{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "Define" -1 256 "Times" 1 12 163 163 163 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "Emphasis" -1 257 "Times" 1 12 128 0 128 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "normal C" -1 258 "Times" 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Menu" -1 259 "" 0 0 163 163 163 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 260 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 280 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 285 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 290 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Geneva" 1 10 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 "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Monaco" 1 9 255 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 "R3 Font 0" -1 257 1 {CSTYLE "" -1 -1 "Geneva" 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 "R3 Font 2" -1 258 1 {CSTYLE "" -1 -1 "Gene va" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple Worksheets, 2001" }{TEXT 289 144 "\n(c) John H. Mathews Rus sell W. Howell\nmathews@fullerton.edu howell@westmont.edu\n\nCompl imentary software to accompany the textbook:" }}{PARA 257 "" 0 "" {TEXT 257 82 "COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9" }{TEXT 288 197 "\nJones and Bartlett Publi shers, Inc., 40 Tall Pine Drive, Sudbury, MA 01776\nTel e. (800) 832-0034; FAX: (508) 443-8000, E-mail: mkt@jbpu b.com, http://www.jbpub.com/" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 286 1 "\n" }{TEXT 256 65 "CHAPTER 4 SEQUENCES, JULIA and MANDELBROT S ETS, and Power Series" }{TEXT 290 2 "\n\n" }{TEXT 256 54 "Section 4.3 \+ Geometric Series and Convergence Theorems" }{TEXT 282 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 53 "We begin this section by studying series of the form " }{XPPEDIT 18 0 "Sum(z^n,n = 0 .. infinity)" "6#-%$SumG6$)%\"zG %\"nG/F(;\"\"!%)infinityG" }{TEXT -1 20 ", which is called a " }{TEXT 291 16 "geometric series" }{TEXT -1 50 ", one of the most important se ries in mathematics." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 32 "Theorem 4.11, (Geometric series)" }{TEXT -1 7 " If " }{XPPEDIT 18 0 "`|z|<1`;" "6#%&|grz|gr<1G" }{TEXT -1 14 ", \+ the series " }{XPPEDIT 18 0 "Sum(z^n,n = 0 .. infinity)" "6#-%$SumG6$) %\"zG%\"nG/F(;\"\"!%)infinityG" }{TEXT -1 16 " converges to " } {XPPEDIT 18 0 "f(z) = 1/(1-z);" "6#/-%\"fG6#%\"zG*&\"\"\"F),&F)F)F'!\" \"F+" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "That is, if " }{XPPEDIT 18 0 "`|z|<1`;" "6#%&|grz|g r<1G" }{TEXT -1 10 ", then " }{XPPEDIT 18 0 "Sum(z^n,n = 0 .. infin ity)" "6#-%$SumG6$)%\"zG%\"nG/F(;\"\"!%)infinityG" }{TEXT -1 5 " = \+ " }{XPPEDIT 18 0 "1+z+z^2+z^3+`...`+z^n+`...`;" "6#,0\"\"\"F$%\"zGF$*$ F%\"\"#F$*$F%\"\"$F$%$...GF$)F%%\"nGF$F*F$" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "1/(1-z)" "6#*&\"\"\"F$,&F$F$%\"zG!\"\"F'" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "`|z|>1`;" "6#%&|grz|gr>1G" }{TEXT -1 21 ", then t he series " }{XPPEDIT 18 0 "Sum(z^n,n = 0 .. infinity)" "6#-%$SumG6$ )%\"zG%\"nG/F(;\"\"!%)infinityG" }{TEXT -1 13 " diverges. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 13 "Corollary 4.2" }{TEXT -1 7 " \+ If " }{XPPEDIT 18 0 "`|z|>1`;" "6#%&|grz|gr>1G" }{TEXT -1 16 ", the series " }{XPPEDIT 18 0 "Sum(1/(z^n),n = 1 .. infinity);" "6#-%$Sum G6$*&\"\"\"F')%\"zG%\"nG!\"\"/F*;F'%)infinityG" }{TEXT -1 16 " conver ges to " }{XPPEDIT 18 0 "f(z) = 1/(z-1);" "6#/-%\"fG6#%\"zG*&\"\"\"F) ,&F'F)F)!\"\"F+" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 13 "Tha t is, if " }{XPPEDIT 18 0 "`|z|>1`;" "6#%&|grz|gr>1G" }{TEXT -1 10 ", then " }{XPPEDIT 18 0 "Sum(1/(z^n),n = 1 .. infinity);" "6#-%$SumG 6$*&\"\"\"F')%\"zG%\"nG!\"\"/F*;F'%)infinityG" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "1/z+1/(z^2)+1/(z^3)+`...`+1/(z^n)+`...`;" "6#,.*&\"\"\" F%%\"zG!\"\"F%*&F%F%*$F&\"\"#F'F%*&F%F%*$F&\"\"$F'F%%$...GF%*&F%F%)F&% \"nGF'F%F.F%" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "1/(z-1);" "6#*&\"\" \"F$,&%\"zGF$F$!\"\"F'" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "`|z|<1`;" "6# %&|grz|gr<1G" }{TEXT -1 20 ", then the series " }{XPPEDIT 18 0 "Sum( 1/(z^n),n = 1 .. infinity);" "6#-%$SumG6$*&\"\"\"F')%\"zG%\"nG!\"\"/F* ;F'%)infinityG" }{TEXT -1 13 " diverges. " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 13 "Corollary 4.3" }{TEXT -1 7 " If " }{XPPEDIT 18 0 "`|z|<1`;" "6#%&|grz|gr<1G" }{TEXT -1 17 ", then for all " } {XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "1/(1-z) = 1+z+z^2+z^3+`...`+z^`n-1`+z^n/(1-z);" "6#/*&\"\"\"F%,&F%F%%\"zG!\"\"F (,0F%F%F'F%*$F'\"\"#F%*$F'\"\"$F%%$...GF%)F'%$n-1GF%*&)F'%\"nGF%,&F%F% F'F(F(F%" }{TEXT -1 4 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 257 "" 0 "" {TEXT 256 23 "Example 4.13, Page 146." } {TEXT 283 14 " Show that " }{XPPEDIT 18 0 "sum((1- i)^n/2^n, n=0..i nfinity) = 1- i" "6#/-%$sumG6$*&),&\"\"\"F*%\"iG!\"\"%\"nGF*)\"\"#F- F,/F-;\"\"!%)infinityG,&F*F*F+F," }{TEXT 269 3 " ,\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "n:='n': Zn:='Zn': z:='z':\nZ := n -> (1-I)^n/2^n :\nz[n] = Z(n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 260 39 "This is a geometric series with ratio: \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "R := Z(n+1)/Z(n):\n`R = `, z [n+1]/z[n] = R;\nR := simplify(R):\n`R = `, z[n+1]/z[n] = R;\n`|R| = ` , abs(z[n+1]/z[n]) = abs(R);\n`|R| = `, abs(z[n+1]/z[n]) = evalf(abs(R ));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 261 7 "Since " }{XPPEDIT 18 0 "abs(R) < 1" "6#2-%$absG6#%\"R G\"\"\"" }{TEXT 270 41 " , the sum is given by the calculation:\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "S:='S':\nS := 1/(1-R):\n`R = `, z[n +1]/z[n] = R;\n`S = 1/(1 - R) ` = S;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 262 48 "Or we can use \+ Maple to sum the series directly.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "n:='n': S:='S': Zn:='Zn': z:='z':\nZn := (1 - I)^n/2^n:\nz[n] = Z n;\nS := sum(Zn,n=0..infinity):\n`Find the sum of the series:`;\nSum( z[n], n=1..infinity) = Sum(Zn, n=1..infinity);\nSum(z[n], n=1..infinit y) = S;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 " " 0 "" {TEXT 256 23 "Example 4.14, Page 146." }{TEXT 284 13 " Evalua te " }{XPPEDIT 18 0 "sum((i^n / 2^n), n=3..infinity)" "6#-%$sumG6$*&) %\"iG%\"nG\"\"\")\"\"#F)!\"\"/F);\"\"$%)infinityG" }{TEXT 271 4 " .\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "n:='n': Z:='Z': Zn:='Zn': z:='z ':\nZ := n -> I^n/2^n:\nz[n] = Z(n);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 263 39 "This is a geometric series with ratio:\n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 166 "R := Z(n+1)/Z(n):\n`R = `, z[n+1]/z[n] = R;\n R := simplify(R):\n`R = `, z[n+1]/z[n] = R;\n`|R| = `, abs(z[n+1]/z[n] ) = abs(R);\n`|R| = `, abs(z[n+1]/z[n]) = evalf(abs(R));" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 264 53 "Since |R| < 1, the sum is given by th e calculation:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "S:='S':\nS := 1 /(1-R) - 1 - R - R^2:\n`R = `, z[n+1]/z[n] = R;\n`S = 1/(1 - R) - 1 \+ - R - R^2 ` = S;" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 265 48 "Or we can use Maple to sum the series directly.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "n:='n': S:='S': Zn:='Zn': z:='z':\nZn := I^n/2^n:\nz [n] = Zn;\nS := sum(Zn, n=3..infinity):\n`Find the sum of the series: `;\nSum(z[n], n=1..infinity) = Sum(Zn, n=1..infinity);\nSum(z[n], n=1. .infinity) = S;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 " The geometric series is used in the pro of of the following theorem, known as the " }{TEXT 292 10 "ratio test " }{TEXT -1 196 ". It is one of the most commonly used tests for dete rmining the convergence or divergence of series. The proof is similar \+ to the one used for real series, and is left for the reader to establi sh." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 39 "Theorem 4.12 (d'Alembert's ratio test)" }{TEXT 293 3 " " } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "If " }{XPPEDIT 18 0 "Sum(zeta[n],n = 0 .. infinity);" "6 #-%$SumG6$&%%zetaG6#%\"nG/F);\"\"!%)infinityG" }{TEXT -1 48 " is a c omplex series with the property that " }{XPPEDIT 18 0 "Limit(abs(zet a[n+1])/abs(zeta[n]),n = infinity) = L;" "6#/-%&LimitG6$*&-%$absG6#&%% zetaG6#,&%\"nG\"\"\"F0F0F0-F)6#&F,6#F/!\"\"/F/%)infinityG%\"LG" } {TEXT -1 5 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "then the series is absolutely convergent if " }{XPPEDIT 18 0 "`L < 1`;" "6#%&L~<~1G" }{TEXT -1 21 ", and divergent if " } {XPPEDIT 18 0 "`1 < L`;" "6#%&1~<~LG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 23 "Example 4.15, Page 147." }{TEXT 285 14 " Show that " }{XPPEDIT 18 0 "sum((1 - i)^n/n!, n=0..infinity)" "6#-%$sumG6$*&),&\"\"\"F)%\"iG !\"\"%\"nGF)-%*factorialG6#F,F+/F,;\"\"!%)infinityG" }{TEXT 272 13 " \+ converges.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "L:='L': n:='n': R: ='R': Z:='Z': Zn:='Zn': z:='z':\nZ := n -> (1 - I)^n / n!:\nz[n] = Z(n );\nR := Z(n+1)/Z(n):\nz[n+1]/z[n] = R;\nR := simplify(R):\nz[n+1]/z[n ] = R;\nL := limit(Z(n+1)/Z(n) , n=infinity):\nlimit(z[n+1]/z[n], n=in finity) = L;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 266 7 "Since " }{XPPEDIT 18 0 "abs(L) < 1" "6#2-%$a bsG6#%\"LG\"\"\"" }{TEXT 273 62 " , the series converges. Let us use Maple to find the sum. \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "n:=' n': S:='S': Zn:='Zn': z:='z':\nZn := (1-I)^n/n!:\nz[n] = Zn;\nS := su m(Zn, n=0..infinity):\n`Find the sum of the series:`;\nSum(z[n], n=1.. infinity) = Sum(Zn, n=1..infinity);\nSum(z[n], n=1..infinity) = S;\nSu m(z[n], n=1..infinity) = evalc(S);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 23 "Example 4.16, Page 1 47." }{TEXT 287 13 " Show that " }{XPPEDIT 18 0 "sum((z - i)^n/2^n, \+ n=0..infinity)" "6#-%$sumG6$*&),&%\"zG\"\"\"%\"iG!\"\"%\"nGF*)\"\"#F-F ,/F-;\"\"!%)infinityG" }{TEXT 274 20 " converges for all " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 276 13 " in the disk " }{XPPEDIT 18 0 "abs(z - i) < 2" "6#2-%$absG6#,&%\"zG\"\"\"%\"iG!\"\"\"\"#" }{TEXT 275 3 " . \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "L:='L': n:='n': R:='R': Z:=' Z': Zn:='Zn': z:='z':\nZ := n -> (z-I)^n/2^n:\n\n\nz[n] = Z(n);\nR := \+ Z(n+1)/Z(n):\nz[n+1]/z[n] = R;\nR := simplify(R):\nz[n+1]/z[n] = R;\nL := limit(Z(n+1)/Z(n) , n=infinity):\n`L = `, limit(z[n+1]/z[n], n=inf inity) = L;\n`|L| ` = abs(L);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 257 "" 0 "" {TEXT 267 7 "Hence " }{XPPEDIT 18 0 "ab s(L) = abs((z - i)/2)" "6#/-%$absG6#%\"LG-F%6#*&,&%\"zG\"\"\"%\"iG!\" \"F-\"\"#F/" }{TEXT 279 15 " , now solve " }{XPPEDIT 18 0 "abs((z - \+ i)/2) < 1" "6#2-%$absG6#*&,&%\"zG\"\"\"%\"iG!\"\"F*\"\"#F,F*" }{TEXT 277 14 " and obtain " }{XPPEDIT 18 0 "abs(z - i) < 2" "6#2-%$absG6#, &%\"zG\"\"\"%\"iG!\"\"\"\"#" }{TEXT 280 46 " .\nTherefore, the series \+ converges the disk " }{XPPEDIT 18 0 "abs(z - i) < 2" "6#2-%$absG6#,& %\"zG\"\"\"%\"iG!\"\"\"\"#" }{TEXT 281 2 " ." }}}{EXCHG {PARA 257 "" 0 "" {TEXT 278 34 "Let us use Maple to find the sum.\n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 205 "n:='n': S:='S': Zn:='Zn': z:='z':\nZn := (z-I )^n/2^n:\nz[n] = Zn;\nS := sum(Zn, n=0..infinity):\n`Find the sum of \+ the series:`;\nSum(z[n], n=1..infinity) = Sum(Zn, n=1..infinity);\nSum (z[n], n=1..infinity) = S;" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 268 19 " End of Section 4.3." }}}}{MARK "0 0 0" 27 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }