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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 43 "Alternating series and absolute c
onvergence" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.
C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:  13.1.2008" }}
{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 18 "Alternating series" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "A series of
 the form" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)
^(n-1)*a[i],n = 1 .. infinity) = a[1]-a[2]+a[3]-a[4];" "6#/-%$SumG6$*&
),$\"\"\"!\"\",&%\"nGF*F*F+F*&%\"aG6#%\"iGF*/%\"nG;F*%)infinityG,*&F/6
#F*F*&F/6#\"\"#F+&F/6#\"\"$F*&F/6#\"\"%F+" }{TEXT -1 10 " + . . . ," }
}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "0 < a[n];" "6#2\"
\"!&%\"aG6#%\"nG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "1 <= n;" "6#1\"
\"\"%\"nG" }{TEXT -1 15 ", is called an " }{TEXT 265 18 "alternating s
eries" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "The sign of the \+
terms alternates between" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"
" }{TEXT -1 6 "and +." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 268 
34 "__________________________________" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "For such a series to
 " }{TEXT 265 8 "converge" }{TEXT -1 23 " it is sufficient that:" }}
{PARA 0 "" 0 "" {TEXT -1 6 "(1)   " }{XPPEDIT 18 0 "a[i+1]<=a[i]" "6#1
&%\"aG6#,&%\"iG\"\"\"F)F)&F%6#F(" }{TEXT -1 10 ", for all " }{XPPEDIT 
18 0 "i >=1" "6#1\"\"\"%\"iG" }{TEXT -1 88 ", that is, the magnitude o
f the terms steadily decreases, or at least does not increase." }}
{PARA 0 "" 0 "" {TEXT -1 6 "(2)   " }{XPPEDIT 18 0 "Limit(a[i],i=infin
ity)=0" "6#/-%&LimitG6$&%\"aG6#%\"iG/F*%)infinityG\"\"!" }{TEXT -1 2 "
. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 34 "________________
__________________" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 32 "These two criteria comprise the " }
{TEXT 265 23 "alternating series test" }{TEXT -1 29 " for convergence \+
of a series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 4 "Let " }{XPPEDIT 18 0 "S[n]=Sum((-1)^(n-1)*a[i],i=1..n)" "6#/&%\"
SG6#%\"nG-%$SumG6$*&),$\"\"\"!\"\",&F'F.F.F/F.&%\"aG6#%\"iGF./F4;F.F'
" }{XPPEDIT 18 0 " ``=a[1]-a[2]+a[3]-a[4] + ` . . . `+(-1)^(n-1)*a[n]
" "6#/%!G,.&%\"aG6#\"\"\"F)&F'6#\"\"#!\"\"&F'6#\"\"$F)&F'6#\"\"%F-%(~.
~.~.~GF)*&),$F)F-,&%\"nGF)F)F-F)&F'6#F9F)F)" }{TEXT -1 8 " be the " }
{TEXT 272 1 "n" }{TEXT -1 30 " th partial sum of the series." }}{PARA 
0 "" 0 "" {TEXT -1 34 "The sequence of even partial sums " }{XPPEDIT 
18 0 "S[2],S[4],S[6],` . . . `;" "6&&%\"SG6#\"\"#&F$6#\"\"%&F$6#\"\"'%
(~.~.~.~G" }{TEXT -1 96 " is an increasing sequence (or at least non-d
ecreasing), while the sequence of odd partial sums " }{XPPEDIT 18 0 "S
[1],S[3],S[5],` . . . `;" "6&&%\"SG6#\"\"\"&F$6#\"\"$&F$6#\"\"&%(~.~.~
.~G" }{TEXT -1 43 "is decreasing (or at least non-increasing)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "For examp
le, consider the alternating series: " }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "Sum((-1)^(n-1)*``(1/n),n = 1 .. infinity) = 1-1/2+
1/3-1/4+1/5-1/6;" "6#/-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF*F*F+F*-%!G6#*&
F*F*F-F+F*/F-;F*%)infinityG,.F*F**&F*F*\"\"#F+F+*&F*F*\"\"$F+F**&F*F*
\"\"%F+F+*&F*F*\"\"&F+F**&F*F*\"\"'F+F+" }{TEXT -1 10 " + . . . ." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The first
 few odd partial sums are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "seq(evalf(sum((-1)^(i-1)*1/i
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!$\"+LLLL$)!#5$\"+LLLLyF($\"+&4Q_f(F($\"+1#\\jX(F($\"+:,WltF($\"+^vL,t
F($\"+/&=PD(F($\"+)z`p@(F($\"+KSr(=(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 53 " . . . and the first few even partia
l sums are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 49 "seq(evalf(sum((-1)^(i-1)*1/i,i=1..2*k)),k=1..
10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$\"+++++]!#5$\"+LLLLeF%$\"+nmm
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}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "In this
 example, the even partial sums are all less than the odd partial sums
, and this is true in general." }}{PARA 0 "" 0 "" {TEXT -1 43 "This fo
rces the two sequences to converge. " }}{PARA 256 "" 0 "" {TEXT -1 1 "
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mmmmmhFail7$$\"\")F($\"1&4Q_4Q_M'Fail7$$\"#5F($\"12#\\j?\\jX'Fail7$$\"
#7F($\"1#y1@y1@`'Fail7$$\"#9F($\"1P=0P=0(e'Fail7$$\"#;F($\"1/&=P]=(GmF
ail7$$\"#=F($\"1&f!GU#)RhmFail7$F[]l$\"1zUvJSr(o'Fail-F`]l6&Fb]l$\"*++
++\"FiuF(F(-Ff]l6#%&POINTGFi]l-F$6&7,7$$\"\"\"F(Fhfm7$$FjbmF($\"1MLLLL
LL$)Fail7$$\"\"&F($\"1LLLLLLLyFail7$$FfbmF($\"1&4Q_4Q_f(Fail7$$\"\"*F(
$\"12#\\j?\\jX(Fail7$$\"#6F($\"1:,W:,WltFail7$$\"#8F($\"1^vL^vL,tFail7
$$\"#:F($\"1/&=P]=PD(Fail7$$\"#<F($\"1^h$yz`p@(Fail7$$\"#>F($\"1!Ga<.9
x=(Fail-F`]l6&Fb]lF(F(F_fmFafmFi]l-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6
$%\"nG%!G-%%VIEWG6$;F(F[]l;F($FhdmFgbm" 1 2 4 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 136 "At present we cannot be sure that the two sequ
ences converge to the same number, so suppose that they converge to tw
o different numbers:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "Limit(S[2*k-1],k=infinity)=S[odd]" "6#/-%&LimitG6$&%\"SG6#,&*&\"\"#
\"\"\"%\"kGF-F-F-!\"\"/F.%)infinityG&F(6#%$oddG" }}{PARA 0 "" 0 "" 
{TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Li
mit(S[2*k],k=infinity)=S[even" "6#/-%&LimitG6$&%\"SG6#*&\"\"#\"\"\"%\"
kGF,/F-%)infinityG&F(6#%%evenG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 26 "Now we use the hypothesis " }{XPPEDIT 18 0 "Limit(a[n],n \+
= infinity) = 0;" "6#/-%&LimitG6$&%\"aG6#%\"nG/F*%)infinityG\"\"!" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 
18 0 "S[2*k-1]-S[2*k]=a[2*k]" "6#/,&&%\"SG6#,&*&\"\"#\"\"\"%\"kGF+F+F+
!\"\"F+&F&6#*&F*F+F,F+F-&%\"aG6#*&F*F+F,F+" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Thus on the one
 hand: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``(S
[2*k-1]-S[2*k]),k = infinity) = Limit(S[2*k-1],k = infinity)-Limit(S[2
*k],k = infinity);" "6#/-%&LimitG6$-%!G6#,&&%\"SG6#,&*&\"\"#\"\"\"%\"k
GF1F1F1!\"\"F1&F,6#*&F0F1F2F1F3/F2%)infinityG,&-F%6$&F,6#,&*&F0F1F2F1F
1F1F3/F2F8F1-F%6$&F,6#*&F0F1F2F1/F2F8F3" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
``=S[odd]-S[even]" "6#/%!G,&&%\"SG6#%$oddG\"\"\"&F'6#%%evenG!\"\"" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 21 "and on the other hand" 
}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``(S[2*k-1]-S
[2*k]),k = infinity) = Limit(a[2*k],k = infinity);" "6#/-%&LimitG6$-%!
G6#,&&%\"SG6#,&*&\"\"#\"\"\"%\"kGF1F1F1!\"\"F1&F,6#*&F0F1F2F1F3/F2%)in
finityG-F%6$&%\"aG6#*&F0F1F2F1/F2F8" }{TEXT -1 5 " = 0." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Therefore " }
{XPPEDIT 18 0 "S[odd]-S[even]=0" "6#/,&&%\"SG6#%$oddG\"\"\"&F&6#%%even
G!\"\"\"\"!" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "S[odd]=S[even]
" "6#/&%\"SG6#%$oddG&F%6#%%evenG" }{TEXT -1 81 ", the two sequences co
nverge to the same limit. The series therfore converges to " }
{XPPEDIT 18 0 "S=S[odd]" "6#/%\"SG&F$6#%$oddG" }{XPPEDIT 18 0 "`` = S[
even];" "6#/%!G&%\"SG6#%%evenG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 10 "Note that " }{TEXT 273 1 "S" }{TEXT -1 14 " lies between \+
" }{XPPEDIT 18 0 "S[n]" "6#&%\"SG6#%\"nG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "S[n+1]" "6#&%\"SG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 11 " fo
r every " }{TEXT 274 1 "n" }{TEXT -1 4 ", so" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(S-S[n])<=abs(S[n]-S[n+1])" "6#1-%$a
bsG6#,&%\"SG\"\"\"&F(6#%\"nG!\"\"-F%6#,&&F(6#F,F)&F(6#,&F,F)F)F)F-" }
{TEXT -1 3 " = " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 12 
", for every " }{TEXT 275 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 9 "Thus the " }{TEXT 266 5 "
error" }{TEXT 265 86 " made in using a partial sum to approximate the \+
sum of the series does not exceed the " }{TEXT 266 18 "first omitted t
erm" }{TEXT 265 14 " in magnitude." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 27 "Alternating series examples" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" 
{TEXT 258 8 "Question" }{TEXT 270 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 
21 "Show that the series " }{XPPEDIT 18 0 "Sum((-1)^(n-1)*``(1/n),n = \+
1 .. infinity);" "6#-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF)F)F*F)-%!G6#*&F)
F)F,F*F)/F,;F)%)infinityG" }{TEXT -1 11 " converges." }}{PARA 0 "" 0 "
" {TEXT -1 95 "Estimate the error when using a partial sum of 1000 ter
ms to approximate the sum of the series." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 257 "" 0 "" {TEXT 267 8 "Solution" }{TEXT 276 4 ":   " }}
{PARA 0 "" 0 "" {TEXT -1 77 "The terms of the series alternate in sign
 and steadily decrease in magnitude." }}{PARA 0 "" 0 "" {TEXT -1 12 "F
urthermore " }{XPPEDIT 18 0 "Limit((-1)^(n-1)*``(1/n),n = infinity) = \+
0;" "6#/-%&LimitG6$*&),$\"\"\"!\"\",&%\"nGF*F*F+F*-%!G6#*&F*F*F-F+F*/F
-%)infinityG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 114 "Thu
s the two criteria for the alternating series test are satisfied, whic
h demonstrates that the series converges." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "T
he error in using a partial sum of 1000 terms to approximate the sum o
f the series is no greater in magnitude than that of the first omitted
 term: " }{XPPEDIT 18 0 "1/1001;" "6#*&\"\"\"F$\"%,5!\"\"" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "evalf(1/1001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+!**4+***!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 256 4 "Note" }{TEXT -1 32 ": The series converges to ln(2)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "Sum((-1)^(n-1)*(1/n),n=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%$SumG6$*&)!\"\",&%\"nG\"\"\"F(F+F+F*F(/F*;F+%)infin
ityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#\"\"#" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The actual error in \+
using a partial sum of 1000 terms to approximate the sum of the series
 is . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "abs(ln(2)-Sum((-1)^(n-1)/n,n=1..1000));\nevalf(%);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$absG6#,&-%#lnG6#\"\"#\"\"\"-%$SumG
6$*&)!\"\",&%\"nGF+F1F+F+F3F1/F3;F+\"%+5F1" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"(+v*\\!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 234 "The following expression for the tail of the s
eries gives an alternative way to calculate the error, and also provid
es a way to draw curves which connect the the points for the two seque
nces of odd and even partial sums of the series." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sum((-1)^(n-
1)/n,n=x+1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&)!\"\",&
%\"xG\"\"\"F)F)F),&-%$PsiG6#,&F)F)F(#F)\"\"#F)-F,6#,&F(F/F/F)F&F)#F&F0
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "e:=x->1/2*(Psi(1+1/2*x)-Psi(1/2*x+1/2));\nevalf(e(100
0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6#%\"xG6\"6$%)operator
G%&arrowGF(,&-%$PsiG6#,&\"\"\"F19$#F1\"\"#F3-F.6#,&F2F3F3F1#!\"\"F4F(F
(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'](*\\!\"*" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "e:=x->1/
2*(Psi(1+1/2*x)-Psi(1/2*x+1/2)):\nf := x -> ln(2)-e(x):\ng := x -> ln(
2)+e(x):\nptsE := [seq([2*k,sum((-1)^(i-1)*1/i,i=1..2*k)],k=1..10)]:\n
ptsO := [seq([2*k-1,sum((-1)^(i-1)*1/i,i=1..2*k-1)],k=1..10)]:\nplot([
f(x),g(x),ln(2),ptsE,ptsO],x=0..20,\n  style=[line$3,point$2],color=[g
ray$2,COLOR(RGB,0,.7,0),red,blue],\n  linestyle=[2,2,3],symbol=circle,
view=[0..20,0..1.2],labels=[`n`,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 
520 309 309 {PLOTDATA 2 "6*-%'CURVESG6&7en7$$\"\"!F)F(7$$\"+S&)G\\a!#6
$\"*[h'GU!#57$$\"+3x&)*3\"F0$\"*.BJ+)F07$$\"+ilyM;F0$\"+N!R!R6F07$$\"+
<arz@F0$\"+v[^W9F07$$\"+DJdpKF0$\"+a\"oH(>F07$$\"+M3VfVF0$\"+b_V8CF07$
$\"+j&*)fD'F0$\"+;KlDIF07$$\"+#H[D:)F0$\"+3ol(\\$F07$$\"+%pU&G5!\"*$\"
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$\"+<q\"*HgF07$$\"+]Z/NaFW$\"+66![4'F07$$\"+]$fC&eFW$\"+Y.6\\hF07$$\"+
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dt$\"+OL$3Z(F07$Fit$\"+c:1ZuF07$F^u$\"+c8+FuF07$Fcu$\"+O&RqS(F07$Fhu$
\"+Jg$zQ(F07$F^v$\"+@9^stF07$Fcv$\"+E,)pN(F07$Fhv$\"+E9/UtF07$F]w$\"+m
mSGtF07$Fbw$\"+,$[gJ(F07$Fgw$\"+En>.tF07$F\\x$\"+\"RfBH(F07$Fax$\"+\"*
QY\"G(F07$Ffx$\"+wL9ssF07$F[y$\"+^\\]isF07$F`y$\"+O0#RD(F07$Fey$\"+w-T
XsF07$Fjy$\"+EH^PsF07$F_z$\"+Y&f'HsF07$Fdz$\"+ctYAsF07$Fiz$\"+1AY:sF07
$F^[l$\"+O,%)3sF07$Fc[l$\"+;:-.sF07$Fh[l$\"+'pXm>(F07$F]\\l$\"+Y8>\">(
F07$Fb\\l$\"+'e<c=(F07$Fg\\l$\"+1A]!=(F07$F\\]l$\"+y&H_<(F0F`]lFf]lFj]
l-F$6&7S7$F($\"3'GX*f0=ZJp!#=7$$\"39LLLL3VfVFbilF`il7$$\"3%pmm;H[D:)Fb
ilF`il7$$\"3MLLLe0$=C\"!#<F`il7$$\"3iLLL3RBr;F\\jlF`il7$$\"3imm;zjf)4#
F\\jlF`il7$$\"3ULL$e4;[\\#F\\jlF`il7$$\"3!)****\\i'y]!HF\\jlF`il7$$\"3
oLL$ezs$HLF\\jlF`il7$$\"3=++]7iI_PF\\jlF`il7$$\"3Onmm;_M(=%F\\jlF`il7$
$\"3%QLL$3y_qXF\\jlF`il7$$\"3]+++]1!>+&F\\jlF`il7$$\"3J+++]Z/NaF\\jlF`
il7$$\"3;+++]$fC&eF\\jlF`il7$$\"3qLL$ez6:B'F\\jlF`il7$$\"3/nmm;=C#o'F
\\jlF`il7$$\"3Mnmmm#pS1(F\\jlF`il7$$\"3<++]i`A3vF\\jlF`il7$$\"3Rmmmm(y
8!zF\\jlF`il7$$\"3K,+]i.tK$)F\\jlF`il7$$\"3!3++v3zMu)F\\jlF`il7$$\"3Yo
mm\"H_?<*F\\jlF`il7$$\"3-nm;zihl&*F\\jlF`il7$$\"39LLL3#G,***F\\jlF`il7
$$\"3WLLezw5V5!#;F`il7$$\"3/++v$Q#\\\"3\"F_^mF`il7$$\"3]LL$e\"*[H7\"F_
^mF`il7$$\"3-+++qvxl6F_^mF`il7$$\"31++]_qn27F_^mF`il7$$\"37++Dcp@[7F_^
mF`il7$$\"3+++]2'HKH\"F_^mF`il7$$\"3`mmmwanL8F_^mF`il7$$\"35+++v+'oP\"
F_^mF`il7$$\"3CLLeR<*fT\"F_^mF`il7$$\"3C+++&)Hxe9F_^mF`il7$$\"3gmm\"H!
o-*\\\"F_^mF`il7$$\"3:++DTO5T:F_^mF`il7$$\"3emmmT9C#e\"F_^mF`il7$$\"3
\"****\\i!*3`i\"F_^mF`il7$$\"3_LLL$*zym;F_^mF`il7$$\"3fLL$3N1#4<F_^mF`
il7$$\"3!pm;HYt7v\"F_^mF`il7$$\"3A+++q(G**y\"F_^mF`il7$$\"3'pmmT6KU$=F
_^mF`il7$$\"3eLLL`v&Q(=F_^mF`il7$$\"30++DOl5;>F_^mF`il7$$\"3A++v.Uac>F
_^mF`il7$F\\]lF`il-%&COLORG6&Fc]lF)$\"\"(!\"\"F)Ff]l-F[^l6#\"\"$-F$6&7
,7$$F]^lF)$\"3++++++++]Fbil7$$\"\"%F)$\"3qLLLLLLLeFbil7$$\"\"'F)$\"3'p
mmmmmm;'Fbil7$$\"\")F)$\"3!\\4Q_4Q_M'Fbil7$$\"#5F)$\"3s1#\\j?\\jX'Fbil
7$$\"#7F)$\"3U#y1@y1@`'Fbil7$$\"#9F)$\"3;P=0P=0(e'Fbil7$$\"#;F)$\"3p.&
=P]=(GmFbil7$$\"#=F)$\"33&f!GU#)RhmFbil7$F\\]l$\"3OzUvJSr(o'Fbil-Fa]l6
&Fc]l$\"*++++\"FjuF(F(-Fg]l6#%&POINTGFj]l-F$6&7,7$$\"\"\"F)Fifm7$$F[cm
F)$\"3qLLLLLLL$)Fbil7$$\"\"&F)$\"3ELLLLLLLyFbil7$$FgbmF)$\"3!\\4Q_4Q_f
(Fbil7$$\"\"*F)$\"3]1#\\j?\\jX(Fbil7$$\"#6F)$\"3-:,W:,WltFbil7$$\"#8F)
$\"39^vL^vL,tFbil7$$\"#:F)$\"3o.&=P]=PD(Fbil7$$\"#<F)$\"3(3:Oyz`p@(Fbi
l7$$\"#>F)$\"3#)zUvJSr(=(Fbil-Fa]l6&Fc]lF(F(F`fmFbfmFj]l-%'SYMBOLG6#%'
CIRCLEG-%+AXESLABELSG6$%\"nG%!G-%%VIEWG6$;F(F\\]l;F($FidmFhbm" 1 2 4 
1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" 
"Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 277 2 ": " }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Show that the series " }
{XPPEDIT 18 0 "Sum((-1)^(n-1)*``(1/(n*(n+1))),n = 1 .. infinity);" "6#
-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF)F)F*F)-%!G6#*&F)F)*&F,F),&F,F)F)F)F)
F*F)/F,;F)%)infinityG" }{TEXT -1 11 " converges." }}{PARA 0 "" 0 "" 
{TEXT -1 95 "Estimate the error when using a partial sum of 1000 terms
 to approximate the sum of the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 257 "" 0 "" {TEXT 267 8 "Solution" }{TEXT 278 2 ": " }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The terms of the series alterna
te in sign and steadily decrease in magnitude." }}{PARA 0 "" 0 "" 
{TEXT -1 12 "Furthermore " }{XPPEDIT 18 0 "Limit((-1)^(n-1)*``(1/(n*(n
+1))),n = infinity) = 0;" "6#/-%&LimitG6$*&),$\"\"\"!\"\",&%\"nGF*F*F+
F*-%!G6#*&F*F**&F-F*,&F-F*F*F*F*F+F*/F-%)infinityG\"\"!" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 114 "Thus the two criteria for the alter
nating series test are satisfied, which demonstrates that the series c
onverges." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "The error in using a partial sum \+
of 1000 terms to approximate the sum of the series is no greater in ma
gnitude than that of the first omitted term: " }{XPPEDIT 18 0 "1/(1001
*`.`*1002);" "6#*&\"\"\"F$*(\"%,5F$%\".GF$\"%-5F$!\"\"" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "evalf(1/(1001*1002));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]
)p+(**!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 4 "Note" }{TEXT -1 26 ": The series converges to " }{XPPEDIT 18 0 
"2*ln(2)-1" "6#,&*&\"\"#\"\"\"-%#lnG6#F%F&F&F&!\"\"" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 60 "Sum((-1)^(n-1)/(n*(n+1)),n=1..infinity);\nvalue(%);\nevalf(%);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\",&%\"nG\"\"\"F(F+F+
*&F*F+,&F*F+F+F+F+F(/F*;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,&!\"\"\"\"\"-%#lnG6#\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
*hVH'Q!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 99 "The actual error in using a partial sum of 1000 terms to approx
imate the sum of the series is . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "abs(-1+2*ln(2)-Sum((-1)^(n
-1)/(n*(n+1)),n=1..1000));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%$absG6#,(!\"\"\"\"\"-%#lnG6#\"\"#F,-%$SumG6$*&)F',&%\"nGF(F'F(F(*
&F3F(,&F3F(F(F(F(F'/F3;F(\"%+5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"%*)\\!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 234 "The following expression for the tail of the series gives an a
lternative way to calculate the error, and also provides a way to draw
 curves which connect the the points for the two sequences of odd and \+
even partial sums of the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sum((-1)^(n-1)/(n*(n+1)),n=x
+1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)!\"\",&%\"xG
\"\"\"F*F*F*,(*(,&\"\"#F*F)F*F*,&-%$PsiG6#,&#\"\"$F.F*F)#F*F.F*-F16#,&
F*F*F)F6F'F*,&!\"#F*F)F;F*F6F.F*F)F*F*F**&F(F*F-F*F'F'" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "e:=x->(
1/2*(2+x)*(Psi(3/2+1/2*x)-Psi(1+1/2*x))*(-2-2*x)+2+x)/\n       ((1+x)*
(2+x));\nevalf(e(1000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6#
%\"xG6\"6$%)operatorG%&arrowGF(*&,(*(,&\"\"#\"\"\"9$F1F1,&-%$PsiG6#,&#
\"\"$F0F1F2#F1F0F1-F56#,&F1F1F2F:!\"\"F1,&!\"#F1F2F@F1F:F0F1F2F1F1*&,&
F1F1F2F1F1F/F1F>F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(!***)\\!
#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 459 "e:=x->(1/2*(2+x)*(Psi(3/2+1/2*x)-Psi(1+1/2*x))*(-2-2
*x)+2+x)/\n       ((1+x)*(2+x)):\nf := x->-1+2*ln(2)+e(x):\ng := x->-1
+2*ln(2)-e(x):\nptsE := [seq([2*k,sum((-1)^(i-1)*1/(i*(i+1)),i=1..2*k)
],k=1..10)]:\nptsO := [seq([2*k-1,sum((-1)^(i-1)*1/(i*(i+1)),i=1..2*k-
1)],k=1..10)]:\nplot([f(x),g(x),2*ln(2)-1,ptsE,ptsO],x=0..20,style=[li
ne$3,point$2],\n  color=[gray$2,COLOR(RGB,0,.7,0),red,blue],linestyle=
[2,2,3],\n  symbol=circle,view=[0..20,0.2..0.5],labels=[`n`,``]);" }}
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;&QFbjl-F]^l6&F_^l$\"*++++\"FfvF(F(-Fc^l6#%&POINTGFf^l-F$6&7,7$$\"\"\"
F)$\"3++++++++]Fbjl7$$F[dmF)$\"3'ommmmmm;%Fbjl7$$\"\"&F)$\"3A+++++++SF
bjl7$$FgcmF)$\"3N!>w/>w/%RFbjl7$$\"\"*F)$\"3n7%)p7%)p7RFbjl7$$\"#6F)$
\"3U(*oa(*oa(*QFbjl7$$\"#8F)$\"3I)Q*Q)Q*Q))QFbjl7$$\"#:F)$\"3Q2qV2qV#)
QFbjl7$$\"#<F)$\"3'fu;,/_$yQFbjl7$$\"#>F)$\"3ie&3N1Ga(QFbjl-F]^l6&F_^l
F(F(F`gmFbgmFf^l-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$%\"nG%!G-%%VIEWG6
$;F(Fh]l;$Fi^lFhcm$FchmFhcm" 1 2 4 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" 
{TEXT 258 8 "Question" }{TEXT 279 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 21 "Show that the series " }{XPPEDIT 18 0 "Sum((-1)^(n-1
)*``(5^n/n!),n = 1 .. infinity);" "6#-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF
)F)F*F)-%!G6#*&)\"\"&F,F)-%*factorialG6#F,F*F)/F,;F)%)infinityG" }
{TEXT -1 11 " converges." }}{PARA 0 "" 0 "" {TEXT -1 93 "Estimate the \+
error when using a partial sum of 20 terms to approximate the sum of t
he series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 
267 8 "Solution" }{TEXT 280 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 42 "The terms of the series alternate in sign." }}{PARA 0 "" 
0 "" {TEXT -1 109 "Initially the terms increase in magnitude but from \+
the 5 th term onwards they steadily decrease in magnitude." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eval
f(seq(5^n/n!,n=1..15));" }}{PARA 12 "" 1 "" {XPPMATH 20 "61$\"\"&\"\"!
$\"++++]7!\")$\"+LLL$3#F($\"+nm;/EF(F+$\"+*))Q,<#F($\"+1#*4]:F($\"+S+7
)o*!\"*$\"+6*)G#Q&F3$\"+bW9\"p#F3$\"+![ZKA\"F3$\"+*\\ko4&!#5$\"++DLg>F
<$\"+*\\(=,q!#6$\"+m\"HPL#FA" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 12 "Furthermore " }{XPPEDIT 18 0 "Limit((-1)^
(n-1)*``(5^n/n!),n = infinity) = 0;" "6#/-%&LimitG6$*&),$\"\"\"!\"\",&
%\"nGF*F*F+F*-%!G6#*&)\"\"&F-F*-%*factorialG6#F-F+F*/F-%)infinityG\"\"
!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 46 "Limit((-1)^(n-1)*5^n/n!,n=infinity);\nvalue(%
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&*&)!\"\",&%\"nG\"\"
\"F)F,F,)\"\"&F+F,F,-%*factorialG6#F+F)/F+%)infinityG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 
";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Th
e two criteria for the alternating series test are satisfied for the t
ail of the series: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "Sum((-1)^(n-1)*``(5^n/n!),n = 5 .. infinity);" "6#-%$SumG6$*&),$\"
\"\"!\"\",&%\"nGF)F)F*F)-%!G6#*&)\"\"&F,F)-%*factorialG6#F,F*F)/F,;F2%
)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "The error in usi
ng a partial sum of 20 terms to approximate the sum of the series is n
o greater in magnitude than that of the first omitted term: " }
{XPPEDIT 18 0 "5^21/21!;" "6#*&\"\"&\"#@-%*factorialG6#F%!\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "evalf((5^21)/21!);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+%f0JL*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 256 4 "Note" }{TEXT -1 26 ": The series converges to " }
{XPPEDIT 18 0 "1-exp(-5);" "6#,&\"\"\"F$-%$expG6#,$\"\"&!\"\"F*" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 80 "Sum((-1)^(n-1)*5^n/n!,n=1..infinity);\nsimplify(
expand(value(%)));\nS := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%$SumG6$*&*&)!\"\",&%\"nG\"\"\"F)F,F,)\"\"&F+F,F,-%*factorialG6#F+F)/
F+;F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#!\"&!\"
\"\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG$\"+I0iK**!#5" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "The actua
l error in using a partial sum of 20 terms to approximate the sum of t
he series is . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 51 "1-exp(-5)-Sum((-1)^(n-1)*5^n/n!,n=1..20);\ne
valf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$-%$expG6#!\"&!\"
\"-%$SumG6$*&*&)F),&%\"nGF$F)F$F$)\"\"&F1F$F$-%*factorialG6#F1F)/F1;F$
\"#?F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&Jf(!#5" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 234 "The following express
ion for the tail of the series gives an alternative way to calculate t
he error, and also provides a way to draw curves which connect the the
 points for the two sequences of odd and even partial sums of the seri
es." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "simplify(sum((-1)^(n-1)*5^n/n!,n=x+1..infinity));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&-%$expG6#!\"&\"\"\",&-%&GAMMAG6#
,&%\"xGF*F*F*F*-F-6$F/F)!\"\"F*F*F,F3F3" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "e:=x->abs(exp(-5)*(GA
MMA(x+1,-5)-GAMMA(x+1))/GAMMA(x+1));\nevalf(e(20));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"eGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$absG6#*&*
&-%$expG6#!\"&\"\"\",&-%&GAMMAG6$,&9$F5F5F5F4F5-F86#F:!\"\"F5F5F<F>F(F
(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+H(*4$f(!#:" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "e:=x->
abs(exp(-5)*(GAMMA(x+1,-5)-GAMMA(x+1))/GAMMA(x+1)):\nf := x->1-exp(-5)
+e(x):\ng := x->1-exp(-5)-e(x):\nptsE := [seq([2*k,sum((-1)^(i-1)*5^i/
i!,i=1..2*k)],k=1..10)]:\nptsO := [seq([2*k-1,sum((-1)^(i-1)*5^i/i!,i=
1..2*k-1)],k=1..10)]:\nplot([f(x),g(x),1-exp(-5),ptsE,ptsO],x=0..20,\n
  style=[line$3,point$2],color=[gray$2,COLOR(RGB,0,.7,0),red,blue],\n \+
 linestyle=[2,2,3],symbol=circle,labels=[`n`,``]);" }}{PARA 13 "" 1 "
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3!)****\\i'y]!HFc_mFg^m7$$\"3oLL$ezs$HLFc_mFg^m7$$\"3=++]7iI_PFc_mFg^m
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^m7$$\"3J+++]Z/NaFc_mFg^m7$$\"3;+++]$fC&eFc_mFg^m7$$\"3qLL$ez6:B'Fc_mF
g^m7$$\"3/nmm;=C#o'Fc_mFg^m7$$\"3Mnmmm#pS1(Fc_mFg^m7$$\"3<++]i`A3vFc_m
Fg^m7$$\"3Rmmmm(y8!zFc_mFg^m7$$\"3K,+]i.tK$)Fc_mFg^m7$$\"3!3++v3zMu)Fc
_mFg^m7$$\"3Yomm\"H_?<*Fc_mFg^m7$$\"3-nm;zihl&*Fc_mFg^m7$$\"39LLL3#G,*
**Fc_mFg^m7$$\"3WLLezw5V5!#;Fg^m7$$\"3/++v$Q#\\\"3\"FfcmFg^m7$$\"3]LL$
e\"*[H7\"FfcmFg^m7$$\"3-+++qvxl6FfcmFg^m7$$\"31++]_qn27FfcmFg^m7$$\"37
++Dcp@[7FfcmFg^m7$$\"3+++]2'HKH\"FfcmFg^m7$$\"3`mmmwanL8FfcmFg^m7$$\"3
5+++v+'oP\"FfcmFg^m7$$\"3CLLeR<*fT\"FfcmFg^m7$$\"3C+++&)Hxe9FfcmFg^m7$
$\"3gmm\"H!o-*\\\"FfcmFg^m7$$\"3:++DTO5T:FfcmFg^m7$$\"3emmmT9C#e\"Ffcm
Fg^m7$$\"3\"****\\i!*3`i\"FfcmFg^m7$$\"3_LLL$*zym;FfcmFg^m7$$\"3fLL$3N
1#4<FfcmFg^m7$$\"3!pm;HYt7v\"FfcmFg^m7$$\"3A+++q(G**y\"FfcmFg^m7$$\"3'
pmmT6KU$=FfcmFg^m7$$\"3eLLL`v&Q(=FfcmFg^m7$$\"30++DOl5;>FfcmFg^m7$$\"3
A++v.Uac>FfcmFg^m7$Fd`lFg^m-%&COLORG6&F[alF)$\"\"(!\"\"F)F^al-Fcal6#\"
\"$-F$6&7,7$$FealF)$!3++++++++vFc_m7$$\"\"%F)$!3SLLLLL$3F\"Ffcm7$$\"\"
'F)$!3Obbbbb0o$)Fc_m7$$\"\")F)$!3kJguJN=bDFc_m7$$\"#5F)$\"3)QS8sB4'f8F
i^m7$$\"#7F)$\"3'G8,e`>_\\)Fi^m7$$\"#9F)$\"3S8<b&GLav*Fi^m7$$\"#;F)$\"
3\"=HGd;xe\"**Fi^m7$$\"#=F)$\"354!=xio8$**Fi^m7$Fd`l$\"3m'*G-*fWD$**Fi
^m-Fi`l6&F[al$\"*++++\"FjnF(F(-F_al6#%&POINTGFbal-F$6&7,7$$\"\"\"F)$\"
\"&F)7$$FbhmF)$\"3SLLLLLLL8Ffcm7$Fb\\nFf\\n7$$F^hmF)$\"3\"z]Oz]OH8(Fc_
m7$$\"\"*F)$\"3T7()=z`5FGFc_m7$$\"#6F)$\"3)oBzMS3#f8Fc_m7$$\"#8F)$\"3q
98a._bX5Fc_m7$$\"#:F)$\"3G\">s@?1)))**Fi^m7$$\"#<F)$\"31Q%HB)oKP**Fi^m
7$$\"#>F)$\"3=fyX*eOH$**Fi^m-Fi`l6&F[alF(F(Fg[nFi[nFbal-%'SYMBOLG6#%'C
IRCLEG-%+AXESLABELSG6$%\"nG%!G-%%VIEWG6$;F(Fd`l%(DEFAULTG" 1 2 4 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv
e 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }
}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 281 2 ": " }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Show that the series " }
{XPPEDIT 18 0 "Sum((-1)^(n-1)*``(n/(2*n^2-1)),n = 1 .. infinity);" "6#
-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF)F)F*F)-%!G6#*&F,F),&*&\"\"#F)*$F,F3F
)F)F)F*F*F)/F,;F)%)infinityG" }{TEXT -1 11 " converges." }}{PARA 0 "" 
0 "" {TEXT -1 95 "Estimate the error when using a partial sum of 1000 \+
terms to approximate the sum of the series." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 257 "" 0 "" {TEXT 267 8 "Solution" }{TEXT 282 2 ": " }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The terms of the series a
lternate in sign and steadily decrease in magnitude." }}{PARA 0 "" 0 "
" {TEXT -1 12 "Furthermore " }{XPPEDIT 18 0 "Limit((-1)^(n-1)*``(n/(2*
n^2-1)),n = infinity) = 0;" "6#/-%&LimitG6$*&),$\"\"\"!\"\",&%\"nGF*F*
F+F*-%!G6#*&F-F*,&*&\"\"#F**$F-F4F*F*F*F+F+F*/F-%)infinityG\"\"!" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 114 "Thus the two criteria f
or the alternating series test are satisfied, which demonstrates that \+
the series converges." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 146 "The error in using a partial sum of 20 terms to appro
ximate the sum of the series is no greater in magnitude than that of t
he first omitted term: " }{XPPEDIT 18 0 "1001/(2*`.`*1001^2-1);" "6#*&
\"%,5\"\"\",&*(\"\"#F%%\".GF%F$F(F%F%!\"\"F*" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ev
alf(1001/(2*1001^2-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)[2]*\\
!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 4 "N
ote" }{TEXT -1 10 ": Maple's " }{TEXT 0 3 "sum" }{TEXT -1 55 " procedu
re can find a value for the sum of this series." }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Sum((-1)^(n
-1)*n/(2*n^2-1),n=1..infinity);\nvalue(%);\nS := evalf(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*()!\"\",&%\"nG\"\"\"F+F(F+F*F+,&*&
\"\"#F+)F*F.F+F+F+F(F(/F*;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%*hypergeomG6%7%\"\"#,&\"\"\"F)*&F'!\"\"F'#F)F'F+,&F)F)*&F'F+F'
F,F)7$,&F'F)*&F'F+F'F,F+,&F'F)*&F'F+F'F,F)F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"SG$\"+>;Vz\")!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 52 "The sum of the series is approximately \+
0.8179431620." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "The actual error is approximately . . ." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "S-Sum((-1)
^(n-1)*n/(2*n^2-1),n=1..1000);\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&$\"+>;Vz\")!#5\"\"\"-%$SumG6$*()!\"\",&%\"nGF'F'F-F
'F/F',&*&\"\"#F')F/F2F'F'F'F-F-/F/;F'\"%+5F-" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+`-v)\\#!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 242 "The following expression for the partial
 sums of the series gives an alternative way to calculate the error, a
nd also provides a way to draw curves which connect the the points for
 the two sequences of odd and even partial sums of the series." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
68 "sum((-1)^(n-1)*n/(2*n^2-1),n=x+1..infinity):\ne := unapply(abs(%),
x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6#%\"xG6\"6$%)operatorG
%&arrowGF(,$*&\"\")\"\"\"-%$absG6#*2,&*&,(*$)9$\"\"#F/!\"\"*&F:F/F9F/F
;#F/F:F;F/-%*hypergeomG6%7%F:,(F:F/*&F:F;F:#F/F:F/F9F/,(F:F/*&F:F;F:FD
F;F9F/7$,(\"\"$F/*&F:F;F:FDF/F9F/,(FIF/*&F:F;F:FDF;F9F/F;F/F/*&,**&#\"
#:F:F/F9F/F/*$)F9FIF/F/#\"\"(F:F/*&\"\"&F/F8F/F/F/-F?6%7%F/,(F/F/*&F:F
;F:FDF;F9F/,(F/F/*&F:F;F:FDF/F9F/7$FBFEF;F/F/F/)F;F9F/,&F7F/#F/F:F;F/,
&F9F/F/F/F/,(*&F:F/F8F/F/*&\"\"%F/F9F/F/F/F/F;,**&F:F/FSF/F/*&\"#5F/F8
F/F/*&FQF/F9F/F/FUF/F;,&*$F:FDF/*&F:F/F9F/F/F;,&FhoF;*&F:F/F9F/F/F;F/F
/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "evalf(evalf(e(1000),15));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+\\7v)\\#!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 345 "f := x->S+e(x):\ng := x->S-
e(x):\nptsE:=[seq([2*k,sum((-1)^(i-1)*i/(2*i^2-1),i=1..2*k)],k=1..10)]
:\nptsO:=[seq([2*k-1,sum((-1)^(i-1)*i/(2*i^2-1),i=1..2*k-1)],k=1..10)]
:\nplot([f(x),g(x),S,ptsE,ptsO],x=0..20,style=[line$3,point$2],\n   co
lor=[gray$2,COLOR(RGB,0,.7,0),red,blue],linestyle=[2,2,3],\n   symbol=
circle,view=[0..20,0..1.2],labels=[`n`,``]);" }}{PARA 13 "" 1 "" 
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$$\"\"\"F)Fahm7$$Fc_lF)$\"3e%35_-jv!*)F[[m7$$\"\"&F)$\"3OG-$yg[wj)F[[m
7$$F`dmF)$\"3N+6$)pwA9&)F[[m7$$\"\"*F)$\"3%[qo6i7LW)F[[m7$$\"#6F)$\"3o
WVi$fJsR)F[[m7$$\"#8F)$\"3O^KG.)p[O)F[[m7$$\"#:F)$\"3#GLH6E*)3M)F[[m7$
$\"#<F)$\"32rm:)40CK)F[[m7$$\"#>F)$\"3'G`,L(3s2$)F[[m-Fa^l6&Fc^lF(F(Fh
gmFjgmFj^l-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$%\"nG%!G-%%VIEWG6$;F(F
\\^l;F($FafmFadm" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 
0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }
{TEXT 283 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Show th
at the series " }{XPPEDIT 18 0 "Sum((-1)^(n-1)/sqrt(n),n = 1 .. infini
ty);" "6#-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF)F)F*F)-%%sqrtG6#F,F*/F,;F)%
)infinityG" }{TEXT -1 12 "  converges." }}{PARA 0 "" 0 "" {TEXT -1 47 
"Estimate the error when using a partial sum of " }{XPPEDIT 18 0 "10^6
" "6#*$\"#5\"\"'" }{TEXT -1 44 " terms to approximate the sum of the s
eries." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 267 
8 "Solution" }{TEXT 284 3 ":  " }}{PARA 257 "" 0 "" {TEXT -1 77 "The t
erms of the series alternate in sign and steadily decrease in magnitud
e." }}{PARA 0 "" 0 "" {TEXT -1 12 "Furthermore " }{XPPEDIT 18 0 "Limit
((-1)^(n-1)/sqrt(n),n = infinity) = 0;" "6#/-%&LimitG6$*&),$\"\"\"!\"
\",&%\"nGF*F*F+F*-%%sqrtG6#F-F+/F-%)infinityG\"\"!" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 114 "Thus the two criteria for the alternatin
g series test are satisfied, which demonstrates that the series conver
ges." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 36 "The error in using a partial sum of " }
{XPPEDIT 18 0 "10^6" "6#*$\"#5\"\"'" }{TEXT -1 108 " terms to approxim
ate the sum of the series is no greater in magnitude than that of the \+
first omitted term: " }{XPPEDIT 18 0 "1/sqrt(1000001);" "6#*&\"\"\"F$-
%%sqrtG6#\"(,++\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(1/sqrt(1000001));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++]******!#8" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 4 "Note" }{TEXT -1 10 ": M
aple's " }{TEXT 0 3 "sum" }{TEXT -1 68 " procedure can find an approxi
mate value for the sum of this series." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Sum((-1)^(n-1)/sqrt(n),n
=1..infinity);\nS := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$S
umG6$*&)!\"\",&%\"nG\"\"\"F(F+F+*$-%%sqrtG6#F*F+F(/F*;F+%)infinityG" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG$\"+Mk)*[g!#5" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The following procedur
e can be used to calculate partial sums " }{XPPEDIT 18 0 "Sum((-1)^(i-
1)/(sqrt(i)),i = 1 .. n)" "6#-%$SumG6$*&),$\"\"\"!\"\",&%\"iGF)F)F*F)-
%%sqrtG6#F,F*/F,;F)%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "PS := proc(n)\n   lo
cal i,sum,even,term;\n   sum := 0;\n   even := true;\n   for i from 1 \+
to n do\n      term := 1/sqrt(i);\n      if even then\n         sum :=
 sum + term;\n      else\n         sum := sum - term;\n      fi;\n    \+
  even := not even;\n   end do;\n   sum;\nend: " }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "We can use hardware floa
ting point arithmetic to speed things up when calculating a partial su
m with a large number of terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Sn := evalf(evalhf(PS(100000
0)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SnG$\"+Nk)R/'!#5" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The actua
l error in using a partial sum of " }{XPPEDIT 18 0 "10^6" "6#*$\"#5\"
\"'" }{TEXT -1 61 " terms to approximate the sum of the series is appr
oximately:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "abs(S-Sn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(*
****\\!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 74 "The following picture illustrates the first 50 partial sums of \+
the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 323 "ptsE := [seq([2*k,sum((-1)^(i-1)*1/sqrt(i),i=1..2*
k)],k=1..25)]:\nptsO := [seq([2*k-1,sum((-1)^(i-1)*1/sqrt(i),i=1..2*k-
1)],k=1..25)]:\nplot([ptsE,ptsO,.6048986434,ptsE,ptsO],x=0..50,style=[
line$3,point$2],\n   color=[gray$2,COLOR(RGB,0,.7,0),red,blue],linesty
le=[2,2,3],\n   symbol=circle,view=[0..50,0..1.2],labels=[`n`,``]);" }
}{PARA 13 "" 1 "" {GLPLOT2D 519 267 267 {PLOTDATA 2 "6*-%'CURVESG6&7;7
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!=t\"RIz3#4%F-7$$\"\")F*$\"3Ol7ba()>OVF-7$$\"#5F*$\"3a=irFWD2XF-7$$\"#
7F*$\"3<EdaFlhNYF-7$$\"#9F*$\"3]Gwa*30lt%F-7$$\"#;F*$\"3.S#>q)R\\=[F-7
$$\"#=F*$\"3!>TFMjFo)[F-7$$\"#?F*$\"3(*RKj%*p\"\\%\\F-7$$\"#AF*$\"3geq
VL()3&*\\F-7$$\"#CF*$\"3B\"\\)o@9**Q]F-7$$\"#EF*$\"3Y^mIq+$y2&F-7$$\"#
GF*$\"34o#fDg3D6&F-7$$\"#IF*$\"36L#zf7?P9&F-7$$\"#KF*$\"3%\\h!G$>1?<&F
-7$$\"#MF*$\"3s/DT,pz(>&F-7$$\"#OF*$\"3!)pG?W(Q9A&F-7$$\"#QF*$\"3P=)[f
S9KC&F-7$$\"#SF*$\"3)**\\:R6dLE&F-7$$\"#UF*$\"3YS1=L71#G&F-7$$\"#WF*$
\"3?ryh8-\\*H&F-7$$\"#YF*$\"3()e(oqj#y:`F-7$$\"#[F*$\"37SO2zo0J`F-7$$
\"#]F*$\"3et>FXZTX`F--%'COLOURG6&%$RGBG$\")=THv!\")FdsFds-%&STYLEG6#%%
LINEG-%*LINESTYLEG6#F)-F$6&7;7$$\"\"\"F*Fbt7$$\"\"$F*$\"3e\"yI+)[V-()F
-7$$\"\"&F*$\"3Uh..N3du\")F-7$$\"\"(F*$\"3A.S[gEtryF-7$$\"\"*F*$\"3])f
%)y3K&pwF-7$$\"#6F*$\"3$[&Q\\tyOAvF-7$$\"#8F*$\"3\">(=n3v64uF-7$$\"#:F
*$\"3.S#>q)R\\=tF-7$$\"#<F*$\"3spDQP-&QC(F-7$$\"#>F*$\"3!*HI8s\\)4=(F-
7$$\"#@F*$\"3/jJ*p*e4FrF-7$$\"#BF*$\"3w0y+uGB!3(F-7$$\"#DF*$\"3,$\\)o@
9**QqF-7$$\"#FF*$\"3&oS0w'4L-qF-7$$\"#HF*$\"3#*)yHV)>YppF-7$$\"#JF*$\"
3s%)pCYJxRpF-7$$\"#LF*$\"3\"HfPGv#y7pF-7$$\"#NF*$\"3?P&p3T0\"))oF-7$$
\"#PF*$\"3#\\WcshGa'oF-7$$\"#RF*$\"3(**o**R%f\\WoF-7$$\"#TF*$\"3)4c,Gt
%4DoF-7$$\"#VF*$\"3o(o1l$p/2oF-7$$\"#XF*$\"3yIxh)>--z'F-7$$\"#ZF*$\"3c
0x/_DVunF-7$$\"#\\F*$\"3Gq]k2$G'fnF-F`sFgsF[t-F$6&7S7$$F*F*$\"3#******
RV')*[gF-7$$\"3SLLL3x&)*3\"!#<Fa\\l7$$\"3zmm\"H2P\"Q?Ff\\lFa\\l7$$\"3X
LL$eRwX5$Ff\\lFa\\l7$$\"3=ML$3x%3yTFf\\lFa\\l7$$\"3gmm\"z%4\\Y_Ff\\lFa
\\l7$$\"34LLeR-/PiFf\\lFa\\l7$$\"3;++DcmpisFf\\lFa\\l7$$\"3vLLe*)>VB$)
Ff\\lFa\\l7$$\"3o++DJbw!Q*Ff\\lFa\\l7$$\"3%ommTIOo/\"!#;Fa\\l7$$\"3^LL
3_>jU6Fb^lFa\\l7$$\"3E++]i^Z]7Fb^lFa\\l7$$\"3/++](=h(e8Fb^lFa\\l7$$\"3
A++]P[6j9Fb^lFa\\l7$$\"3[L$e*[z(yb\"Fb^lFa\\l7$$\"3+nm;a/cq;Fb^lFa\\l7
$$\"3mmmm;t,m<Fb^lFa\\l7$$\"3=+]iSj0x=Fb^lFa\\l7$$\"3#omm;pW`(>Fb^lFa
\\l7$$\"3M+]i!f#=$3#Fb^lFa\\l7$$\"37+](=xpe=#Fb^lFa\\l7$$\"3-nm\"H28IH
#Fb^lFa\\l7$$\"3%om\"zpSS\"R#Fb^lFa\\l7$$\"3cLL3_?`(\\#Fb^lFa\\l7$$\"3
fL$e*)>pxg#Fb^lFa\\l7$$\"3D+]Pf4t.FFb^lFa\\l7$$\"3ZLLe*Gst!GFb^lFa\\l7
$$\"39+++DRW9HFb^lFa\\l7$$\"3:++DJE>>IFb^lFa\\l7$$\"35+]i!RU07$Fb^lFa
\\l7$$\"3$)***\\(=S2LKFb^lFa\\l7$$\"3nmmm\"p)=MLFb^lFa\\l7$$\"3U++](=]
@W$Fb^lFa\\l7$$\"36L$e*[$z*RNFb^lFa\\l7$$\"3e++]iC$pk$Fb^lFa\\l7$$\"3S
m;H2qcZPFb^lFa\\l7$$\"3Y+]7.\"fF&QFb^lFa\\l7$$\"3amm;/OgbRFb^lFa\\l7$$
\"3I+]ilAFjSFb^lFa\\l7$$\"3)RLLL)*pp;%Fb^lFa\\l7$$\"3WLL3xe,tUFb^lFa\\
l7$$\"3Wn;HdO=yVFb^lFa\\l7$$\"3a+++D>#[Z%Fb^lFa\\l7$$\"3)om;aG!e&e%Fb^
lFa\\l7$$\"3wLLL$)Qk%o%Fb^lFa\\l7$$\"3m+]iSjE!z%Fb^lFa\\l7$$\"3u+]P40O
\"*[Fb^lFa\\l7$F\\sFa\\l-%&COLORG6&FcsF*$F`u!\"\"F*Fgs-F\\t6#Fft-F$6&F
&-Fas6&Fcs$\"*++++\"FfsF`\\lF`\\l-Fhs6#%&POINTGF[t-F$6&F`t-Fas6&FcsF`
\\lF`\\lF^flF`flF[t-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$%\"nG%!G-%%VIE
WG6$;F`\\lF\\s;F`\\l$FDFgel" 1 2 4 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" }}}}{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 20 "Absolute convergence" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Le
t " }{XPPEDIT 18 0 "Sum(a[n],n=1..infinity)" "6#-%$SumG6$&%\"aG6#%\"nG
/F);\"\"\"%)infinityG" }{TEXT -1 23 " be a series such that " }
{XPPEDIT 18 0 "Sum(abs(a[n]),n = 1 .. infinity)" "6#-%$SumG6$-%$absG6#
&%\"aG6#%\"nG/F,;\"\"\"%)infinityG" }{TEXT -1 12 " converges. " }}
{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. i
nfinity)" "6#-%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 
11 " converges." }}{PARA 0 "" 0 "" {TEXT -1 13 "In this case " }
{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity)" "6#-%$SumG6$&%\"aG6#%\"nG/
F);\"\"\"%)infinityG" }{TEXT -1 12 " is said to " }{TEXT 265 19 "conve
rge absolutely" }{TEXT -1 7 " or be " }{TEXT 265 21 "absolutely conver
gent" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{TEXT 271 
31 "_______________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "For every n, " }
{XPPEDIT 18 0 "0 <= a[n] + abs(a[n])" "6#1\"\"!,&&%\"aG6#%\"nG\"\"\"-%
$absG6#&F'6#F)F*" }{XPPEDIT 18 0 "``<= 2*abs(a[n])" "6#1%!G*&\"\"#\"\"
\"-%$absG6#&%\"aG6#%\"nGF'" }{TEXT -1 8 ", since " }{XPPEDIT 18 0 "-ab
s(a[n])<=a[n]" "6#1,$-%$absG6#&%\"aG6#%\"nG!\"\"&F)6#F+" }{XPPEDIT 18 
0 "``<=abs(a[n])" "6#1%!G-%$absG6#&%\"aG6#%\"nG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 8 "Suppose " }{XPPEDIT 18 0 "Sum(abs(a[n]),n \+
= 1 .. infinity)" "6#-%$SumG6$-%$absG6#&%\"aG6#%\"nG/F,;\"\"\"%)infini
tyG" }{TEXT -1 22 " converges to S. Then " }{XPPEDIT 18 0 "Sum(2*abs(a
[n]),n = 1 .. infinity)" "6#-%$SumG6$*&\"\"#\"\"\"-%$absG6#&%\"aG6#%\"
nGF(/F/;F(%)infinityG" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "2
*S" "6#*&\"\"#\"\"\"%\"SGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 10 "Therefore " }{XPPEDIT 18 0 "Sum(a[n]+abs(a[n]),n=1..infinity)" 
"6#-%$SumG6$,&&%\"aG6#%\"nG\"\"\"-%$absG6#&F(6#F*F+/F*;F+%)infinityG" 
}{TEXT -1 20 " converges to a sum " }{TEXT 259 1 "T" }{TEXT -1 8 ", wh
ere " }{XPPEDIT 18 0 "T<= S" "6#1%\"TG%\"SG" }{TEXT -1 24 " by the com
parison test." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity) = Sum(a[n
]+abs(a[n]),n = 1 .. infinity)-Sum(abs(a[n]),n = 1 .. infinity);" "6#/
-%$SumG6$&%\"aG6#%\"nG/F*;\"\"\"%)infinityG,&-F%6$,&&F(6#F*F--%$absG6#
&F(6#F*F-/F*;F-F.F--F%6$-F66#&F(6#F*/F*;F-F.!\"\"" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "S-T" "6#,&%\"SG\"\"\"%\"TG!\"\"" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Thus if w
e " }{TEXT 265 16 "change the signs" }{TEXT -1 27 " of some of the ter
ms of a " }{TEXT 265 35 "convergent series of positive terms" }{TEXT 
-1 23 ", the resulting series " }{TEXT 265 15 "still converges" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Absol
ute convergence examples" }}{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 10 "The series
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n^2),n = 1
 .. infinity) = 1+1/4+1/9+1/16+1/25+` . . . `;" "6#/-%$SumG6$*&\"\"\"F
(*$%\"nG\"\"#!\"\"/F*;F(%)infinityG,.F(F(*&F(F(\"\"%F,F(*&F(F(\"\"*F,F
(*&F(F(\"#;F,F(*&F(F(\"#DF,F(%(~.~.~.~GF(" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 13 "converges to " }{XPPEDIT 18 0 "Pi^2/6" "6#*&%#PiG
\"\"#\"\"'!\"\"" }{TEXT -1 2 "  " }{TEXT 261 1 "~" }{TEXT -1 13 " 1.64
4934068." }}{PARA 0 "" 0 "" {TEXT -1 26 "It follows that the series" }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^(n-1)/(n^2)
,n = 1 .. infinity) = 1-1/4+1/9-1/16+1/25-1/36+` . . . `;" "6#/-%$SumG
6$*&),$\"\"\"!\"\",&%\"nGF*F*F+F**$F-\"\"#F+/F-;F*%)infinityG,0F*F**&F
*F*\"\"%F+F+*&F*F*\"\"*F+F**&F*F*\"#;F+F+*&F*F*\"#DF+F**&F*F*\"#OF+F+%
(~.~.~.~GF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "converges \+
(absolutely)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 80 "The series can also be seen to converge by applying the a
lternating series test." }}{PARA 0 "" 0 "" {TEXT -1 25 "In fact, it co
nverges to " }{XPPEDIT 18 0 "Pi^2/12" "6#*&%#PiG\"\"#\"#7!\"\"" }
{TEXT -1 31 " = 0.8224670334, approximately." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Sum((-1)^(i-1)/i^2
,i=1..infinity);\nevalf(%);\nevalf(evalf(Pi^2/12,15));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\",&%\"iG\"\"\"F(F+F+*$)F*\"\"#F+F(
/F*;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M.nC#)!#5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M.nC#)!#5" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "pts := [seq
([n,sum((-1)^(i-1)/i^2,i=1..n)],n=1..20)]:\nplot([Pi^2/12,pts],x=0..20
,style=[line,point],\n   color=[COLOR(RGB,0,.7,0),blue],linestyle=3,\n
   symbol=circle,view=[0..20,0.5..1],labels=[`n`,``]);" }}{PARA 13 "" 
1 "" {GLPLOT2D 555 225 225 {PLOTDATA 2 "6(-%'CURVESG6%7S7$$\"\"!F)$\"3
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S1(F6F*7$$\"3<++]i`A3vF6F*7$$\"3Rmmmm(y8!zF6F*7$$\"3K,+]i.tK$)F6F*7$$
\"3!3++v3zMu)F6F*7$$\"3Yomm\"H_?<*F6F*7$$\"3-nm;zihl&*F6F*7$$\"39LLL3#
G,***F6F*7$$\"3WLLezw5V5!#;F*7$$\"3/++v$Q#\\\"3\"FcpF*7$$\"3]LL$e\"*[H
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G6#%%LINEG-F$6%767$$\"\"\"F)Fhu7$$\"\"#F)$\"3++++++++vF,7$$\"\"$F)$\"3
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'\\;c\")F,7$$\"\"*F)$\"3u^)4hv@'z#)F,7$$\"#5F)$\"3m^)4hv@'z\")F,7$$\"#
6F)$\"3%\\-4UQmAE)F,7$$\"#7F)$\"3s!ek(R>#G>)F,7$$\"#8F)$\"3-;xRPN*>D)F
,7$$\"#9F)$\"3@^WBHJ(4?)F,7$$\"#:F)$\"3%f*)yOd<aC)F,7$$\"#;F)$\"3%f*)y
O2bj?)F,7$$\"#<F)$\"3%GY.)\\r&4C)F,7$$\"#=F)$\"3])fsA&H45#)F,7$$\"#>F)
$\"3)*GvH$y$zP#)F,7$Fgt$\"3_HvH$y$z7#)F,-%'COLOURG6&F\\uF(F($\"*++++\"
!\")-Fau6#%&POINTG-%*LINESTYLEG6#Fav-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELS
G6$%\"nG%!G-%%VIEWG6$;F(Fgt;$F[wF_uFhu" 1 2 4 3 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 22 "We can use the series " }{XPPEDIT 18 0 "S
um(a[n],n = 1 .. infinity) = 1-1/4+1/9-1/16+1/25-1/36+` . . . `;" "6#/
-%$SumG6$&%\"aG6#%\"nG/F*;\"\"\"%)infinityG,0F-F-*&F-F-\"\"%!\"\"F2*&F
-F-\"\"*F2F-*&F-F-\"#;F2F2*&F-F-\"#DF2F-*&F-F-\"#OF2F2%(~.~.~.~GF-" }
{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a[n] = (-1)^(n-1)/(n^2);" "6#/&%
\"aG6#%\"nG*&),$\"\"\"!\"\",&F'F+F+F,F+*$F'\"\"#F," }{TEXT -1 53 ", to
 illustrate the argument in the previous section." }}{PARA 0 "" 0 "" 
{TEXT -1 11 "The series " }{XPPEDIT 18 0 "Sum(2*abs(a[n]),n = 1 .. inf
inity) = Sum(2/n^2,n = 1 .. infinity)" "6#/-%$SumG6$*&\"\"#\"\"\"-%$ab
sG6#&%\"aG6#%\"nGF)/F0;F)%)infinityG-F%6$*&F(F)*$F0F(!\"\"/F0;F)F3" }
{TEXT -1 15 "  converges to " }{XPPEDIT 18 0 "Pi^2/3" "6#*&%#PiG\"\"#
\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 260 1 "~" }{TEXT -1 31 " 3.28986813
4, while the series " }{XPPEDIT 18 0 "Sum(a[n]+abs(a[n]),n = 1 .. infi
nity) = Sum(((-1)^(n-1)+1)/(n^2),n = 1 .. infinity);" "6#/-%$SumG6$,&&
%\"aG6#%\"nG\"\"\"-%$absG6#&F)6#F+F,/F+;F,%)infinityG-F%6$*&,&),$F,!\"
\",&F+F,F,F;F,F,F,F,*$F+\"\"#F;/F+;F,F4" }{TEXT -1 4 "  = " }{XPPEDIT 
18 0 "2+2/9+2/25+2/49+2/81+2/121+` . . . `;" "6#,0\"\"#\"\"\"*&F$F%\"
\"*!\"\"F%*&F$F%\"#DF(F%*&F$F%\"#\\F(F%*&F$F%\"#\")F(F%*&F$F%\"$@\"F(F
%%(~.~.~.~GF%" }{TEXT -1 46 ", converges to approximately   0.82246703
34 + " }{XPPEDIT 18 0 "Pi^2/6" "6#*&%#PiG\"\"#\"\"'!\"\"" }{TEXT -1 
15 " = 2.467401101." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 50 "sum((1+(-1)^(n-1))/(n^2),n=1..infinity);\ne
valf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sumG6$*&,&\"\"\"F()!\"
\",&%\"nGF(F*F(F(F(*$)F,\"\"#F(F*/F,;F(%)infinityG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"++6SnC!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "PSA := n -> sum(2/(i^2),i =
1..n):\nPSB := n -> sum((1+(-1)^(i-1))/i^2,i =1 .. n):\nptsA := [seq([
n,PSA(n)],n=1..20)]:\nptsB := [seq([n,PSB(n)],n=1..20)]:\nplot([3.2898
68134,2.467401101,ptsA,ptsB],x=0..20,style=[line$2,point$2],\n   color
=[cyan,brown,blue,red],linestyle=3,symbol=circle,\n   view=[0..20,1.5.
.3.5],labels=[`n`,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 529 242 242 
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F,7$Fgt$\"3mYg#y[EB>$F,-Fjt6&F\\uF(F(F]u-Fau6#%&POINTG-F$6%76Fcy7$FfyF
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EW<CF,7$FgtF]bl-Fjt6&F\\uF]uF(F(Fe_l-%*LINESTYLEG6#F]z-%'SYMBOLG6#%'CI
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 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }
}}{PARA 0 "" 0 "" {TEXT -1 41 "This example is again based on the seri
es" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n^2),n =
 1 .. infinity) = 1+1/4+1/9+1/16+1/25+` . . . `;" "6#/-%$SumG6$*&\"\"
\"F(*$%\"nG\"\"#!\"\"/F*;F(%)infinityG,.F(F(*&F(F(\"\"%F,F(*&F(F(\"\"*
F,F(*&F(F(\"#;F,F(*&F(F(\"#DF,F(%(~.~.~.~GF(" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 20 " which converges to " }{XPPEDIT 18 0 "Pi^2/6" "
6#*&%#PiG\"\"#\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "sigma[n]" "6#
&%&sigmaG6#%\"nG" }{TEXT -1 38 " be the sequence of \"signs\" defined \+
by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sigma(n) = 2*fl
oor(`mod`(n+2,4)/2)-1;" "6#/-%&sigmaG6#%\"nG,&*&\"\"#\"\"\"-%&floorG6#
*&-%$modG6$,&F'F+F*F+\"\"%F+F*!\"\"F+F+F+F5" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "si
gma := n -> floor((n+2 mod 4)/2)*2-1;\nseq(sigma(n),n=1..20);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6#%\"nG6\"6$%)operatorG%&ar
rowGF(,&-%&floorG6#,$-%$modG6$,&9$\"\"\"\"\"#F6\"\"%#F6F7F7!\"\"F6F(F(
F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "66\"\"\"!\"\"F$F#F#F$F$F#F#F$F$F#F
#F$F$F#F#F$F$F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The serie
s" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Sum(sigma(n)/(n^2),n = 1 .. infi
nity) = 1-1/4-1/9+1/16+1/25-1/36-1/49+1/64+` . . . `;" "6#/-%$SumG6$*&
-%&sigmaG6#%\"nG\"\"\"*$F+\"\"#!\"\"/F+;F,%)infinityG,4F,F,*&F,F,\"\"%
F/F/*&F,F,\"\"*F/F/*&F,F,\"#;F/F,*&F,F,\"#DF/F,*&F,F,\"#OF/F/*&F,F,\"#
\\F/F/*&F,F,\"#kF/F,%(~.~.~.~GF," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 23 "converges (absolutely)." }}{PARA 0 "" 0 "" {TEXT -1 71 "N
ote that we could use the following alternative definitions for sigma.
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sigma(n)=(-1)^(n*
(n-1)/2)" "6#/-%&sigmaG6#%\"nG),$\"\"\"!\"\"*(F'F*,&F'F*F*F+F*\"\"#F+
" }{TEXT -1 18 "       ------- (i)" }}{PARA 256 "" 0 "" {TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "sigma(n) = sqrt(2)*sin((2*i+1)*Pi/4);" "6#/-%&sigmaG
6#%\"nG*&-%%sqrtG6#\"\"#\"\"\"-%$sinG6#*(,&*&F,F-%\"iGF-F-F-F-F-%#PiGF
-\"\"%!\"\"F-" }{TEXT -1 14 "  ------- (ii)" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "sigma2 := n -> (-1)^(n*(n-1)/2);\nseq(sigma2(n),n=1..
20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma2Gf*6#%\"nG6\"6$%)oper
atorG%&arrowGF()!\"\",$*&9$\"\"\",&F0F1F-F1F1#F1\"\"#F(F(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "66\"\"\"!\"\"F$F#F#F$F$F#F#F$F$F#F#F$F$F#F#F$F
$F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "sigma3 := n -> sqrt(2)*sin((2*n+1)*Pi/4);\nseq(sigma3
(n),n=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma3Gf*6#%\"nG6
\"6$%)operatorG%&arrowGF(*&-%%sqrtG6#\"\"#\"\"\"-%$sinG6#,$*&,&9$F0F1F
1F1%#PiGF1#F1\"\"%F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "66\"\"\"!
\"\"F$F#F#F$F$F#F#F$F$F#F#F$F$F#F#F$F$F#" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "We can estimate the sum of the las
t series by calculating some partial sums." }}{PARA 0 "" 0 "" {TEXT 
-1 10 "Note that " }{TEXT 0 3 "sum" }{TEXT -1 59 " does not work corre
ctly if we use the first definition of " }{XPPEDIT 18 0 "sigma(n)" "6#
-%&sigmaG6#%\"nG" }{TEXT -1 12 ", so we use " }{TEXT 0 3 "add" }{TEXT 
-1 9 " instead." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 146 "sigma := n -> floor((n+2 mod 4)/2)*2-1:\nPS :
= n -> add(sigma(i)/(i^2),i =1..n);\nevalf(PS(100));\nevalf(PS(1000));
\nevalf(PS(2000));\nevalf(PS(5000));" }}{PARA 7 "" 1 "" {TEXT -1 42 "W
arning, `i` in call to `add` is not local" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#PSGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$addG6$*&-%&
sigmaG6#%\"iG\"\"\"*$)F3\"\"#F4!\"\"/F3;F49$F(F(F(" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+7&yM5(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+
[$)[.r!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+d$)[.r!#5" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+e$)[.r!#5" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "PS := n -> sum(sqrt(
2)*sin((2*i+1)*Pi/4)/i^2,i =1..n);\nevalf(PS(100));\nevalf(PS(1000));
\nevalf(PS(2000));\nevalf(PS(5000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#PSGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$sumG6$*&*&-%%sqrtG6#\"\"
#\"\"\"-%$sinG6#,$*&,&%\"iGF4F5F5F5%#PiGF5#F5\"\"%F5F5*$)F<F4F5!\"\"/F
<;F59$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7&yM5(!#5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[$)[.r!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+d$)[.r!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+e$
)[.r!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
32 "These calculations suggest that " }{XPPEDIT 18 0 "Sum(sigma(n)/(n^
2),n = 1 .. infinity);" "6#-%$SumG6$*&-%&sigmaG6#%\"nG\"\"\"*$F*\"\"#!
\"\"/F*;F+%)infinityG" }{TEXT -1 1 " " }{TEXT 262 1 "~" }{TEXT -1 14 "
 0.7103488358." }}{PARA 0 "" 0 "" {TEXT -1 48 "A more accurate value i
s 0.71034883582119071050." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "PS := n -> sum(sqrt(2)*sin((2*i+1)
*Pi/4)/i^2,i =1..n):\npts := [seq([n,PS(n)],n=1..20)]:\nplot([.7103488
358,pts],x=0..20,style=[line,point],\n   color=[COLOR(RGB,0,.7,0),blue
],linestyle=3,\n   symbol=circle,view=[0..20,0.5..1],labels=[`n`,``]);
" }}{PARA 13 "" 1 "" {GLPLOT2D 555 225 225 {PLOTDATA 2 "6(-%'CURVESG6%
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 22 "We can use the series " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infin
ity) = 1-1/4-1/9+1/16+1/25-1/36-1/49+1/64+` . . . `;" "6#/-%$SumG6$&%
\"aG6#%\"nG/F*;\"\"\"%)infinityG,4F-F-*&F-F-\"\"%!\"\"F2*&F-F-\"\"*F2F
2*&F-F-\"#;F2F-*&F-F-\"#DF2F-*&F-F-\"#OF2F2*&F-F-\"#\\F2F2*&F-F-\"#kF2
F-%(~.~.~.~GF-" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a[n] = sigma(n)
/(n^2);" "6#/&%\"aG6#%\"nG*&-%&sigmaG6#F'\"\"\"*$F'\"\"#!\"\"" }{TEXT 
-1 53 ", to illustrate the argument in the previous section." }}{PARA 
0 "" 0 "" {TEXT -1 11 "The series " }{XPPEDIT 18 0 "Sum(2*abs(a[n]),n \+
= 1 .. infinity) = Sum(2/n^2,n = 1 .. infinity)" "6#/-%$SumG6$*&\"\"#
\"\"\"-%$absG6#&%\"aG6#%\"nGF)/F0;F)%)infinityG-F%6$*&F(F)*$F0F(!\"\"/
F0;F)F3" }{TEXT -1 15 "  converges to " }{XPPEDIT 18 0 "Pi^2/3" "6#*&%
#PiG\"\"#\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 263 1 "~" }{TEXT -1 31 " 3
.289868134, while the series " }{XPPEDIT 18 0 "Sum(a[n]+abs(a[n]),n = \+
1 .. infinity) = Sum((sigma(n)+1)/(n^2),n = 1 .. infinity);" "6#/-%$Su
mG6$,&&%\"aG6#%\"nG\"\"\"-%$absG6#&F)6#F+F,/F+;F,%)infinityG-F%6$*&,&-
%&sigmaG6#F+F,F,F,F,*$F+\"\"#!\"\"/F+;F,F4" }{TEXT -1 31 "  converges \+
to  0.7103488358 + " }{XPPEDIT 18 0 "Pi^2/6" "6#*&%#PiG\"\"#\"\"'!\"\"
" }{TEXT -1 16 "  = 2.355282904." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "PSA := n -> sum(2/(i^2),i =
1..n):\nPSB := n -> sum((1+sqrt(2)*sin((2*i+1)*Pi/4))/(i^2),i =1..n):
\nptsA := [seq([n,PSA(n)],n=1..20)]:\nptsB := [seq([n,PSB(n)],n=1..20)
]:\nplot([3.289868134,2.355282904,ptsA,ptsB],x=0..20,\nstyle=[line,lin
e,point,point],\ncolor=[cyan,brown,blue,red],linestyle=3,\nsymbol=circ
le,view=[0..20,1.5..3.5],labels=[`n`,``]);" }}{PARA 13 "" 1 "" 
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\"1+++q(G**y\"FbpF)7$$\"1nm;9@BM=FbpF)7$$\"1MLL`v&Q(=FbpF)7$$\"1++DOl5
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1Yg#y[Et=$F+7$Fft$\"1Zg#y[EB>$F+-Fit6&F[uF(F(F\\u-F`u6#%&POINTG-F$6%76
Fby7$FeyFey7$F[zFey7$F`z$\"1++++++D@F+7$Fez$\"1++++++0AF+7$FjzF``l7$F_
[lF``l7$Fd[l$\"1+++++DOAF+7$Fi[l$\"1#pC!e8%4E#F+7$F^\\lFh`l7$Fc\\lFh`l
7$Fh\\l$\"1!e8pCI[F#F+7$F]]l$\"13-WmXm'G#F+7$Fb]lF`al7$Fg]lF`al7$F\\^l
$\"13-WmqZ%H#F+7$Fa^l$\"1@^m\"[(R,BF+7$Ff^lFhal7$F[_lFhal7$Fft$\"1@^m
\"[(R1BF+-Fit6&F[uF\\uF(F(Fd_l-%*LINESTYLEG6#F\\z-%'SYMBOLG6#%'CIRCLEG
-%+AXESLABELSG6$%\"nG%!G-%%VIEWG6$;F(Fft;$Fh]l!\"\"$\"#NFccl" 1 2 4 3 
10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C
urve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Ex
ample 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 
0 "" {TEXT -1 35 "This example is based on the series" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n^3),n = 1 .. infinity) = 1
+1/8+1/27+1/64+1/125+1/216+1/343+1/512+1/729+1/1000+` . . . `;" "6#/-%
$SumG6$*&\"\"\"F(*$%\"nG\"\"$!\"\"/F*;F(%)infinityG,8F(F(*&F(F(\"\")F,
F(*&F(F(\"#FF,F(*&F(F(\"#kF,F(*&F(F(\"$D\"F,F(*&F(F(\"$;#F,F(*&F(F(\"$
V$F,F(*&F(F(\"$7&F,F(*&F(F(\"$H(F,F(*&F(F(\"%+5F,F(%(~.~.~.~GF(" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 47 " which converges to appr
oximately 1.202056903. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Sum(1/n^3,n = 1 .. infinity);\neval
f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*$)%\"nG\"
\"$F'!\"\"/F*;F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.p0
-7!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 
"Let " }{XPPEDIT 18 0 "sigma[n]" "6#&%&sigmaG6#%\"nG" }{TEXT -1 47 " b
e the sequence of \"signs\" defined as follows." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sigma := n -
> ((3*n mod 5) mod 2)*2-1;\nseq(sigma(n),n=1..20);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&sigmaGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,&-%$modG
6$-F.6$,$9$\"\"$\"\"&\"\"#F6\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "66\"\"\"F#!\"\"F$F$F#F#F$F$F$F#F#F$F$F$F#F#F$F$F$" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The serie
s" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Sum(sigma(n)/(n^3),n = 1 .. infi
nity) = 1+1/8-1/27-1/64-1/125+1/216+1/343-1/512-1/729-1/1000;" "6#/-%$
SumG6$*&-%&sigmaG6#%\"nG\"\"\"*$F+\"\"$!\"\"/F+;F,%)infinityG,6F,F,*&F
,F,\"\")F/F,*&F,F,\"#FF/F/*&F,F,\"#kF/F/*&F,F,\"$D\"F/F/*&F,F,\"$;#F/F
,*&F,F,\"$V$F/F,*&F,F,\"$7&F/F/*&F,F,\"$H(F/F/*&F,F,\"%+5F/F/" }{TEXT 
-1 8 " + . . ." }}{PARA 0 "" 0 "" {TEXT -1 23 "converges (absolutely).
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "We ca
n estimate the sum of the last series by calculating some partial sums
." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{TEXT 0 3 "sum" }{TEXT 
-1 30 " does not work correctly here." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "PS := n -> add(evalf(si
gma(i)/i^3,15),i =1..n);\nevalf(evalf(PS(100),15));\nevalf(evalf(PS(10
00),15));\nevalf(evalf(PS(10000),15));\nevalf(evalf(PS(50000),15));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PSGf*6#%\"nG6\"6$%)operatorG%&arro
wGF(-%$addG6$-%&evalfG6$*&-%&sigmaG6#%\"iG\"\"\"*$)F6\"\"$F7!\"\"\"#:/
F6;F79$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G\"*on5!\"*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+80on5!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+;/on5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+:/
on5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "These calculations suggest that " 
}{XPPEDIT 18 0 "Sum(sigma(n)/(n^3),n = 1 .. infinity);" "6#-%$SumG6$*&
-%&sigmaG6#%\"nG\"\"\"*$F*\"\"$!\"\"/F*;F+%)infinityG" }{TEXT -1 1 " \+
" }{TEXT 264 1 "~" }{TEXT -1 13 " 1.067680415." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "sigma := n \+
-> ((3*n mod 5) mod 2)*2-1:\nPS := n -> add(evalf(sigma(i)/i^3,15),i =
1..n):\npts := [seq([n,PS(n)],n=1..15)]:\nplot([1.067680415,pts],x=0..
15,style=[line,point],\n   color=[COLOR(RGB,0,.7,0),blue],linestyle=3,
\n   symbol=circle,view=[0..15,.9..1.2],labels=[`n`,``]);" }}{PARA 13 
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$\"3X****\\(=HPJ*F0F*7$$\"3;++DJaU`7F,F*7$$\"3)***\\P%GZRd\"F,F*7$$\"3
%)**\\(=276(=F,F*7$$\"3'***\\(o**3)y@F,F*7$$\"3/+](ofHq\\#F,F*7$$\"3.+
]Pf'HU\"GF,F*7$$\"33++]7*309$F,F*7$$\"3:++Dce*yU$F,F*7$$\"3;++]([D9v$F
,F*7$$\"3c****\\iNGwSF,F*7$$\"37++]7XM*Q%F,F*7$$\"3/+](o%QjtYF,F*7$$\"
32++]i8o6]F,F*7$$\"3i******\\>0)H&F,F*7$$\"3Y**\\(=-p6j&F,F*7$$\"3d***
**\\2Mg#fF,F*7$$\"35+](=xZ&\\iF,F*7$$\"3;+]i:$4wb'F,F*7$$\"3-++v=#R!zo
F,F*7$$\"3q+]P4A@urF,F*7$$\"3I++Dchf#\\(F,F*7$$\"3))**\\(of2L#yF,F*7$$
\"3M**\\7yG>6\")F,F*7$$\"3w++voo6A%)F,F*7$$\"3q*****\\xJLu)F,F*7$$\"3W
++v$*ydd!*F,F*7$$\"3#***\\(=<F;O*F,F*7$$\"35***\\i0A#*p*F,F*7$$\"3*)**
**\\2mD+5!#;F*7$$\"3*****\\i0XE.\"FhqF*7$$\"3%**\\(o/Q*>1\"FhqF*7$$\"3
=++vQ(zS4\"FhqF*7$$\"3***\\(=-,FC6FhqF*7$$\"33+v$4tFe:\"FhqF*7$$\"3!**
**\\73\"o'=\"FhqF*7$$\"3-+voz;)*=7FhqF*7$$\"31+++&*44]7FhqF*7$$\"35+]7
jZ!>G\"FhqF*7$$\"34+v=(4bMJ\"FhqF*7$$\"3;++]xlWU8FhqF*7$$\"39+]i&3ucP
\"FhqF*7$$\"3\"******\\;$R09FhqF*7$$\"38+v=-*zqV\"FhqF*7$$\"33+D\"G:3u
Y\"FhqF*7$$\"#:F)F*-%&COLORG6&%$RGBGF)$\"\"(!\"\"F)-%&STYLEG6#%%LINEG-
F$6%717$$\"\"\"F)Fhu7$$\"\"#F)$\"3+++++++D6F,7$$\"\"$F)$\"31+++jH'z3\"
F,7$$\"\"%F)$\"31+++jzLs5F,7$$\"\"&F)$\"31+++jzLk5F,7$$\"\"'F)$\"3++++
$fn*o5F,7$$F^uF)$\"33+++XI)=2\"F,7$$\"\")F)$\"33+++?*H*p5F,7$$\"\"*F)$
\"3'******z<e&o5F,7$$\"#5F)$\"33+++y\"ev1\"F,7$$\"#6F)$\"37+++$\\4$o5F
,7$$\"#7F)$\"3#******p>)))o5F,7$$\"#8F)$\"33+++JIVo5F,7$$\"#9F)$\"3#**
******fo!o5F,7$Fgt$\"3++++/Bxn5F,-%'COLOURG6&F\\uF(F($\"*++++\"!\")-Fa
u6#%&POINTG-%*LINESTYLEG6#Fav-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$%\"n
G%!G-%%VIEWG6$;F(Fgt;$F^xF_u$F]yF_u" 1 2 4 3 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}
{PARA 0 "" 0 "" {TEXT -1 21 "Show that the series " }{XPPEDIT 18 0 "Su
m((-1)^(n-1)*``(1/(n*(n+1)*(n+2))),n = 1 .. infinity);" "6#-%$SumG6$*&
),$\"\"\"!\"\",&%\"nGF)F)F*F)-%!G6#*&F)F)*(F,F),&F,F)F)F)F),&F,F)\"\"#
F)F)F*F)/F,;F)%)infinityG" }{TEXT -1 11 " converges." }}{PARA 0 "" 0 "
" {TEXT -1 94 "Estimate the error when using a partial sum of 100 term
s to approximate the sum of the series." }}{PARA 0 "" 0 "" {TEXT -1 
37 "Does this series converge absolutely?" }}{PARA 0 "" 0 "" {TEXT -1 
40 "________________________________________" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "____________________
____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 
21 "Show that the series " }{XPPEDIT 18 0 "Sum((-1)^(n-1)*2^(2*n-1)/(2
*n-1)!,n = 1 .. infinity);" "6#-%$SumG6$*(),$\"\"\"!\"\",&%\"nGF)F)F*F
))\"\"#,&*&F.F)F,F)F)F)F*F)-%*factorialG6#,&*&F.F)F,F)F)F)F*F*/F,;F)%)
infinityG" }{TEXT -1 11 " converges." }}{PARA 0 "" 0 "" {TEXT -1 47 "E
stimate the error when using a partial sum of " }{XPPEDIT 18 0 "5;" "6
#\"\"&" }{TEXT -1 44 " terms to approximate the sum of the series." }}
{PARA 0 "" 0 "" {TEXT -1 37 "Does this series converge absolutely?" }}
{PARA 0 "" 0 "" {TEXT -1 40 "________________________________________
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 40 "________________________________________" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" 
}}{PARA 0 "" 0 "" {TEXT -1 21 "Show that the series " }{XPPEDIT 18 0 "
Sum((-1)^n/ln(n),n = 2 .. infinity);" "6#-%$SumG6$*&),$\"\"\"!\"\"%\"n
GF)-%#lnG6#F+F*/F+;\"\"#%)infinityG" }{TEXT -1 11 " converges." }}
{PARA 0 "" 0 "" {TEXT -1 47 "Estimate the error when using a partial s
um of " }{XPPEDIT 18 0 "10^6" "6#*$\"#5\"\"'" }{TEXT -1 44 " terms to \+
approximate the sum of the series." }}{PARA 0 "" 0 "" {TEXT -1 37 "Doe
s this series converge absolutely?" }}{PARA 0 "" 0 "" {TEXT -1 40 "___
_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "____________________
____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 16 "Code for picture
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 
0 "" {TEXT -1 28 "Code for convergence picture" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "e:=x->1/2*(
Psi(1+1/2*x)-Psi(1/2*x+1/2)):\nf := x -> ln(2)-e(x):\ng := x -> ln(2)+
e(x):\nptsE := [seq([2*k,sum((-1)^(i-1)*1/i,i=1..2*k)],k=1..10)]:\npts
O := [seq([2*k-1,sum((-1)^(i-1)*1/i,i=1..2*k-1)],k=1..10)]:\nplot([f(x
),g(x),ln(2),ptsE,ptsO],x=0..20,style=[line$3,point$2],\n   color=[gra
y$2,COLOR(RGB,0,.7,0),red,blue],linestyle=[2,2,3],\n   symbol=circle,v
iew=[0..20,0..1.2],labels=[`n`,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
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