{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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{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 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 1 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 }{CSTYLE "" -1 294 "" 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 "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 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple Worksheets, 2001" }{TEXT 282 144 "\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 281 197 "\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 278 1 "\n" }{TEXT 256 65 "CHAPTER 4 SEQUENCES, JULIA and MANDELBROT S ETS, and Power Series" }{TEXT 271 2 "\n\n" }{TEXT 256 33 "Section 4.1 \+ Sequences and Series" }{TEXT 272 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 23 " In formal terms, a " }{TEXT 283 16 "complex sequence" }{TEXT -1 147 " is a function whose domain is the positive integers and whose range is a subset of the complex numbers. For convenience, we at time s use the term " }{TEXT 284 8 "sequence" }{TEXT -1 14 " rather than c " }{TEXT 285 15 "omplex sequence" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "If we wish a function " } {XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 68 " to represent an arbitrary \+ sequence, we could specify it by writing " }{XPPEDIT 18 0 "s(1) = z[1] ;" "6#/-%\"sG6#\"\"\"&%\"zG6#F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "s(2) = z[2];" "6#/-%\"sG6#\"\"#&%\"zG6#F'" }{TEXT -1 24 ", and so on. The \+ values " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "z[2]" "6#&%\"zG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "z[3];" "6#&%\"zG6#\"\"$" }{TEXT -1 22 ", ..., are called the " } {TEXT 286 5 "terms" }{TEXT -1 107 " of a sequence, and mathematicians, being generally lazy when it comes to things like this, often refer t o " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "z[2]" "6#&%\"zG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "z[3];" "6#&%\"zG6#\"\"$" }{TEXT -1 183 ", etc., as the sequence its elf, even though they are really speaking of the range of the sequence when they do this. Mathematicians are also not so fussy about startin g a sequence at " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6#\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "z[1];" "6#&%\"zG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\"#" }{TEXT -1 84 ", ..., etc., wou ld also be acceptable notation, provided all terms were defined. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 36 "Definition 4.1: Limit of a sequence" }{TEXT 287 3 " " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The expression " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "Limit(z[n],n = infinity) = zeta;" "6#/-%&LimitG6$&%\"zG6#%\"nG/F*%)infinityG%%zetaG" }{TEXT -1 14 " me ans that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "for any real number " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" } {TEXT -1 41 ">0, there corresponds a positive integer " }{XPPEDIT 18 0 "N[epsilon];" "6#&%\"NG6#%(epsilonG" }{TEXT -1 18 " which depends on " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 12 " such that \+ " }{XPPEDIT 18 0 "z[n]" "6#&%\"zG6#%\"nG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "D[epsilo n];" "6#&%\"DG6#%(epsilonG" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "zeta" "6#% %zetaG" }{TEXT -1 12 ") whenever " }{XPPEDIT 18 0 "n;" "6#%\"nG" } {TEXT -1 1 ">" }{XPPEDIT 18 0 "N[epsilon]" "6#&%\"NG6#%(epsilonG" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Theorem 4.1" }{TEXT 288 3 " " }{TEXT -1 4 "Let " } {XPPEDIT 18 0 "z[n] = x[n]+i*y[n];" "6#/&%\"zG6#%\"nG,&&%\"xG6#F'\"\" \"*&%\"iGF,&%\"yG6#F'F,F," }{TEXT -1 5 " and " }{XPPEDIT 18 0 "zeta = \+ u+i*v;" "6#/%%zetaG,&%\"uG\"\"\"*&%\"iGF'%\"vGF'F'" }{TEXT -1 9 ". Th en, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 18 0 "Limit(z[n],n = infinity) = zeta;" "6#/-%&LimitG6 $&%\"zG6#%\"nG/F*%)infinityG%%zetaG" }{TEXT -1 23 " if and only if " }{XPPEDIT 18 0 "Limit(x[n],n = infinity) = u;" "6#/-%&LimitG6$&% \"xG6#%\"nG/F*%)infinityG%\"uG" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "Limit(y[n],n = infinity) = v;" "6#/-%&LimitG6$&%\"yG6#%\"nG/F*%)inf inityG%\"vG" }{TEXT -1 4 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 22 "Example 4.1, Page 128." } {TEXT 273 10 " Find " }{XPPEDIT 18 0 "limit((sqrt(n) + i*(n+1))/n , n=infinity)" "6#-%&limitG6$*&,&-%%sqrtG6#%\"nG\"\"\"*&%\"iGF,,&F+F,F ,F,F,F,F,F+!\"\"/F+%)infinityG" }{TEXT 262 4 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "L:='L': n:='n': Xn:='Xn': Yn:='Yn': Zn:='Zn': z:= 'z':\nXn := sqrt(n)/n:\nYn := (n+1)/n:\nZn := Xn + I*Yn:\nL := limit( Zn, n=infinity):\nz[n] = Zn; ` `;\nlimit( z[n], n=infinity) = L;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 22 "Example 4.2, Page 128. " }{TEXT 274 27 " Show that the sequence " }{XPPEDIT 18 0 "z[n] = ( 1 + i)^n" "6#/&%\"zG6#%\"nG),&\"\"\"F*%\"iGF*F'" }{TEXT 263 12 " dive rges.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "n:='n': L:='L': n:='n': Zn:='Zn': z:='z':\nZn := (1 + I)^n:\nL := limit(Zn, n=infinity):\nz[ n] = Zn; ` `;\nlimit( z[n], n=infinity) = L;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 10 "Maple did " }{TEXT 257 3 "NOT" }{TEXT -1 16 " find the \+ limit!" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 260 45 "Now look at the real and imaginary parts of " }{XPPEDIT 18 0 "z[n]" "6#&%\"zG6#%\"nG" } {TEXT 264 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "L:='L': n:='n ': Xn:='Xn': Yn:='Yn': Zn:='Zn':\nx:='x': y:='y': z:='z':\nXn := sqrt( 2)^n * cos(n*pi/4):\nYn := sqrt(2)^n * sin(n*pi/4):\nZn := Xn + I*Yn: \nL := limit(Xn, n=infinity) + I*limit(Yn, n=infinity):\n`The general term is:`;\nz[n], ` = `,x[n] + y[n] = Zn; ` `;\nlimit( z[n], n=infini ty) = L;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Theorem 4.2" } {TEXT 294 3 " " }{TEXT -1 3 "If " }{XPPEDIT 18 0 "\{z[n]\};" "6#<#&% \"zG6#%\"nG" }{TEXT -1 28 " is a Cauchy sequence, then " }{XPPEDIT 18 0 "\{z[n]\}" "6#<#&%\"zG6#%\"nG" }{TEXT -1 11 " converges." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 32 "Definition 4.3: Infinite Series" } {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 22 "The formal expression \+ " }{XPPEDIT 18 0 "Sum(z[k],k = 1 .. infinity)" "6#-%$SumG6$&%\"zG6#%\" kG/F);\"\"\"%)infinityG" }{TEXT -1 16 " is called an " }{TEXT 289 15 "infinite series" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "z[1];" "6#& %\"zG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\" #" }{TEXT -1 23 ", etc., are called the " }{TEXT 290 5 "terms" }{TEXT -1 17 " of the series. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 30 "If there is a complex number " }{XPPEDIT 18 0 "S; " "6#%\"SG" }{TEXT -1 13 " for which " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S" "6#%\"SG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "Limit(S[n],n = infinity);" "6#-%&L imitG6$&%\"SG6#%\"nG/F)%)infinityG" }{TEXT -1 6 " = " }{XPPEDIT 18 0 "Limit(Sum(z[k],k = 1 .. n),n = infinity);" "6#-%&LimitG6$-%$SumG6$& %\"zG6#%\"kG/F,;\"\"\"%\"nG/F0%)infinityG" }{TEXT -1 4 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "we will say tha t the infinite series " }{XPPEDIT 18 0 "Sum(z[k],k = 1 .. infinity)" "6#-%$SumG6$&%\"zG6#%\"kG/F);\"\"\"%)infinityG" }{TEXT -1 2 " " } {TEXT 291 9 "converges" }{TEXT -1 5 " to " }{XPPEDIT 18 0 "S" "6#%\"S G" }{TEXT -1 13 ", and that " }{XPPEDIT 18 0 "S" "6#%\"SG" }{TEXT -1 9 " is the " }{TEXT 292 3 "sum" }{TEXT -1 25 " of the infinite ser ies. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " When this happens, we write " }{XPPEDIT 18 0 "S = Sum(z[k],k = 1 .. i nfinity);" "6#/%\"SG-%$SumG6$&%\"zG6#%\"kG/F+;\"\"\"%)infinityG" } {TEXT -1 4 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Theorem \+ 4.3" }{TEXT -1 8 " Let " }{XPPEDIT 18 0 "z[n] = x[n]+i*y[n];" "6#/& %\"zG6#%\"nG,&&%\"xG6#F'\"\"\"*&%\"iGF,&%\"yG6#F'F,F," }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "S = U+i*V;" "6#/%\"SG,&%\"UG\"\"\"*&%\"iGF'%\"V GF'F'" }{TEXT -1 10 ". Then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "S;" "6#%\"SG" } {TEXT -1 5 " = " }{XPPEDIT 18 0 "Sum(z[n],n = 1 .. infinity)" "6#-%$ SumG6$&%\"zG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "Sum(x[n]+i*y[n],n = 1 .. infinity);" "6#-%$SumG6$,&&%\" xG6#%\"nG\"\"\"*&%\"iGF+&%\"yG6#F*F+F+/F*;F+%)infinityG" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "if and only if " }{XPPEDIT 18 0 "U = Sum(x[n],n = 1 .. infinity);" "6# /%\"UG-%$SumG6$&%\"xG6#%\"nG/F+;\"\"\"%)infinityG" }{TEXT -1 11 " a nd " }{XPPEDIT 18 0 "V = Sum(y[n],n = 1 .. infinity);" "6#/%\"VG-%$ SumG6$&%\"yG6#%\"nG/F+;\"\"\"%)infinityG" }{TEXT -1 4 " . " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 28 "T heorem 4.4 (n-th term test)" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 293 2 " " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " } {XPPEDIT 18 0 "Sum(z[n],n = 1 .. infinity);" "6#-%$SumG6$&%\"zG6#%\"nG /F);\"\"\"%)infinityG" }{TEXT -1 40 " is a convergent complex series , then " }{XPPEDIT 18 0 "Limit(z[n],n = infinity) = 0;" "6#/-%&LimitG6 $&%\"zG6#%\"nG/F*%)infinityG\"\"!" }{TEXT -1 4 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 22 "Exam ple 4.3, Page 130." }{TEXT 275 24 " Show that the series " } {XPPEDIT 18 0 "sum((1 + i*n*(-1)^n )/n^2, n=1..infinity)" "6#-%$sumG6$ *&,&\"\"\"F(*(%\"iGF(%\"nGF(),$F(!\"\"F+F(F(F(*$F+\"\"#F./F+;F(%)infin ityG" }{TEXT 265 13 " converges.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "n:='n': S:='S': Xn:='Xn': Yn:='Yn': Zn:='Zn': z:='z':\nXn := 1/n^ 2:\nYn := (-1)^n/n:\nZn := Xn + I*Yn:\nz[n] = Zn;\nS := sum(Xn, n=1.. infinity) + I*sum(Yn, n=1..infinity):\n`Find the sum of the series:`; \nSum(z[n], n=1..infinity) = \n Sum(Xn, n=1..infinity)+ I*Sum(Yn, n=1 ..infinity);\nSum(z[n], n=1..infinity) = S;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 22 "Example 4.4, Page 130." }{TEXT 277 24 " Show that the series " }{XPPEDIT 18 0 "sum(((-1)^n + i)/n, n=1..infinity)" "6#-%$sumG6$*&,&),$\"\"\"! \"\"%\"nGF*%\"iGF*F*F,F+/F,;F*%)infinityG" }{TEXT 266 12 " diverges. \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "n:='n': S:='S': Xn:='Xn': Yn :='Yn': Zn:='Zn': z:='z':\nXn := (-1)^n /n:\nYn := 1/n:\nZn := Xn + I* Yn:\nz[n] = Zn;\nS := sum(Xn, n=1..infinity) + I*sum(Yn, n=1..infinit y):\n`Find the sum of the series:`;\nSum(z[n], n=1..infinity) = \n Su m(Xn, n=1..infinity)+ I*Sum(Yn, n=1..infinity);\nSum(z[n], n=1..infini ty) = S;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 22 "Example 4.5, Page 131." }{TEXT 276 24 " Show th at the series " }{XPPEDIT 18 0 "sum((1+ i)^n, n=1..infinity)" "6#-%$s umG6$),&\"\"\"F(%\"iGF(%\"nG/F*;F(%)infinityG" }{TEXT 267 12 " diverg es.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "n:='n': S:='S': Zn:='Zn': z:='z':\nZn := (1 + i)^n:\nz[n] = Zn;\nS := sum(Zn, n=1..infinity): \n`Find the sum of the series:`;\nSum(z[n], n=1..infinity) = Sum(Zn, n =1..infinity);\nSum(z[n], n=1..infinity) = S;\n`But this is NOT the ri ght answer !`;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 257 9 "BAD news!" }{TEXT 279 73 " MAPLE subst ituted into a divergent geometric series ! It substituted " } {XPPEDIT 18 0 "Z = 1+i" "6#/%\"ZG,&\"\"\"F&%\"iGF&" }{TEXT 268 21 " i nto the \nformula " }{XPPEDIT 18 0 "1/(1-Z) - 1" "6#,&*&\"\"\"F%,&F%F %%\"ZG!\"\"F(F%F%F(" }{TEXT 269 40 " . You should always check to se e if " }{XPPEDIT 18 0 "1 < abs(Z)" "6#2\"\"\"-%$absG6#%\"ZG" }{TEXT 270 22 " , or risk getting a " }{TEXT 257 5 "wrong" }{TEXT 280 8 " an swer!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "n:='n': S:='S': Z:='Z': Zn:='Zn': z:='z':\nZn := (1 + I)^n:\nZ \+ := 1 + I:\nS := 1/(1-Z) - 1:\n`1/(1-Z) - 1 ` = S;\n`where Z ` = Z; \n`The general term is:`;\nz[n] = Zn; ` `;\n`But`;\n1 < `|1+I| `;\n1 < abs(Z);\n1 < evalf(abs(Z));\n`Therefore the series diverges.`;" }}} {EXCHG {PARA 257 "" 0 "" {TEXT 261 19 "End of Section 4.1." }}}}{MARK "0 0 0" 26 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }