{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{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 0 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 0 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 "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 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 Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "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 "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 32 "Module 10 : Serious About Series " }{TEXT 256 0 "" }{TEXT 257 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT -1 30 "1003 : Power Series Expansions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 17 "O B J E C T I V E " }}{PARA 0 "" 0 "" {TEXT -1 557 "In this project, we will explore met hods of expanding functions into power series. The basic idea hinges o n the geometric series expansion of 1/(1-x). However, using differenti ation and integration we can expand many more functions into power ser ies also. In addition, we will examine the interval of convergence and how it is affected by the location of the expansion and features of t he function such as vertical asymptotes. In general, this module will \+ reinforce methods one might uses by hand and not rely on the automated expansions Maple can generate." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 9 "S E T U P" }}{PARA 0 "" 0 "" {TEXT -1 252 "In this project we will use the following command package. Type a nd execute this line before beginning the project below. If you re-ent er the worksheet for this project, be sure to re-execute this statemen t before jumping to any point in the worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 5 "> " 0 "" {MPLTEXT 1 0 21 "restart; wit h(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoor ds has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 " " 0 "" {TEXT -1 83 "__________________________________________________ _________________________________" }}{PARA 4 "" 0 "" {TEXT -1 33 "A. E xpand A Function As A Series" }}{PARA 0 "" 0 "" {TEXT -1 83 "________ ______________________________________________________________________ _____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 48 "An infinite geometric series can be s iimplified." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "Sum( a*r^k, k = 0..infinity): % = value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&%\"aG\"\"\")%\"rG%\"kGF)/F ,;\"\"!%)infinityG,$*&F(F),&F+F)F)!\"\"F4F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can turn this process around . Staring with an expression, we can expand it iinto a series." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "a/(1 - r): % = series( %, r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ *&%\"aG\"\"\",&F&F&%\"rG!\"\"F)+1F(F%\"\"!F%F&F%\"\"#F%\"\"$F%\"\"%F% \"\"&-%\"OG6#F&\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Given an expression in x, we can compute the series ex pansion. You can also do these by hand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series( 1/(1-x), x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!F%F%F%\"\"#F%\"\"$F %\"\"%F%\"\"&-%\"OG6#F%\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "series( 1/(1-x), x, 15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+C% \"xG\"\"\"\"\"!F%F%F%\"\"#F%\"\"$F%\"\"%F%\"\"&F%\"\"'F%\"\"(F%\"\")F% \"\"*F%\"#5F%\"#6F%\"#7F%\"#8F%\"#9-%\"OG6#F%\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 205 "Maple will expand to a c ertain number of terms as a default. We can also specify to what power of x we wnat to expand. Maple then uses the \"big Oh\" notation to in dicate the order of the error or remainder." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "All of the previous examples e xpanded the power series about x = 0. We can also expand about other v alues." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "series( 1/(1-x), x = 3, 12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+=,&%\"xG\"\"\"\"\"$!\"\"#F(\"\"#\"\"!#F&\"\"%F&#F(\"\" )F*#F&\"#;F'#F(\"#KF-#F&\"#k\"\"&#F(\"$G\"\"\"'#F&\"$c#\"\"(#F(\"$7&F/ #F&\"%C5\"\"*#F(\"%[?\"#5#F&\"%'4%\"#6-%\"OG6#F&\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "series( 1/(2 + x^2), x = -1,12);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#+=,&%\"xG\"\"\"F&F&#F&\"\"$\"\"!#\"\"# \"\"*F&#F&\"#FF+#!\"%\"#\")F(#!#6\"$V#\"\"%#!#5\"$H(\"\"&#\"#8\"%(=#\" \"'#\"#c\"%hl\"\"(#\"#t\"&$o>\"\")#!#A\"&\\!fF,#!$j#\"'Zr<\"#5#!$g%\"' T9`\"#6-%\"OG6#F&\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 83 "_________________________________________________________ __________________________" }}{PARA 4 "" 0 "" {TEXT -1 37 "B. Integrat e / Series / Differentiate" }}{PARA 0 "" 0 "" {TEXT -1 83 "___________ ______________________________________________________________________ __" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 331 "The method we used above to expand a ser ies into a geometric series works only in certain cases. If an express ion does not lend itself readily to this method, there are other trick s. One is to intergrate the function, expand the anti-derivative into \+ a series, then differentiate the result. In this indirect way we find \+ the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Lets consider an example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 127 "We start with a funciton, integrate it, \+ expand it into a power series, then differentiate to get back to the o riginal function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> 3/(x-5)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.*$) ,&9$F.\"\"&!\"\"\"\"#F.F4\"\"$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F := int( f(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"FG,$*&\"\"\"F',&%\"xGF'\"\"&!\"\"F+!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "series( %, x, 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"xG#\"\"$\"\"&\"\"!#F&\"#D\"\"\"#F&\"$D\"\"\"##F&\"$D'F&#F& \"%DJ\"\"%#F&\"&Dc\"F'#F&\"&D\"y\"\"'#F&\"'D1R\"\"(#F&\"(DJ&>\"\")#F& \"(Dcw*\"\"*-%\"OG6#F+\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff( %, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+7%\"xG#\"\"$\"#D \"\"!#\"\"'\"$D\"\"\"\"#\"\"*\"$D'\"\"##\"#7\"%DJF&#F&F3\"\"%#\"#=\"&D \"y\"\"&#\"#@\"'D1RF*#\"#C\"(DJ&>\"\"(#\"#F\"(Dcw*\"\")-%\"OG6#F,F." } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 224 "We can convert the new found series into a polynomial and evaluate it to dem onstrate that it take values close to original funciton. the reason fo r he slight error is that we are only taking a finite number of terms \+ of here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g := unapply(convert( %, polynom), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\" 6$%)operatorG%&arrowGF(,4#\"\"$\"#D\"\"\"*&#\"\"'\"$D\"F09$F0F0*&#\"\" *\"$D'F0)F5\"\"#F0F0*&#\"#7\"%DJF0)F5F.F0F0*&#F.F?F0)F5\"\"%F0F0*&#\"# =\"&D\"yF0)F5\"\"&F0F0*&#\"#@\"'D1RF0)F5F3F0F0*&#\"#C\"(DJ&>F0)F5\"\"( F0F0*&#\"#F\"(Dcw*F0)F5\"\")F0F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f(1); g(1); f(1) - g(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"(Z5$= \"(Dcw*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$B\"\"*++Dc\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 83 "____________________ _______________________________________________________________" }} {PARA 4 "" 0 "" {TEXT -1 37 "C. Differentiate / Series / Integrate" }} {PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________ __________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 212 "We can use the same trick in reverse too. Take a function, differ entiate it, expand into a power series, and integrate. When we integra te, the constant of integration will become the constant term of the s eries. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Lets consider an example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 127 "We start with a funciton, integrate it, \+ expand it into a power series, then differentiate to get back to the o riginal function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> arctan(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%'arctanG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff( f(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$, &F$F$*$)%\"xG\"\"#F$F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "series( %, x, 14);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\" \"\"\"!!\"\"\"\"#F%\"\"%F'\"\"'F%\"\")F'\"#5F%\"#7-%\"OG6#F%\"#9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Int( %, x) : % = int( %%, x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$+3%\"xG\"\"\"\"\"!!\"\" \"\"#F)\"\"%F+\"\"'F)\"\")F+\"#5F)\"#7-%\"OG6#F)\"#9F(+3F(F)F)#F+\"\"$ F8#F)\"\"&F:#F+\"\"(F<#F)\"\"*F>#F+\"#6F@#F)\"#8FBF2\"#:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 392 "Every indefinite \+ integral has a constant of integration. In particular, when we integra te the series, we can't disregard the constant of integration. It turn s out to be constant term of the series. Lets look at an example. We s tart with a function , and compute its series expansion by the method \+ described above. Then we convert the series to a polynomial so we can \+ graph it along with f(x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := x -> 1/(1-x); c := -1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowG F(*&\"\"\"F-,&F-F-9$!\"\"F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"cG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "series( D(f)( x), x=c, 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1,&%\"xG\"\"\"F&F&#F& \"\"%\"\"!F'F&#\"\"$\"#;\"\"##F&\"\")F+#\"\"&\"#kF(#F+F2F1-%\"OG6#F&\" \"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "g := unapply( conver t( int(%, x), polynom), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR 6#%\"xG6\"6$%)operatorG%&arrowGF(,09$#\"\"\"\"\"%F.F/*&#F/\"\")F/),&F- F/F/F/\"\"#F/F/*&#F/\"#;F/)F5\"\"$F/F/*&#F/\"#KF/)F5F0F/F/*&#F/\"#kF/) F5\"\"&F/F/*&#F/\"$G\"F/)F5\"\"'F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 78 "plot( [f(x), g(x)], x = -5..1, y = -2..5, thickness =[3,2], color=[red,coral]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7ep7$$!\"&\"\"!$\"3emmmmmmm;!#=7$$!3!****** \\2<#p[!#<$\"3q^zshZ!Qq\"F-7$$!3!)*****4bBav%F1$\"3)fi*3(e\"\\P)[5Q^?'=F-7$$!3q*****4/>F-7$$!3N+++ ,kZGTF1$\"3S))f(R&o*)\\>F-7$$!3(******4;)=,SF1$\"3Y$H6&[[_**>F-7$$!3y* ****f83V(QF1$\"3!RnNA@t:0#F-7$$!3:+++NkzVPF1$\"3*RdrfD;!3@F-7$$!3\")** ***zlT)GOF1$\"3o9li)pn.;#F-7$$!37+++0)H%*\\$F1$\"3hTKoNQ]AAF-7$$!3#)** ***\\d'[pLF1$\"3-&)yd.%)f)G#F-7$$!38+++&>iUC$F1$\"3[YgS5@7cBF-7$$!3y** ***4YY08$F1$\"3p\\h5tr)4U#F-7$$!3)******\\XF`*HF1$\"3!\\LqfvBH]#F-7$$! 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For example, xk = 1/(1-x) only for | x | < 1. Every po wer series has an interval of convergence - although in some cases it \+ is all real numbers or just a single number. In the process we underwe nt to find his series, one of the steps included expanding a power ser ies. Thus that interval of convergence effects the final series outcom e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "We can examine this concept in further de tail using a customized plot function." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" } }{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been \+ redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1041 "conv_plot \+ := proc( f, g1,g2, a,b, c1, c2, eps)\nlocal x1,x2,y1,y2,delta, m, M, A , B, i, n, PNG, CL;\nn := 100: delta := (b-a)/n: x2 := a: \+ PNG := 'style = patchnogrid'; \nM := maximize( f(x), x = a..b): \+ m := minimize(f(x), x = a..b): \n\nfor i from 1 to n do\nx1 : = x2: x2:= x1 + delta: y1 := evalf( f(x1)): y2 := evalf( f(x2)): \nif( abs(evalf( y1 - g1(x1))) < eps ) then \n CL := maro on: \nelse \n CL := black: \nfi:\n#print ( [[x1, m],[x1,y1],[x2,y2 ],[x2,m]]); \nA[i] := polygonplot( [[x1, m],[x1,y1],[x2,y2],[x2,m]], c olor = CL, PNG):\nif( abs(evalf( y1 - g2(x1))) < eps ) then \n CL : = coral: \nelse \n CL := black: \nfi: \nB[i] := polygonplot( [[x1 , M],[x1,y1],[x2,y2],[x2,M]], color = CL, PNG):\nod:\n\ndisplay( [plot ([f(x), g1(x), g2(x)], x = a..b, y = n..M, thickness = [4,2,2], color \+ = [red, blue, green] ),\n plot( [[[c1, m],[c1,g1(c1)]], [[ c2, g2(c2)],[c2,M]] ] , x =a..b, color = [blue, green], linestyle = 3) ,\n seq( A[i], i = 1..n), seq( B[i], i = 1..n)], axes = framed ); \+ \nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 295 "This special plot funcion will show where the series is \+ within a small tolerance of e of the rigival function. This will provi de us a convenient way to see where the series converges to the functi on and where it doesn't. It allows us to compare two approximate funct ions to an original function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> ln( 1+x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowG F(-%#lnG6#,&\"\"\"F09$F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "g := unapply( convert(series( f(x), x, 7), polynom), x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowG F(,.9$\"\"\"*&#F.\"\"#F.*$)F-F1F.F.!\"\"*&#F.\"\"$F.)F-F7F.F.*&#F.\"\" %F.*$)F-F;F.F.F4*&#F.\"\"&F.)F-F@F.F.*&#F.\"\"'F.*$)F-FDF.F.F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "h := unapply( convert(ser ies( f(x), x, 12), polynom), x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"hGR6#%\"xG6\"6$%)operatorG%&arrowGF(,89$\"\"\"*&#F.\"\"#F.*$)F-F1F. 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The value curve is g(x) which is the series taken out to 7 terms. The interval where \+ g differs from f by less than epsilon = .05 is indicated by the purpl e shading on the bottom of the screen, The dashed blue line indicates \+ the center of the expansion. In a similar way, the green curve is the \+ graph of h(x), which is the series taken out to 12 terms. The interval where it is close to f(x) is indicated by the yellow bar above the gr aph. What this diagram shows is that the series with more terms is a b etter fit to the function.Also, both series seem to converge within th e same region between - 1 and +1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 103 "Lets look at another example. We'll cons ider tan-1(x), and look at series expansions of 3 and 15 terms." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := x -> arctan(x); c := 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"fG%'arctanG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "series( D(f)(x), x = c, 3); \+ convert(int(%, x), polynom); g := unapply( %,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"xG\"\"\"\"\"!!\"\"\"\"#-%\"OG6#F%\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"*&#F%\"\"$F%*$)F$F(F%F%! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG %&arrowGF(,&9$\"\"\"*&#F.\"\"$F.*$)F-F1F.F.!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "series( D(f)(x), x = c, 15); conv ert(int(%, x), polynom); h := unapply( %,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"\"\"!!\"\"\"\"#F%\"\"%F'\"\"'F%\"\")F'\"#5 F%\"#7F'\"#9-%\"OG6#F%\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2%\"xG \"\"\"*&#F%\"\"$F%*$)F$F(F%F%!\"\"*&#F%\"\"&F%)F$F.F%F%*&#F%\"\"(F%*$) F$F2F%F%F+*&#F%\"\"*F%)F$F7F%F%*&#F%\"#6F%*$)F$F;F%F%F+*&#F%\"#8F%)F$F @F%F%*&#F%\"#:F%*$)F$FDF%F%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"h GR6#%\"xG6\"6$%)operatorG%&arrowGF(,29$\"\"\"*&#F.\"\"$F.*$)F-F1F.F.! \"\"*&#F.\"\"&F.)F-F7F.F.*&#F.\"\"(F.*$)F-F;F.F.F4*&#F.\"\"*F.)F-F@F.F .*&#F.\"#6F.*$)F-FDF.F.F4*&#F.\"#8F.)F-FIF.F.*&#F.\"#:F.*$)F-FMF.F.F4F (F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "plot( [f(x),g(x), \+ h(x) ], x = -5..5, y = -4..4, color = [red, blue, green], thickness = \+ [3,1,2] );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-% 'CURVESG6%7S7$$!\"&\"\"!$!3!f,Xpw+MP\"!#<7$$!3YLLLe%G?y%F-$!3'zWwsn]YO \"F-7$$!3OmmT&esBf%F-$!3]t4++4Rc8F-7$$!3ALL$3s%3zVF-$!3GA&=')y(GY8F-7$ $!3_LL$e/$QkTF-$!3r1g*)3p7N8F-7$$!3ommT5=q]RF-$!37,?HYP)GK\"F-7$$!3ILL 3_>f_PF-$!3Re3)Q!fO58F-7$$!3K++vo1YZNF-$!3/];MSU.'H\"F-7$$!3;LL3-OJNLF -$!3yIjn^H]z7F-7$$!3p***\\P*o%Q7$F-$!3S(fujB')4E\"F-7$$!3Kmmm\"RFj!HF- $!3*Q6Z>'zSR7F-7$$!33LL$e4OZr#F-$!3PBD>d(fy@\"F-7$$!3u*****\\n\\!*\\#F -$!3q$R/u#)e,>\"F-7$$!3%)*****\\ixCG#F-$!3spaOy_'y:\"F-7$$!3#******\\K qP2#F-$!3Gv]]:*y97\"F-7$$!39LL3-TC%)=F-$!3AC7'oIyG3\"F-7$$!3[mmm\"4z)e ;F-$!39uku0$3$G5F-7$$!3Mmmmm`'zY\"F-$!3v:daYK*ys*!#=7$$!3#****\\(=t)eC \"F-$!3'>L*y3@ZW*)F]q7$$!3!ommmh5$\\5F-$!31,y@Adb%4)F]q7$$!3S$***\\(=[ jL)F]q$!3#3yqjvh\"\\pF]q7$$!3)f***\\iXg#G'F]q$!3#*)=Y8y/%4cF]q7$$!3ndm mT&Q(RTF]q$!3)HjwggW\\#RF]q7$$!3%\\mmTg=><#F]q$!3H-^F>BqQ@F]q7$$!3vDML Le*e$\\!#?$!3$G&zR\\d&e$\\F_s7$$\"3!=nm\"zRQb@F]q$\"3xN=5Jr!H7#F]q7$$ \"3_,+](=>Y2%F]q$\"3Up=L$oC#pQF]q7$$\"3summ\"zXu9'F]q$\"3RQCBwh!>^&F]q 7$$\"3#4+++]y))G)F]q$\"3)ysp%)p*3@pF]q7$$\"3H++]i_QQ5F-$\"3gqf'p[rA/)F ]q7$$\"3b++D\"y%3T7F-$\"3h<7%p#4hD*)F]q7$$\"3+++]P![hY\"F-$\"3xW\\`u!G @s*F]q7$$\"3iKLL$Qx$o;F-$\"3nBtu@$H3.\"F-7$$\"3Y+++v.I%)=F-$\"3WbbAt1* G3\"F-7$$\"3?mm\"zpe*z?F-$\"3_EHP(fVE7\"F-7$$\"3;,++D\\'QH#F-$\"3YN^\" RU\"pf6F-7$$\"3%HL$e9S8&\\#F-$\"3'pV98l<'*=\"F-7$$\"3s++D1#=bq#F-$\"3f tLhE^v;7F-7$$\"3\"HLL$3s?6HF-$\"3[k*G&[P#*R7F-7$$\"3a***\\7`Wl7$F-$\"3 !4%Gv'yO7E\"F-7$$\"3enmmm*RRL$F-$\"39'=YHh*Qz7F-7$$\"3%zmmTvJga$F-$\"3 _I#pP,HfH\"F-7$$\"3]MLe9tOcPF-$\"3*e$oC#*fh58F-7$$\"31,++]Qk\\RF-$\"3$ \\ws&G+#GK\"F-7$$\"3![LL3dg6<%F-$\"32[s$z%e\\N8F-7$$\"3%ymmmw(GpVF-$\" 3%*R$\\&)=,eM\"F-7$$\"3C++D\"oK0e%F-$\"3e)=C**e`eN\"F-7$$\"35,+v=5s#y% F-$\"3A^?x*ozYO\"F-7$$\"\"&F*$\"3!f,Xpw+MP\"F--%'COLOURG6&%$RGBG$\"*++ ++\"!\")$F*F*Fa[l-%*THICKNESSG6#\"\"$-F$6%7S7$F($\"3WmmmmmmmO!#;7$F/$ \"3U+gJdb%p;$F\\\\l7$F4$\"3;l.P7K=pFF\\\\l7$F9$\"3%\\&\\l'Qe7O#F\\\\l7 $F>$\"37+7]!ok3*>F\\\\l7$FC$\"3M\"e\"*)zTNg;F\\\\l7$FH$\"3%z9?tv+iQ\"F \\\\l7$FM$\"3(>&o\"G'=NL6F\\\\l7$FR$\"327(40Z\"QK!*F-7$FW$\"3!\\v^$*\\ Au.(F-7$Ffn$\"3[_\\UF)omF&F-7$F[o$\"33')oD@uFaRF-7$F`o$\"3Ug7f%QYLq#F- 7$Feo$\"3_(>O>1+7o\"F-7$Fjo$\"3y/vLUe&**)*)F]q7$F_p$\"3oC^&y23oX$F]q7$ Fdp$!39.?B\"[/?P\"F]q7$Fip$!3$HiqBB\\^8%F]q7$F_q$!3Ln7Sqf]7gF]q7$Fdq$! 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AULTG-%%VIEWG6$;F(Ffz;F]]q$\"$+\"F*" 1 2 0 1 10 0 2 9 1 3 2 1.000000 41.000000 28.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" " Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curv e 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37 " "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "C urve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56 " "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "C urve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75 " "Curve 76" "Curve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "C urve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93" "Curve 94 " "Curve 95" "Curve 96" "Curve 97" "Curve 98" "Curve 99" "Curve 100" " Curve 101" "Curve 102" "Curve 103" "Curve 104" "Curve 105" "Curve 106 " "Curve 107" "Curve 108" "Curve 109" "Curve 110" "Curve 111" "Curve 1 12" "Curve 113" "Curve 114" "Curve 115" "Curve 116" "Curve 117" "Curve 118" "Curve 119" "Curve 120" "Curve 121" "Curve 122" "Curve 123" "Cur ve 124" "Curve 125" "Curve 126" "Curve 127" "Curve 128" "Curve 129" "C urve 130" "Curve 131" "Curve 132" "Curve 133" "Curve 134" "Curve 135" "Curve 136" "Curve 137" "Curve 138" "Curve 139" "Curve 140" "Curve 141 " "Curve 142" "Curve 143" "Curve 144" "Curve 145" "Curve 146" "Curve 1 47" "Curve 148" "Curve 149" "Curve 150" "Curve 151" "Curve 152" "Curve 153" "Curve 154" "Curve 155" "Curve 156" "Curve 157" "Curve 158" "Cur ve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 163" "Curve 164" "C urve 165" "Curve 166" "Curve 167" "Curve 168" "Curve 169" "Curve 170" "Curve 171" "Curve 172" "Curve 173" "Curve 174" "Curve 175" "Curve 176 " "Curve 177" "Curve 178" "Curve 179" "Curve 180" "Curve 181" "Curve 1 82" "Curve 183" "Curve 184" "Curve 185" "Curve 186" "Curve 187" "Curve 188" "Curve 189" "Curve 190" "Curve 191" "Curve 192" "Curve 193" "Cur ve 194" "Curve 195" "Curve 196" "Curve 197" "Curve 198" "Curve 199" "C urve 200" "Curve 201" "Curve 202" "Curve 203" "Curve 204" "Curve 205" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 257 "In t his example we have been using c = 0 as the center for our series expa nsion. We cal also do the same thing for a different center value. Her e is an example with c = Pi/3. Notice how the entire convergence regio n moves along with the center to the right." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := x -> arctan(x); c := Pi/3;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fG%'arctanG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"cG,$%#PiG#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 " series( f(x) , x = c, 3): convert( %, polynom): g := unapply( %, \+ x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "series( f(x) , x = c , 15): convert( %, polynom): h := unapply( %, x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "conv_plot( f, g, h, -4, 4, c, c, .1 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$!\"%,$-%'arctanG6#\"\"%!\"\" 7$F%$!+kw\"eK\"!\"*7$#!#)*\"#D$!+P:-@8F/7$F1F&" }}{PARA 8 "" 1 "" {TEXT -1 45 "Error, (in plot/options2d) bad argument, PNG\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 313 "There is anoth er interesting aspect to this investigation. The function we studied a bove, f(x) = ln( 1 + x ), has a vertical asymptote at x = -1. When we \+ expanded a series about x = 0, ouir interval of convergence was (-1,1) . But what happens when we expand around other values of x, farther fr om the singularity?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f := x -> ln(1+x); c1 := 0; c2 := 2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&ar rowGF(-%#lnG6#,&\"\"\"F09$F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#c1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "g := unapply(convert(series( f(x), \+ x = c1,12), polynom), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6# %\"xG6\"6$%)operatorG%&arrowGF(,89$\"\"\"*&#F.\"\"#F.*$)F-F1F.F.!\"\"* &#F.\"\"$F.)F-F7F.F.*&#F.\"\"%F.*$)F-F;F.F.F4*&#F.\"\"&F.)F-F@F.F.*&#F .\"\"'F.*$)F-FDF.F.F4*&#F.\"\"(F.)F-FIF.F.*&#F.\"\")F.*$)F-FMF.F.F4*&# F.\"\"*F.)F-FRF.F.*&#F.\"#5F.*$)F-FVF.F.F4*&#F.\"#6F.)F-FenF.F.F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "h := unapply(convert(se ries( f(x), x = c2, 12), polynom), x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%\"xG6\"6$%)operatorG%&arrowGF(,<-%#lnG6#\"\"$\"\"\"*&# F1F0F19$F1F1#\"\"#F0!\"\"*&#F1\"#=F1*$),&F4F1F6F7F6F1F1F7*&#F1\"#\")F1 )F=F0F1F1*&#F1\"$C$F1*$)F=\"\"%F1F1F7*&#F1\"%:7F1)F=\"\"&F1F1*&#F1\"%u VF1*$)F=\"\"'F1F1F7*&#F1\"&4`\"F1)F=\"\"(F1F1*&#F1\"&)[_F1*$)F=\"\")F1 F1F7*&#F1\"'ZrF1)F=\"#6F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "pl ot( [ f(x), g(x), h(x) ], x = -0.85..7, y = -2..3, color = [red, blue, green], thickness = [3,1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7W7$$!3w*************\\)!#=$!31\")e)[)* >r*=!#<7$$!3Cn\"z%\\3Bs!)F*$!3'4c\"=R;AY;F-7$$!3gL$e*)phWk(F*$!3o2'3P \"e\"eW\"F-7$$!3(**\\P%[Dp;sF*$!3+YCoT^%*y7F-7$$!3Kmm\"zRB*)y'F*$!3d!o D?#)yf8\"F-7$$!3o\\(=n4CX/'F*$!3s%[f[pT[F*F*7$$!3/L3_&zC,I&F*$!3i[-po8 \\]vF*7$$!3_m;ae]\"ei$F*$!3ZF*$!3e\"Qd$G-Ad@F *7$$!32Q$3x6#4IE!#>$!3k&QHcq(HlEFY7$$\"3oKe9wJ:#H\"F*$\"3!=z)>B)H_@\"F *7$$\"3t*\\7.vLC!HF*$\"35B\"RRj3$[DF*7$$\"3OLekB#)ynXF*$\"3u+uLArFiPF* 7$$\"3N*\\iS)=!yA'F*$\"3dH](*R%39%[F*7$$\"3SomTv*H`$zF*$\"3\"pLsA;u=%e F*7$$\"3/L$3xk@$R%*F*$\"3w]?k2\"Grk'F*7$$\"3;+]70gC86F-$\"3o-Z,%o**[* )F*7$$\"3=$ek)z$oef\"F-$\"3O5F-7$$\"3gmm;(=ZE#>F-$\"3?I/aw(*[s5F-7$$\"3+]7ya%yp4#F-$\"3a7R;WpUI6F -7$$\"3Mmm\"f;\"H^AF-$\"3iJmE*>_!z6F-7$$\"3+]7Gnmf?CF-$\"33WZ(***\\\") H7F-7$$\"31]P%=a:=e#F-$\"3!oack!)peF\"F-7$$\"3'p;zW_I+v#F-$\"3D-.\")zR w@8F-7$$\"3z;Hd*Q/X!HF-$\"3Q%Ga$e38i8F-7$$\"3UL3x@`7rIF-$\"3$fEtIX>RS \"F-7$$\"33$ekBk(>WKF-$\"3EC))\\*y_bW\"F-7$$\"3U\\(=igd[R$F-$\"3wx@dJ^ V![\"F-7$$\"3'Q$ek%\\uvb$F-$\"3*feVHd!z;:F-7$$\"36++DipnDPF-$\"3k`q6B3 ,`:F-7$$\"3Q*\\i5VK,*QF-$\"3]Sp8&Q>se\"F-7$$\"33]7G`:D\\SF-$\"3\"fWK]- S#>;F-7$$\"3_+vV4i#fA%F-$\"3Y7a5Z?j`;F-7$$\"3Qmm\"fCwYQ%F-$\"3=TqL(=dN o\"F-7$$\"3I+]P%zvTb%F-$\"3g;PBS+b9IZA3x\"F-7$$\"3];zW,-oL]F-$\"3mkP!*RrN(z \"F-7$$\"35]i!>zJ))>&F-$\"3+htj`3OC=F-7$$\"3im;aewHg`F-$\"3/iV\"zF -7$$\"3wL3-F\\jeeF-$\"3&>T1F-7$$\"3UM[P-'F-$\"33!f!QFqH\\>F- 7$$\"3)4+]AUqa<'F-$\"3'*4JdB$o1(>F-7$$\"3!p;/\"[5O\\jF-$\"3Q%>:sP8Y*>F -7$$\"3WLL$o*3*[]'F-$\"3zlCeF\\b:?F-7$$\"3%3D\"ya\"=2n'F-$\"3b_$\\EC5u .#F-7$$\"3g](=(**fVHoF-$\"3!o6@nZ!*y0#F-7$$\"\"(\"\"!$\"3uN)z;aT%z?F-- %'COLOURG6&%$RGBG$\"*++++\"!\")$F[\\lF[\\lFe\\l-%*THICKNESSG6#\"\"$-F$ 6%7Z7$F($!3vmgIKHbQ=F-7$F>$!3#Gk(Q.\"3Q8\"F-7$FH$!3QCtCRco\\vF*7$FM$!3 ],-\")[@G.XF*7$FR$!3kQUYD-Ad@F*FV7$Fgn$\"3^A\\@B)H_@\"F*7$F\\o$\"3iJ$* *)o)3$[DF*7$Fao$\"3g-r>5bKiPF*7$Ffo$\"3glDAe7@V[F*7$F[p$\"395m1oX*=(eF *7$F`p$\"3!)HWzruDqoF*7$Fep$\"330X-.w!Q(*)F*7$Fjp$\"3Ajd[W'Rxe\"F-7$F_ q$\"3SfLzvpk4RF-7$Fdq$\"3*z'eqV;)p,\"!#;7$Fiq$\"39.Q2JC:bJFj_l7$F^r$\" 3S(4-'\\Nb!z(Fj_l7$Fcr$\"3k`-n]%Q%p?!#:7$Fhr$\"3z@RP`P[9YFd`l7$F]s$\"3 W%))pU%40Z5!#97$Fbs$\"3'Gk!['>S!p@F[al7$Fgs$\"3N]q(ef\"RAWF[al7$F\\t$ \"3IsOQ$=$R!>)F[al7$Fat$\"3mG'HE[,b`\"!#87$Fft$\"3M'o`)H8CYGFhal7$F[u$ \"3<+:6-,aUZFhal7$F`u$\"3/l1gQQlE!)Fhal7$Feu$\"3g'RN#Q,8[8!#77$Fju$\"3 _'zz;GO\"*=#Febl7$F_v$\"3e+I.x$Q?V$Febl7$Fdv$\"32#)=,s'*GRbFebl7$Fiv$ \"3%GC4Nb*)GP)Febl7$F^w$\"3IQB@zrI!G\"!#67$Fcw$\"3/$)\\U)R*yb=Fecl7$Fh w$\"3![\"Q#Q)ebYFFecl7$F]x$\"3Ab))*3>uK#RFecl7$Fbx$\"3e:R#=Wxyi&Fecl7$ Fgx$\"3!*4jpk`L?zFecl7$F\\y$\"3P8>dH$*G?6!#57$Fay$\"3-DU\"Hj*))[:Fhdl7 $Ffy$\"3qwF#p=6n8#Fhdl7$F[z$\"3dz!y*>qS8HFhdl7$F`z$\"3;83\"eP%)[%QFhdl 7$$\"3Q%3x^t:CE'F-$\"3;7)G1<#\\$\\%Fhdl7$Fez$\"3U-nK[M6S_Fhdl7$$\"3<]( oC(f7FkF-$\"3c+*e6lU;+'Fhdl7$Fjz$\"3[%>z#4)4D'oFhdl7$$\"39#H2e_/ye'F-$ \"3N2n$3G#y-zFhdl7$F_[l$\"3K/CY)*4_%3*Fhdl7$$\"3=,+Dxq2]nF-$\"3Azq@v`Q O5!\"*7$Fd[l$\"3WF.`7h\\!=\"Fbgl7$$\"3%H1*y**p2soF-$\"3U%[*4J#R_E\"Fbg l7$$\"3Iv$f)**zr9pF-$\"3)40To9yaN\"Fbgl7$$\"3k(oH****et&pF-$\"3!zr'yW; `^9Fbgl7$Fi[l$\"3;99RX[t`:Fbgl-F_\\l6&Fa\\lFe\\lFe\\lFb\\l-Fg\\l6#\"\" \"-F$6%7Y7$F($!3_8xHtziL9F-7$F>$!3P?z1XG7,5F-7$FH$!3\\411NH^MqF*7$FM$! 3C12g)*=_BVF*7$FR$!3%\\sE)om&e4#F*7$FW$!3iAj&3_OHY#FY7$Fgn$\"3u&fvI.'3 A7F*7$F\\o$\"319kS4qQ]DF*7$Fao$\"3uQzb2V#Gw$F*7$Ffo$\"3mAycCb`T[F*7$F[ p$\"3OUz(\\#z*=%eF*7$F`p$\"3A9un$eKrk'F*7$Fep$\"3'=K$\\u?D#[(F*7$Fjp$ \"3!p.@M)3-c#)F*7$F_q$\"3)*))**oTo**[*)F*7$Fdq$\"314:Hk4@R&*F*7$Fiq$\" 3]0im&R])>5F-7$F^r$\"3UI/aw(*[s5F-7$Fcr$\"3w7R;WpUI6F-7$Fhr$\"3%4km#*> _!z6F-7$F]s$\"3z***y***\\\")H7F-7$Fbs$\"3g$ocm!)peF\"F-7$Fgs$\"3vR)[Q) Rw@8F-7$F\\t$\"3c1([^*38i8F-7$Fat$\"3H:L;A(>RS\"F-7$Fft$\"3kk*\\2NabW \"F-7$F[u$\"3Syx['3T/[\"F-7$F`u$\"36Shj*>7o^\"F-7$Feu$\"3S\"zF8J#3`:F- 7$Fju$\"3a%=SwqDue\"F-7$F_v$\"3WV+$H'zw>;F-7$Fdv$\"3g1FVH/,b;F-7$Fiv$ \"3q2+H,$>mo\"F-7$F^w$\"3!\\%Hq\">F8s\"F-7$Fcw$\"3-(>z(3Y8bF-7$Fgx$\"3lvrQSC8 5?F-7$F\\y$\"3!Hd'>[-ed@F-7$Fay$\"3)QPg(f%pGP#F-7$Ffy$\"3QH([T,7!3FF-7 $F[z$\"3FH0='\\zP@$F-7$F`z$\"3nL%eXYn?!RF-7$Fez$\"3YcBiFB?u]F-7$F^fl$ \"3?^@M]&>gx&F-7$Fjz$\"3o*Hdh93jh'F-7$Fffl$\"3%erwZt%\\#p(F-7$F_[l$\"3 [3&)fx%f'))*)F-7$F^gl$\"3E5>@;VBZ5Fj_l7$Fd[l$\"31Q]`+u]B7Fj_l7$Fggl$\" 3+2^\"*)3c8L\"Fj_l7$F\\hl$\"3g3(oxH2%\\9Fj_l7$Fahl$\"3#fUZt)z_y:Fj_l7$ Fi[l$\"3`XwD(4]'> " 0 "" {MPLTEXT 1 0 42 "conv_p lot( f, g, h, -0.8, 7, c1, c2, .15);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7&7$$!\")!\"\"$!+7zV4;!\"*F$7$$!++++?s!#5$!+lT8!G\"F*7$F,F(" }} {PARA 8 "" 1 "" {TEXT -1 45 "Error, (in plot/options2d) bad argument, \+ PNG\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 433 "The red curve is the original function. the blue graph is the ser ies expanded around x = 0, and the purple bar indicates the interval o f convergence. The green graph is the series expanded about x = 2, and the yellow bar indicates its interval of convergence. From this diag ram, it appears the expansion about 2, has interval of convergence app roaching (-1, 5), instead of (-1,1) for the other. Lets look at anothe r expansion value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "c1 := 0; c2 := 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G\" \"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "g := unapply( conver t(series( f(x), x = c1, 12), polynom), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(,89$\"\"\"*&#F. \"\"#F.*$)F-F1F.F.!\"\"*&#F.\"\"$F.)F-F7F.F.*&#F.\"\"%F.*$)F-F;F.F.F4* &#F.\"\"&F.)F-F@F.F.*&#F.\"\"'F.*$)F-FDF.F.F4*&#F.\"\"(F.)F-FIF.F.*&#F .\"\")F.*$)F-FMF.F.F4*&#F.\"\"*F.)F-FRF.F.*&#F.\"#5F.*$)F-FVF.F.F4*&#F .\"#6F.)F-FenF.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " h := unapply( convert(series( f(x), x = c2, 12), polynom), x);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%\"xG6\"6$%)operatorG%&arrowG F(,<-%#lnG6#\"\"&\"\"\"*&#F1F0F19$F1F1#\"\"%F0!\"\"*&#F1\"#]F1*$),&F4F 1F6F7\"\"#F1F1F7*&#F1\"$v$F1)F=\"\"$F1F1*&#F1\"%+DF1*$)F=F6F1F1F7*&#F1 \"&Dc\"F1)F=F0F1F1*&#F1\"&]P*F1*$)F=\"\"'F1F1F7*&#F1\"'voaF1)F=\"\"(F1 F1*&#F1\"(+]7$F1*$)F=\"\")F1F1F7*&#F1\")D\"yv\"F1)F=\"\"*F1F1*&#F1\")] il(*F1*$)F=\"#5F1F1F7*&#F1\"*v$4r`F1)F=\"#6F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "conv_plot(f, g, h, -0.8, 12, c1, c2 , .15.2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$$!\")!\"\"$!+7zV4;! \"*F$7$$!++++?n!#5$!+r;u96F*7$F,F(" }}{PARA 8 "" 1 "" {TEXT -1 45 "Err or, (in plot/options2d) bad argument, PNG\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 192 "The expansion about x = 4 seem s to have an interval of convergence approaching (-1, 9 ). Its a litt le hard to see where it cuts off on the right. The following plot help s to demonstrate this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot( \{f(x), g(x), h(x), [[9,0], [ 9,h(9)]] \}, x = -.5..12, y = -1..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7Z7$$!3++++++++]!#=$!35*Q`NX#4J pF*7$$!3]mm;HdNvAF*$!3'*\\L@$=$p\"e#F*7$$\"3.)pm\"H#oU`*!#?$\"3KZW?RR5 *[*F57$$\"3iLLe*)4WhFF*$\"34:sC=7VQCF*7$$\"3YNL3F>@XaF*$\"31fd%oj;vM%F *7$$\"3[m;zptA;\")F*$\"3=#ozv?+7)fF*7$$\"3GLe*)f+Ef5!#<$\"3av[(QaPb1)F *7$$\"3/+D1kTn:8FJ$\"3#yVbt`0S&=FJ7$$\"3WLeR(*z&3e\"FJ$\"3,dh')\\\"*=A #*FJ7$$\"3;+D\"GQ\">X=FJ$\"3-$*\\$HPi!>\\!#;7$$\"36nmTg24<@FJ$\"3nNDq0 ^#[I#!#:7$$\"3wL$3-))zlN#FJ$\"3L='z&o\\hMxF\\o7$$\"3l++D1z=EEFJ$\"3'o[ $f%G$=HE!#97$$\"33++voH!p*GFJ$\"3Oh'o\"HV*>&zFgo7$$\"3c++v$4(ydJFJ$\"3 R!*=n!f&\\+@!#87$$\"3mLeRs[p%R$FJ$\"3rnfypY)*RZFbp7$$\"3YnmTN6SwOFJ$\" 3oMiGi=$4;\"!#77$$\"3lmmm\"HV]\"RFJ$\"3)*pw[#He;N#F]q7$$\"3U+Dc^3k#>%F J$\"3[\"QC<,N&p]F]q7$$\"30nm;HaH#\\gFbr7$$\"37n\"zW<5&yaFJ$\"3d+f!G+D0,\"!#57$$\"3*QL3-8I Qu&FJ$\"34%e)4\"35Lr\"Fbs7$$\"3(R$eR(*HU>gFJ$\"3]Zv*4$z:!*GFbs7$$\"3k+ vV)RF$fiFJ$\"3m+(\\asZ)oWFbs7$$\"3nL$eRsI%=lFJ$\"3*>263x'QBqFbs7$$\"3M ++]7)4hy'FJ$\"3a'oKQ:3(*4\"!\"*7$$\"3O+]7y:)z/(FJ$\"3]/S/Z6`w;F\\u7$$ \"3C+DcwfN,tFJ$\"3')QTF`>7%[#F\\u7$$\"3c**\\(o/&o#e(FJ$\"3u$yAR$Q`$y$F \\u7$$\"3nmm;HF_y7$$ \"3smTNr]R'4\"Ffn$\"3kI-OL9$QF#F_y7$$\"33]P4@!y(36Ffn$\"3?h+H(o[ad#F_y 7$$\"3WLL$3(4;@6Ffn$\"3c.Q$*4C,8HF_y7$$\"3*=z%*zxjV8\"Ffn$\"3E<(H33=oJ $F_y7$$\"3;]i:&emv9\"Ffn$\"3as&)HD**)3x$F_y7$$\"35++DcL?g6Ffn$\"3kZ0h$ f0xD%F_y7$$\"3=]PMF,%G<\"Ffn$\"3bED([[/5![F_y7$$\"3u7y]&4I'z6Ffn$\"3SC y7MNF=^F_y7$$\"35v=nj+U'=\"Ffn$\"3WP_^;7\\aaF_y7$$\"3YPf$=.5K>\"Ffn$\" 3w+D6@,n5eF_y7$$\"#7\"\"!$\"3))>0[d1(y='F_y-%'COLOURG6&%$RGBG$\"#5!\" \"$Fd]lFd]lF^^l-F$6$7Y7$F($!31-^0%=z$y`F*7$F.$!3Ep(pZP1S/#F*7$F3$\"3:O )Ri\\zQ9$!#>7$F9$\"3od4#='Q)z^#F*7$F>$\"3C[#[el&*[P%F*7$FC$\"3#GC6%oa[ ^fF*7$FH$\"3V6!o1F9mA(F*7$FN$\"33PLa`1'zR)F*7$FS$\"3KbF'Gon9[*F*7$FX$ \"3[\"y%)*GgjX5FJ7$Fhn$\"3'ff?*e6!p8\"FJ7$F^o$\"38hg;`F#4@\"FJ7$Fco$\" 3=2`e`>=)G\"FJ7$Fio$\"3MJxZG@=g8FJ7$F^p$\"3I7?<%)H)\\U\"FJ7$Fdp$\"3GN> l-\")R![\"FJ7$Fip$\"3;kw%f#)GDa\"FJ7$F_q$\"3Sr3jh0I#f\"FJ7$Fdq$\"37bJO -CCZ;FJ7$Fiq$\"3)Gc&yiyZ$p\"FJ7$F^r$\"3%[D,Y2h=u\"FJ7$Fdr$\"3'G&e9uW&e y\"FJ7$Fir$\"37+'pOy#yH=FJ7$F^s$\"3s*y^=g!\\o=FJ7$Fds$\"3iGUNO#G'3>FJ7 $Fis$\"3-(HGZ;#o[>FJ7$F^t$\"3![iHJW\"H#)>FJ7$Fct$\"3v`n+8EP.8K)QYDFJ7$Ffx$\"3QKhSQOd\"p#FJ7$F[y$ \"3t0,BEsS7HFJ7$Fay$\"3uMP&=!>TSKFJ7$Ffy$\"3oQGg*pY@o$FJ7$F`z$\"37\\A; f!)pGWFJ7$Fez$\"3!y#=Cx5zt[FJ7$Fjz$\"3cK$*RVg^0aFJ7$F_[l$\"3!eR7*\\2:& 3'FJ7$Fd[l$\"3j4+\\69B-pFJ7$Fi[l$\"3PR\\w]52OyFJ7$F^\\l$\"3#[P?p&o/W*) FJ7$Fc\\l$\"3LwZ+xJI@'*FJ7$Fh\\l$\"3+O`,+r@O5Ffn7$F]]l$\"3R.7o%[.s6\"F fn7$Fb]l$\"3H&y.H7oc?\"Ffn-Fh]l6&Fj]lF^^lF[^lF^^l-F$6$7U7$F($!3'GX*f0= ZJpF*7$$!3(HL$ekynPOF*$!3fc&*[mm\">_%F*7$F.$!3O;f`.Kp\"e#F*7$$!3w*\\PM _1+4\"F*$!3ILYfOe6a6F*7$F3$\"3OeX?RR5*[*F57$F9$\"3$y!fk(3J%QCF*7$F>$\" 3A#oioaRrM%F*7$FC$\"3$**)\\$R$)HA%fF*7$FH$\"3#Ho/0CnMA(F*7$FN$\"37`#4s g3qR)F*7$FS$\"3'=G)y2!>7[*F*7$FX$\"35[r>N.jX5FJ7$Fhn$\"3rwjB=,!p8\"FJ7 $F^o$\"3?'GvFcA4@\"FJ7$Fco$\"3YZ6!H$>=)G\"FJ7$Fio$\"3'*[e3F@=g8FJ7$F^p $\"3gA07%)H)\\U\"FJ7$Fdp$\"3O05l-\")R![\"FJ7$Fip$\"3]jw%f#)GDa\"FJFfal 7$Fdq$\"3MbJO-CCZ;FJ7$Fiq$\"37ZbyiyZ$p\"FJ7$F^r$\"3Lpkfu5'=u\"FJ7$Fdr$ \"3=\\w&RZaey\"FJ7$Fir$\"3Gs7D!y#yH=FJ7$F^s$\"3u5Zes0\\o=FJ7$Fds$\"31w w4K!G'3>FJ7$Fis$\"3wTkb?5o[>FJ7$F^t$\"3k'e,(zrG#)>FJ7$Fct$\"3:!HgeVdt, #FJ7$Fht$\"3)>.HHNTB0#FJ7$F^u$\"35\"RaVK@a3#FJ7$Fcu$\"3b)>H@()=k6#FJ7$ Fhu$\"3#ezpq!ou\\@FJ7$F]v$\"3)[)Rd(\\u(y@FJ7$Fbv$\"3ruM(*\\\\')3AFJ7$F gv$\"3Hckh@3PNAFJ7$F]w$\"3S-Dg*zmNE#FJ7$Fbw$\"33$[#\\t,R*G#FJ7$Fgw$\"3 x#>r5k)o:BFJ7$F\\x$\"3zw:R@%[2M#FJ7$Fax$\"3w$\\fZtFjO#FJ7$Ffx$\"3?^G$G 8g.R#FJ7$F[y$\"3\"\\`ceg`VT#FJ7$Fay$\"3&=pj*pIfPCFJ7$Ffy$\"3KP-)H!=[eC FJ7$F`z$\"3]kc>@!)*=[#FJ7$Fjz$\"3MYun9rQ-DFJ7$Fd[l$\"39Wz3:+yBDFJ7$F^ \\l$\"3EDUO6yR#>)fM#FJ-Fh]l6&Fj]lF^^lF^^lF[^l-%+AX ESLABELSG6$Q\"x6\"Q\"yFhcm-%%VIEWG6$;$!\"&F]^lFb]l;$F]^lFd]l$\"\"$Fd]l " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 365 "From this example, it appears that a vertical \+ asymptote is a like a brick wall which limits the convergence of a pow er series. Also, the interval of convergence must be symmetrical on ei ther side of the center of the expansion. These two factors work toget her to make the convergence intervals smaller as we choose expansion p oints closer to the vertical asymptote." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }