x" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"
\"" }{TEXT -1 16 " -------  (iv). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 45 "Consider the d
egree 10 Taylor polynomial for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%
\"xG" }{TEXT -1 7 " about " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }
{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "p(x
)= 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7! + x^8/8
! + x^9/9! + x^10/10!" "6#/-%\"pG6#%\"xG,8\"\"\"F)F'F)*&F'\"\"#-%*fact
orialG6#F+!\"\"F)*&F'\"\"$-F-6#F1F/F)*&F'\"\"%-F-6#F5F/F)*&F'\"\"&-F-6
#F9F/F)*&F'\"\"'-F-6#F=F/F)*&F'\"\"(-F-6#FAF/F)*&F'\"\")-F-6#FEF/F)*&F
'\"\"*-F-6#FIF/F)*&F'\"#5-F-6#FMF/F)" }{TEXT -1 2 " ." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "p := unap
ply(sum(x^i/i!,i=0..10),x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"p
Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,8\"\"\"F-9$F-*&#F-\"\"#F-)F.F1F-F
-*&#F-\"\"'F-)F.\"\"$F-F-*&#F-\"#CF-)F.\"\"%F-F-*&#F-\"$?\"F-)F.\"\"&F
-F-*&#F-\"$?(F-)F.F5F-F-*&#F-\"%S]F-)F.\"\"(F-F-*&#F-\"&?.%F-)F.\"\")F
-F-*&#F-\"'!)GOF-)F.\"\"*F-F-*&#F-\"(+)GOF-)F.\"#5F-F-F(F(F(" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The polyn
omial " }{XPPEDIT 18 0 "p(x)^2" "6#*$-%\"pG6#%\"xG\"\"#" }{TEXT -1 
134 " has degree 20, but if we compute its Taylor series up as far as \+
the term of degree 12, this will effectively truncate the polynomial.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "taylor(p(x)^2,x,13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+?%\"x
G\"\"\"\"\"!\"\"#F%F'F'#\"\"%\"\"$F*#F'F*F)#F)\"#:\"\"&#F)\"#X\"\"'#\"
\")\"$:$\"\"(#F'F4F3#F)\"%NG\"\"*#F)\"&vT\"\"#5#\"#J\"'+[g\"#6#\"#P\"(
gXN%\"#7-%\"OG6#F%\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 25 "Up as far as the term in " }{XPPEDIT 18 0 "x^10" "6#
*$%\"xG\"#5" }{TEXT -1 53 ", this agrees with the Taylor series for th
e product " }{XPPEDIT 18 0 "p(x)*exp(x)" "6#*&-%\"pG6#%\"xG\"\"\"-%$ex
pG6#F'F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "taylor(p(x)*exp(x),x,13);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#+?%\"xG\"\"\"\"\"!\"\"#F%F'F'#\"\"%\"\"$F*#F
'F*F)#F)\"#:\"\"&#F)\"#X\"\"'#\"\")\"$:$\"\"(#F'F4F3#F)\"%NG\"\"*#F)\"
&vT\"\"#5#\"%Z?\")+o\"*R\"#6#\"%h8\"*+smf\"\"#7-%\"OG6#F%\"#8" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Both of t
he previous expansions agree with the terms of the Maclaurin series fo
r " }{XPPEDIT 18 0 "exp(x)+exp(x)=exp(2*x)" "6#/,&-%$expG6#%\"xG\"\"\"
-F&6#F(F)-F&6#*&\"\"#F)F(F)" }{TEXT -1 26 " up as far as the term in \+
" }{XPPEDIT 18 0 "x^10" "6#*$%\"xG\"#5" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "taylor(
exp(2*x),x,13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+?%\"xG\"\"\"\"\"!
\"\"#F%F'F'#\"\"%\"\"$F*#F'F*F)#F)\"#:\"\"&#F)\"#X\"\"'#\"\")\"$:$\"\"
(#F'F4F3#F)\"%NG\"\"*#F)\"&vT\"\"#5#F3\"'Df:\"#6#F)\"'vxY\"#7-%\"OG6#F
%\"#8" }}}{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 28 "Differentiating power series" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 20 "If the power series " }{XPPEDIT 18 0 "Sum(a[n]*x^n,n=0..infinit
y)" "6#-%$SumG6$*&&%\"aG6#%\"nG\"\"\")%\"xGF*F+/F*;\"\"!%)infinityG" }
{TEXT -1 15 " converges for " }{XPPEDIT 18 0 "-R<x" "6#2,$%\"RG!\"\"%
\"xG" }{XPPEDIT 18 0 "``<R" "6#2%!G%\"RG" }{TEXT -1 19 ", then the fun
ction" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = Sum(a
[n]*x^n,n=0..infinity)" "6#/-%\"fG6#%\"xG-%$SumG6$*&&%\"aG6#%\"nG\"\"
\")F'F/F0/F/;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 33 "has a derivative in the interval " }{XPPEDIT 18 0 "-R<x" 
"6#2,$%\"RG!\"\"%\"xG" }{XPPEDIT 18 0 "``<R" "6#2%!G%\"RG" }{TEXT -1 
27 ", which can be obtained by " }{TEXT 261 47 "differentiating the po
wer series \"term by term\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 5 "Thus " }}{PARA 256 "" 0 "" {TEXT -1 4 "f '(" }{TEXT 260 1 "x" }
{TEXT -1 5 ") =  " }{XPPEDIT 18 0 "Sum(n*a[n]*x^(n-1),n = 1 .. infinit
y);" "6#-%$SumG6$*(%\"nG\"\"\"&%\"aG6#F'F()%\"xG,&F'F(F(!\"\"F(/F';F(%
)infinityG" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 93 "Note that
, since we can repeat this process any number of times, all the higher
 derivatives f" }{XPPEDIT 18 0 "`@@`(``,n);" "6#-%#@@G6$%!G%\"nG" }
{XPPEDIT 18 0 "``(x)" "6#-%!G6#%\"xG" }{TEXT -1 7 " exist." }}{PARA 0 
"" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "For example, we \+
have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x) = x-x^
3/3!+x^5/5!-x^7/7!+x^9/9!-x^11/11!+` . . . `;" "6#/-%$sinG6#%\"xG,0F'
\"\"\"*&F'\"\"$-%*factorialG6#F+!\"\"F/*&F'\"\"&-F-6#F1F/F)*&F'\"\"(-F
-6#F5F/F/*&F'\"\"*-F-6#F9F/F)*&F'\"#6-F-6#F=F/F/%(~.~.~.~GF)" }{TEXT 
-1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 50 "and differentiating the seri
es term by term gives " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-3*x^2/3!+5*x^4/5!-7*x^6/7!+9*x
^8/9!-11*x^10/11!+` . . . `;" "6#,0\"\"\"F$*(\"\"$F$*$%\"xG\"\"#F$-%*f
actorialG6#F&!\"\"F-*(\"\"&F$*$F(\"\"%F$-F+6#F/F-F$*(\"\"(F$*$F(\"\"'F
$-F+6#F5F-F-*(\"\"*F$*$F(\"\")F$-F+6#F;F-F$*(\"#6F$*$F(\"#5F$-F+6#FAF-
F-%(~.~.~.~GF$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "=  " }
{XPPEDIT 18 0 "1-x^2/2!+x^4/4!-x^6/6!+x^8/8!-x^10/10!+` . . . `;" "6#,
0\"\"\"F$*&%\"xG\"\"#-%*factorialG6#F'!\"\"F+*&F&\"\"%-F)6#F-F+F$*&F&
\"\"'-F)6#F1F+F+*&F&\"\")-F)6#F5F+F$*&F&\"#5-F)6#F9F+F+%(~.~.~.~GF$" }
{TEXT -1 3 ",  " }}{PARA 0 "" 0 "" {TEXT -1 35 "which is the Maclaurin
 series for  " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "[sin(x)]= cos(x)" "6#/7#-%$sinG6#%\"xG-
%$cosG6#F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 4 "diff" }{TEXT 
-1 25 "  makes use of this fact." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "taylor(sin(x),x,10);\nDiff(%
,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG\"\"\"F%#!\"
\"\"\"'\"\"$#F%\"$?\"\"\"&#F'\"%S]\"\"(#F%\"'!)GO\"\"*-%\"OG6#F%\"#6" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$+/%\"xG\"\"\"F(#!\"\"\"\"'
\"\"$#F(\"$?\"\"\"&#F*\"%S]\"\"(#F(\"'!)GO\"\"*-%\"OG6#F(\"#6F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG\"\"\"\"\"!#!\"\"\"\"#F)#F%\"#C
\"\"%#F(\"$?(\"\"'#F%\"&?.%\"\")-%\"OG6#F%\"#5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 28 "For another example, we have" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "ln(1+x)=x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/
6+x^7/7-x^8/8+ ` . . . `" "6#/-%#lnG6#,&\"\"\"F(%\"xGF(,4F)F(*&F)\"\"#
F,!\"\"F-*&F)\"\"$F/F-F(*&F)\"\"%F1F-F-*&F)\"\"&F3F-F(*&F)\"\"'F5F-F-*
&F)\"\"(F7F-F(*&F)\"\")F9F-F-%(~.~.~.~GF(" }{TEXT -1 3 ",  " }}{PARA 
0 "" 0 "" {TEXT -1 38 "and differentiating term by term gives" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "1-x+x^2-x^3+x^4-x^5+
x^6-x^7+ ` . . . `" "6#,4\"\"\"F$%\"xG!\"\"*$F%\"\"#F$*$F%\"\"$F&*$F%
\"\"%F$*$F%\"\"&F&*$F%\"\"'F$*$F%\"\"(F&%(~.~.~.~GF$" }{TEXT -1 4 ",  \+
 " }}{PARA 0 "" 0 "" {TEXT -1 35 "which is the Maclaurin series for  \+
" }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "[ln(1+x)]=1/(1+x)" "6#/7#-%#lnG6#,&\"\"\"F)%\"xGF)*&F)F
),&F)F)F*F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "taylor(ln(1+x),x,10);\nDiff(
%,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+7%\"xG\"\"\"F%#!
\"\"\"\"#F(#F%\"\"$F*#F'\"\"%F,#F%\"\"&F.#F'\"\"'F0#F%\"\"(F2#F'\"\")F
4#F%\"\"*F6-%\"OG6#F%\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6
$+7%\"xG\"\"\"F(#!\"\"\"\"#F+#F(\"\"$F-#F*\"\"%F/#F(\"\"&F1#F*\"\"'F3#
F(\"\"(F5#F*\"\")F7#F(\"\"*F9-%\"OG6#F(\"#5F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#+7%\"xG\"\"\"\"\"!!\"\"F%F%\"\"#F'\"\"$F%\"\"%F'\"\"&F%
\"\"'F'\"\"(F%\"\")-%\"OG6#F%\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 
"" {TEXT -1 24 "Integrating power series" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "If the power series
 " }{XPPEDIT 18 0 "Sum(a[n]*x^n,n=0..infinity)" "6#-%$SumG6$*&&%\"aG6#
%\"nG\"\"\")%\"xGF*F+/F*;\"\"!%)infinityG" }{TEXT -1 15 " converges fo
r " }{XPPEDIT 18 0 "-R<x" "6#2,$%\"RG!\"\"%\"xG" }{XPPEDIT 18 0 "``<R
" "6#2%!G%\"RG" }{TEXT -1 7 ", and  " }{XPPEDIT 18 0 "f(x) = Sum(a[n]*
x^n,n=0..infinity)" "6#/-%\"fG6#%\"xG-%$SumG6$*&&%\"aG6#%\"nG\"\"\")F'
F/F0/F/;\"\"!%)infinityG" }{TEXT -1 19 " , then the series " }}{PARA 
256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Sum(a[n]*x^(n+1)/(n+1),n =
 0 .. infinity);" "6#-%$SumG6$*(&%\"aG6#%\"nG\"\"\")%\"xG,&F*F+F+F+F+,
&F*F+F+F+!\"\"/F*;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 20 "also converges for  " }{XPPEDIT 18 0 "-R<x" "6#2,$%\"RG!
\"\"%\"xG" }{XPPEDIT 18 0 "``<R" "6#2%!G%\"RG" }{TEXT -1 6 ", and " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x) = Sum(a[n
]*x^(n+1)/(n+1),n = 0 .. infinity);" "6#/-%$IntG6$-%\"fG6#%\"xGF*-%$Su
mG6$*(&%\"aG6#%\"nG\"\"\")F*,&F2F3F3F3F3,&F2F3F3F3!\"\"/F2;\"\"!%)infi
nityG" }{XPPEDIT 18 0 "``+c" "6#,&%!G\"\"\"%\"cGF%" }{TEXT -1 3 ",  " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "that is
, we can " }{TEXT 261 39 "integrate a power series \"term by term\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 20 "For example, we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "sin*x = x-x^3/3!+x^5/5!-x^7/7!+x^9/9!-x^11/11!+` . . .
 `;" "6#/*&%$sinG\"\"\"%\"xGF&,0F'F&*&F'\"\"$-%*factorialG6#F*!\"\"F.*
&F'\"\"&-F,6#F0F.F&*&F'\"\"(-F,6#F4F.F.*&F'\"\"*-F,6#F8F.F&*&F'\"#6-F,
6#F<F.F.%(~.~.~.~GF&" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 46 
"and integrating the series term by term gives " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2/2-x^4/4!+x^6/6!-x^8/8!+x^10/10!-x^1
2/12!+ ` . . . `" "6#,0*&%\"xG\"\"#F&!\"\"\"\"\"*&F%\"\"%-%*factorialG
6#F*F'F'*&F%\"\"'-F,6#F/F'F(*&F%\"\")-F,6#F3F'F'*&F%\"#5-F,6#F7F'F(*&F
%\"#7-F,6#F;F'F'%(~.~.~.~GF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 25 "which is the series for  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(sin*x,x) = -cos*x+c;" "6#/-%$IntG6$*&%$sinG\"\"\"%
\"xGF)F*,&*&%$cosGF)F*F)!\"\"%\"cGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 
"" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c = 1" "6#/%\"cG\"\"\"" }{TEXT 
-1 1 "." }}{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 1 "\004" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "2 0" 11 
}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }