{VERSION 6 0 "IBM INTEL NT" "6.0" }
{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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 
0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 
0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 
0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 
0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "Pu
rple Emphasis" -1 261 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "Red Emphasis" -1 262 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 
0 0 }{CSTYLE "Dark Red Emphasis" -1 263 "Times" 1 12 128 0 0 1 0 1 0 
0 0 0 0 0 0 0 }{CSTYLE "" 262 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
}{CSTYLE "" 262 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 
0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim
es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 
1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" 
-1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 
1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" 
-1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 
2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 
0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal
" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }
3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Bracketed series" }}{PARA 0 "" 0 
"" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 
"" {TEXT -1 19 "Version:  26.3.2007" }}{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 51 "P
reliminary example of bracketing terms of a series" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 32 "The altern
ating harmonic series " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Sum((-1)^(n-1)/n,n = 1 .. infinity) = 1-1/2+1/3-1/4+1/5-1/6+` . \+
. . `;" "6#/-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF*F*F+F*F-F+/F-;F*%)infini
tyG,0F*F**&F*F*\"\"#F+F+*&F*F*\"\"$F+F**&F*F*\"\"%F+F+*&F*F*\"\"&F+F**
&F*F*\"\"'F+F+%(~.~.~.~GF*" }{TEXT -1 15 " --------- (i)." }}{PARA 0 "
" 0 "" {TEXT -1 41 "converges by the alternating series test." }}
{PARA 0 "" 0 "" {TEXT -1 24 "Now consider the series " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(1-1/2)+``(1/3-1/4)+``(1/5-1/6)+
` . . . `+``(1/(2*n-1)-1/(2*n))+` . . . `" "6#,.-%!G6#,&\"\"\"F(*&F(F(
\"\"#!\"\"F+F(-F%6#,&*&F(F(\"\"$F+F(*&F(F(\"\"%F+F+F(-F%6#,&*&F(F(\"\"
&F+F(*&F(F(\"\"'F+F+F(%(~.~.~.~GF(-F%6#,&*&F(F(,&*&F*F(%\"nGF(F(F(F+F+
F(*&F(F(*&F*F(FAF(F+F+F(F:F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "obtained by grouping the terms \+
of the series (i) together in pairs." }}{PARA 0 "" 0 "" {TEXT -1 38 "T
he general term of this new series is" }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "1/(2*n-1) - 1/(2*n) = 1/((2*n-1)*(2*n))" "6#/,&*&
\"\"\"F&,&*&\"\"#F&%\"nGF&F&F&!\"\"F+F&*&F&F&*&F)F&F*F&F+F+*&F&F&*&,&*
&F)F&F*F&F&F&F+F&*&F)F&F*F&F&F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 16 "so the series is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Sum(1/((2*n-1)*2*n),n = 1 .. infinity) = 1/(1*`.`*2)+1/
(3*`.`*4)+1/(5*`.`*6)+1/(7*`.`*8)+` . . . `;" "6#/-%$SumG6$*&\"\"\"F(*
(,&*&\"\"#F(%\"nGF(F(F(!\"\"F(F,F(F-F(F./F-;F(%)infinityG,,*&F(F(*(F(F
(%\".GF(F,F(F.F(*&F(F(*(\"\"$F(F5F(\"\"%F(F.F(*&F(F(*(\"\"&F(F5F(\"\"'
F(F.F(*&F(F(*(\"\"(F(F5F(\"\")F(F.F(%(~.~.~.~GF(" }{TEXT -1 16 " -----
---- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 174 "The partial sums of the series (ii) form a subsequence of the \+
sequence of partial sums of the series (i), and so the series (ii) con
verges to the same sum as the series (i).\n" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 90 "Sum((-1)^(n-1)/n,n=1..infinity);\nvalue(%);\nSum(1/
((2*n-1)*(2*n)),n=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$SumG6$*&)!\"\",&%\"nG\"\"\"F(F+F+F*F(/F*;F+%)infinit
yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#\"\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%$SumG6$,$*&\"\"\"F(*&,&%\"nG\"\"#!\"\"F(F(F+F(F-#F(
F,/F+;F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#\"\"#" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "The p
artial sums of the series (ii) are shown by the red points in the foll
owing picture, which illustrates the convergence of the alternating se
ries (i)." }}{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)]:\nptsO := [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=[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..1.2],labels=[`n`,``]);" }}
{PARA 13 "" 1 "" {GLPLOT2D 439 320 320 {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!\"*$\"+>j#G\"RF07$$\"+e0$=C\"FW$\"+ElLTUF07$$\"+
LA`c9FW$\"+[;%)3XF07$$\"+3RBr;FW$\"+y:BHZF07$$\"+zjf)4#FW$\"+&)y&*o]F0
7$$\"+'4;[\\#FW$\"+twT.`F07$$\"+j'y]!HFW$\"+y&e>\\&F07$$\"+'zs$HLFW$\"
+^FVYcF07$$\"+8iI_PFW$\"++1#4x&F07$$\"+<_M(=%FW$\"+cYHweF07$$\"+4y_qXF
W$\"+L&\\X&fF07$$\"+]1!>+&FW$\"+<q\"*HgF07$$\"+]Z/NaFW$\"+66![4'F07$$
\"+]$fC&eFW$\"+Y.6\\hF07$$\"+'z6:B'FW$\"+r$)o#>'F07$$\"+<=C#o'FW$\"+To
gQiF07$$\"+n#pS1(FW$\"+E-GtiF07$$\"+j`A3vFW$\"+'G-&4jF07$$\"+n(y8!zFW$
\"+1NSQjF07$$\"+j.tK$)FW$\"+\"Q\"=njF07$$\"+)3zMu)FW$\"+w-6#R'F07$$\"+
#H_?<*FW$\"+c?)eT'F07$$\"+!G;cc*FW$\"+cA%fV'F07$$\"+4#G,***FW$\"+wS!fX
'F07$$\"+!o2J/\"!\")$\"+\"e2]Z'F07$$\"+%Q#\\\"3\"Fju$\"+\">K/\\'F07$$
\"+;*[H7\"Fju$\"+'[jf]'F07$$\"+qvxl6Fju$\"+'=-4_'F07$$\"+`qn27Fju$\"+Y
p`MlF07$$\"+cp@[7Fju$\"+6`*oa'F07$$\"+3'HKH\"Fju$\"+')ouflF07$$\"+xanL
8Fju$\"+@UeqlF07$$\"+v+'oP\"Fju$\"+@(z9e'F07$$\"+S<*fT\"Fju$\"+O-!3f'F
07$$\"+&)Hxe9Fju$\"+h'Q/g'F07$$\"+.o-*\\\"Fju$\"+wI-4mF07$$\"+TO5T:Fju
$\"+OL`<mF07$$\"+U9C#e\"Fju$\"+'oIai'F07$$\"+1*3`i\"Fju$\"+mSGLmF07$$
\"+$*zym;Fju$\"+ciZSmF07$$\"+^j?4<Fju$\"+19[ZmF07$$\"+jMF^<Fju$\"+wM5a
mF07$$\"+q(G**y\"Fju$\"+'4A*fmF07$$\"+9@BM=Fju$\"+;zHmmF07$$\"+`v&Q(=F
ju$\"+mAvrmF07$$\"+Ol5;>Fju$\"+EgKxmF07$$\"+/Uac>Fju$\"+19W#o'F07$$\"#
?F)$\"+KSr(o'F0-%'COLOURG6&%$RGBG$\")=THvFjuFd]lFd]l-%&STYLEG6#%%LINEG
-%*LINESTYLEG6#\"\"#-F$6&7en7$F($\"+hVH'Q\"FW7$F+$\"+Yx+W8FW7$F2$\"+JJ
E18FW7$F7$\"+e/Rs7FW7$F<$\"+uG%=C\"FW7$FA$\"+Yv**)=\"FW7$FF$\"+O3&\\9
\"FW7$FK$\"+S!HP3\"FW7$FP$\"+!oGl.\"FW7$FU$\"+$H<,&**F07$Fen$\"+'32;i*
F07$Fjn$\"+k>5a$*F07$F_o$\"+M?rL\"*F07$Fdo$\"+Fd)Rz)F07$Fio$\"+Rf_f&)F
07$F^p$\"+M])4P)F07$Fcp$\"+h3^;#)F07$Fhp$\"+7I-#4)F07$F]q$\"+c*[m)zF07
$Fbq$\"+zSR3zF07$Fgq$\"+&fEI$yF07$F\\r$\"+,D9oxF07$Far$\"+mK$Qr(F07$Ff
r$\"+T_DqwF07$F[s$\"+rnLCwF07$F`s$\"+'Qj'*e(F07$Fes$\"+E8W`vF07$Fjs$\"
+1,aCvF07$F_t$\"+JAw&\\(F07$Fdt$\"+OL$3Z(F07$Fit$\"+c:1ZuF07$F^u$\"+c8
+FuF07$Fcu$\"+O&RqS(F07$Fhu$\"+Jg$zQ(F07$F^v$\"+@9^stF07$Fcv$\"+E,)pN(
F07$Fhv$\"+E9/UtF07$F]w$\"+mmSGtF07$Fbw$\"+,$[gJ(F07$Fgw$\"+En>.tF07$F
\\x$\"+\"RfBH(F07$Fax$\"+\"*QY\"G(F07$Ffx$\"+wL9ssF07$F[y$\"+^\\]isF07
$F`y$\"+O0#RD(F07$Fey$\"+w-TXsF07$Fjy$\"+EH^PsF07$F_z$\"+Y&f'HsF07$Fdz
$\"+ctYAsF07$Fiz$\"+1AY:sF07$F^[l$\"+O,%)3sF07$Fc[l$\"+;:-.sF07$Fh[l$
\"+'pXm>(F07$F]\\l$\"+Y8>\">(F07$Fb\\l$\"+'e<c=(F07$Fg\\l$\"+1A]!=(F07
$F\\]l$\"+y&H_<(F0F`]lFf]lFj]l-F$6&7S7$F($\"3'GX*f0=ZJp!#=7$$\"39LLLL3
VfVFbilF`il7$$\"3%pmm;H[D:)FbilF`il7$$\"3MLLLe0$=C\"!#<F`il7$$\"3iLLL3
RBr;F\\jlF`il7$$\"3imm;zjf)4#F\\jlF`il7$$\"3ULL$e4;[\\#F\\jlF`il7$$\"3
!)****\\i'y]!HF\\jlF`il7$$\"3oLL$ezs$HLF\\jlF`il7$$\"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(y8!zF\\jlF`il7$$\"3K,+]i.tK$)F\\jlF`il7$$
\"3!3++v3zMu)F\\jlF`il7$$\"3Yomm\"H_?<*F\\jlF`il7$$\"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$$\"3`mm
mwanL8F_^mF`il7$$\"35+++v+'oP\"F_^mF`il7$$\"3CLLeR<*fT\"F_^mF`il7$$\"3
C+++&)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$$\"\"'F)$\"3'pmmmmmm;'Fbil7$$\"\")F)$\"3!\\4Q_4Q_M'Fbi
l7$$\"#5F)$\"3s1#\\j?\\jX'Fbil7$$\"#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$$F[cmF)$\"3qLLLLLLL$)Fbil7$$\"\"&F)$\"3ELLLLL
LLyFbil7$$FgbmF)$\"3!\\4Q_4Q_f(Fbil7$$\"\"*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@(Fbil7$$\"#>F)$\"3#)zUvJSr(=(Fbil-Fa]l6&Fc]l
F(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 26 "Bracketing and converge
nce" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 25 "In general, suppose that " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity) = a[1]+a[2]+a[3]+` .
 . . `+a[n]+` . . . `" "6#/-%$SumG6$&%\"aG6#%\"nG/F*;\"\"\"%)infinityG
,.&F(6#F-F-&F(6#\"\"#F-&F(6#\"\"$F-%(~.~.~.~GF-&F(6#F*F-F8F-" }{TEXT 
-1 15 " ------- (iii) " }}{PARA 0 "" 0 "" {TEXT -1 5 "is a " }{TEXT 
261 17 "convergent series" }{TEXT -1 10 ", and let " }}{PARA 256 "" 0 
"" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[1]+b[2]+b[3]+` . . . `+b[k]+` . . \+
. `" "6#,.&%\"bG6#\"\"\"F'&F%6#\"\"#F'&F%6#\"\"$F'%(~.~.~.~GF'&F%6#%\"
kGF'F.F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(a[1]+` . . . `+a[n[1]])
+``(a[``(n[1]+1)]+` . . . `+a[n[2]])+``(a[``(n[2]+1)]+` . . . `+a[n[3]
])+` . . . `;" "6#/%!G,*-F$6#,(&%\"aG6#\"\"\"F,%(~.~.~.~GF,&F*6#&%\"nG
6#F,F,F,-F$6#,(&F*6#-F$6#,&&F16#F,F,F,F,F,F-F,&F*6#&F16#\"\"#F,F,-F$6#
,(&F*6#-F$6#,&&F16#FAF,F,F,F,F-F,&F*6#&F16#\"\"$F,F,F-F," }{TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "``+``(a[``(n[k]+1)]+` . . . `+a[n[k]])+` . . . `;" "
6#,(%!G\"\"\"-F$6#,(&%\"aG6#-F$6#,&&%\"nG6#%\"kGF%F%F%F%%(~.~.~.~GF%&F
*6#&F06#F2F%F%F3F%" }{TEXT -1 14 " ------- (iv) " }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "be a " }{TEXT 261 16 "bra
cketed series" }{TEXT -1 20 " in which each term " }{XPPEDIT 18 0 "b[k
]=``(a[``(n[k]+1)]+` . . . `+a[n[k]])" "6#/&%\"bG6#%\"kG-%!G6#,(&%\"aG
6#-F)6#,&&%\"nG6#F'\"\"\"F5F5F5%(~.~.~.~GF5&F-6#&F36#F'F5" }{TEXT -1 
58 " is a finite sum of consecutive terms of the series (iii)." }}
{PARA 0 "" 0 "" {TEXT -1 27 "If the partial sums of the " }{TEXT 261 
11 "unbracketed" }{TEXT -1 17 " series (iii) are" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "s[1],s[2],` . . . `,s[n],` . . . `;" "
6'&%\"sG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nGF*" }{TEXT -1 14 " -----
-- (v), " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
6 "where " }{XPPEDIT 18 0 "s[n] = a[1]+a[2]+a[3]+` . . . `+a[n];" "6#/
&%\"sG6#%\"nG,,&%\"aG6#\"\"\"F,&F*6#\"\"#F,&F*6#\"\"$F,%(~.~.~.~GF,&F*
6#F'F," }{TEXT -1 57 ", then the partial sums of the bracketed series \+
(iv) are " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[n[1]],
s[n[2]],` . . . `,s[n[k]],` . . . `;" "6'&%\"sG6#&%\"nG6#\"\"\"&F$6#&F
'6#\"\"#%(~.~.~.~G&F$6#&F'6#%\"kGF/" }{TEXT -1 14 " ------- (vi)," }}
{PARA 0 "" 0 "" {TEXT -1 14 "and so form a " }{TEXT 261 11 "subsequenc
e" }{TEXT -1 21 " of the sequence (v)." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 48 "Thus the bracketed series (iv) conver
ges to the " }{TEXT 261 8 "same sum" }{TEXT -1 42 " as the unbracketed
 series (iii), that is," }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "Sum(b[n],n=1..infinity)=Sum(a[n],n=1..infinity)" "6#/-%
$SumG6$&%\"bG6#%\"nG/F*;\"\"\"%)infinityG-F%6$&%\"aG6#F*/F*;F-F." }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 261 93 "Introducing brackets in
 a convergent series produces a series which converges to the same sum
" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 25 "__
_______________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 46 "Now suppose that all the terms in e
ach group  " }{XPPEDIT 18 0 "b[k]=``(a[``(n[k]+1)]+` . . . `+a[n[k]])
" "6#/&%\"bG6#%\"kG-%!G6#,(&%\"aG6#-F)6#,&&%\"nG6#F'\"\"\"F5F5F5%(~.~.
~.~GF5&F-6#&F36#F'F5" }{TEXT -1 25 " of the bracketed series " }{TEXT 
261 18 "have the same sign" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 62 "In this special case, the convergence of the bracketed series \+
" }{XPPEDIT 18 0 "Sum(b[n],n = 1 .. infinity)" "6#-%$SumG6$&%\"bG6#%\"
nG/F);\"\"\"%)infinityG" }{TEXT -1 35 " implies the convergence of ser
ies " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity);" "6#-%$SumG6$&%\"aG
6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 13 " obtained by " }{TEXT 261 
21 "dropping the brackets" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "If all the terms " }{XPPEDIT 18 
0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "n[k-1]
<n" "6#2&%\"nG6#,&%\"kG\"\"\"F)!\"\"F%" }{XPPEDIT 18 0 "``<=n[k]" "6#1
%!G&%\"nG6#%\"kG" }{TEXT -1 37 " are greater than equal to zero, then
" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "s[n[k-1]] <=s[n]
" "6#1&%\"sG6#&%\"nG6#,&%\"kG\"\"\"F,!\"\"&F%6#F(" }{XPPEDIT 18 0 "``<
=s[n[k]]" "6#1%!G&%\"sG6#&%\"nG6#%\"kG" }{TEXT -1 1 "," }}{PARA 0 "" 
0 "" {TEXT -1 71 "that is, the intermediate partial sums of the unbrac
keted series (iii) " }{TEXT 261 17 "steadily increase" }{TEXT -1 84 " \+
(or at least don't decrease) between the partial sums of the bracketed
 series (iv)." }}{PARA 0 "" 0 "" {TEXT -1 17 "If all the terms " }
{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 6 " with " }{XPPEDIT 
18 0 "n[k-1]<n" "6#2&%\"nG6#,&%\"kG\"\"\"F)!\"\"F%" }{XPPEDIT 18 0 "``
<=n[k]" "6#1%!G&%\"nG6#%\"kG" }{TEXT -1 34 " are less than equal to ze
ro, then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[n[k]] <
=s[n]" "6#1&%\"sG6#&%\"nG6#%\"kG&F%6#F(" }{XPPEDIT 18 0 "``<=s[n[k-1]]
" "6#1%!G&%\"sG6#&%\"nG6#,&%\"kG\"\"\"F-!\"\"" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 71 "that is, the intermediate partial sums of
 the unbracketed series (iii) " }{TEXT 261 17 "steadily decrease" }
{TEXT -1 84 " (or at least don't increase) between the partial sums of
 the bracketed series (iv)." }}{PARA 0 "" 0 "" {TEXT -1 20 "If the par
tial sums " }{XPPEDIT 18 0 "s[n[k]]" "6#&%\"sG6#&%\"nG6#%\"kG" }{TEXT 
-1 50 " of the bracketed series (iv) converge to a limit " }{TEXT 266 
1 "S" }{TEXT -1 137 ", then the partial sums of the unbracketed series
 (iii) are sandwiched between them and as such are forced to converge \+
to the same limit " }{TEXT 267 1 "S" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 162 "Removing brackets fro
m a convergent bracketed series, in which all the terms in each bracke
t have the same sign, produces a series which converges to the same su
m" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 265 25 "_
________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Examples of bracketed ser
ies" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 
"" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }
{XPPEDIT 18 0 "sigma[n]" "6#&%&sigmaG6#%\"nG" }{TEXT -1 38 " be the se
quence of \"signs\" defined by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "sigma(n) = 2*floor(`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 "sigma := n -> floor((n+2 mod 4)/2)*2-1;\nse
q(sigma(n),n=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6#%
\"nG6\"6$%)operatorG%&arrowGF(,&-%&floorG6#,$-%$modG6$,&9$\"\"\"\"\"#F
6\"\"%#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 11 "The series " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Sum(sigma(n)/(n^2),n = 1 .. infinity) = 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 55 "converges ab
solutely to a sum which is no greater than " }{XPPEDIT 18 0 "Sum(1/(n^
2),n = 1 .. infinity)=Pi^2/6" "6#/-%$SumG6$*&\"\"\"F(*$%\"nG\"\"#!\"\"
/F*;F(%)infinityG*&%#PiGF+\"\"'F," }{TEXT -1 1 " " }{TEXT 260 1 "~" }
{TEXT -1 13 " 1.644934067." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 100 "In another worksheet we estimated the sum of thi
s series by direct calculation of some partial sums." }}{PARA 0 "" 0 "
" {TEXT -1 94 "We can also find the sum of this series by grouping tog
ether 4 terms at a time and calculating" }}{PARA 256 "" 0 "" {TEXT -1 
2 "  " }{XPPEDIT 18 0 "Sum(1/((4*n-3)^2)-1/((4*n-2)^2)-1/((4*n-1)^2)+1
/((4*n)^2),n = 1 .. infinity);" "6#-%$SumG6$,**&\"\"\"F(*$,&*&\"\"%F(%
\"nGF(F(\"\"$!\"\"\"\"#F/F(*&F(F(*$,&*&F,F(F-F(F(F0F/F0F/F/*&F(F(*$,&*
&F,F(F-F(F(F(F/F0F/F/*&F(F(*$*&F,F(F-F(F0F/F(/F-;F(%)infinityG" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 79 "evalf(Sum(1/(4*n-3)^2-1/(4*n-2)^2-1/(4*n-1)^2+1/
(4*n)^2,\n   n=1..infinity),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"5]52>@e$)[.r!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Ex
ample 2" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "sigma[n]
" "6#&%&sigmaG6#%\"nG" }{TEXT -1 47 " be the sequence of \"signs\" def
ined 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(si
gma(n),n=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6#%\"nG
6\"6$%)operatorG%&arrowGF(,&-%$modG6$-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$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The \+
series " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(sigma(
n)/(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$*&-%&sigmaG6#%\"nG\"\"\"*$F+\"\"$
!\"\"/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 55 "converges absolutely to a sum which is no greater than " 
}{XPPEDIT 18 0 "Sum(1/(n^3),n = 1 .. infinity);" "6#-%$SumG6$*&\"\"\"F
'*$%\"nG\"\"$!\"\"/F);F'%)infinityG" }{TEXT -1 1 " " }{TEXT 259 1 "~" 
}{TEXT -1 13 " 1.202056903." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 100 "In another worksheet we estimated the sum of t
his series by direct calculation of some partial sums." }}{PARA 0 "" 
0 "" {TEXT -1 94 "We can also find the sum of this series by grouping \+
together 5 terms at a time and calculating" }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "Sum(1/((5*n-4)^3)+1/((5*n-3)^3)-1/((5*n-2)^3
)-1/((5*n-1)^3)-1/((5*n)^3),n = 1 .. infinity);" "6#-%$SumG6$,,*&\"\"
\"F(*$,&*&\"\"&F(%\"nGF(F(\"\"%!\"\"\"\"$F/F(*&F(F(*$,&*&F,F(F-F(F(F0F
/F0F/F(*&F(F(*$,&*&F,F(F-F(F(\"\"#F/F0F/F/*&F(F(*$,&*&F,F(F-F(F(F(F/F0
F/F/*&F(F(*$*&F,F(F-F(F0F/F//F-;F(%)infinityG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
98 "evalf(Sum(1/((5*n-4)^3)+1/((5*n-3)^3)-1/((5*n-2)^3)-\n  1/((5*n-1)
^3)-1/(5*n)^3,n=1..infinity),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"5Z.?Xu9/on5!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exa
mple 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 
"" {TEXT -1 19 "Consider the series" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Sum((-1)^(floor(sqrt(n))-1)/n,n = 1 .. infinity);" "
6#-%$SumG6$*&),$\"\"\"!\"\",&-%&floorG6#-%%sqrtG6#%\"nGF)F)F*F)F2F*/F2
;F)%)infinityG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " 
}{XPPEDIT 18 0 "floor(x)" "6#-%&floorG6#%\"xG" }{TEXT -1 55 " is the l
argest integer which is less than or equal to " }{TEXT 268 1 "x" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 57 "a := n -> (-1)^(floor(sqrt(n))-1)*1/n;\nseq(a(n)
,n=1..17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)op
eratorG%&arrowGF(*&)!\"\",&-%&floorG6#-%%sqrtG6#9$\"\"\"F.F7F7F6F.F(F(
F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "63\"\"\"#F#\"\"##F#\"\"$#!\"\"\"\"
%#F)\"\"&#F)\"\"'#F)\"\"(#F)\"\")#F#\"\"*#F#\"#5#F#\"#6#F#\"#7#F#\"#8#
F#\"#9#F#\"#:#F)\"#;#F)\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "Sum((-1)^(floor(sqrt(n))-1)/n,n = 1 .. infinity);
" "6#-%$SumG6$*&),$\"\"\"!\"\",&-%&floorG6#-%%sqrtG6#%\"nGF)F)F*F)F2F*
/F2;F)%)infinityG" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "``=1+1/2+1/3-1/4-1/5-1/6-1/7-1/8+1/9+1/10+1/11+1/12+1/
13+1/14+1/15-1/16-1/17-` . . . `" "6#/%!G,F\"\"\"F&*&F&F&\"\"#!\"\"F&*
&F&F&\"\"$F)F&*&F&F&\"\"%F)F)*&F&F&\"\"&F)F)*&F&F&\"\"'F)F)*&F&F&\"\"(
F)F)*&F&F&\"\")F)F)*&F&F&\"\"*F)F&*&F&F&\"#5F)F&*&F&F&\"#6F)F&*&F&F&\"
#7F)F&*&F&F&\"#8F)F&*&F&F&\"#9F)F&*&F&F&\"#:F)F&*&F&F&\"#;F)F)*&F&F&\"
#<F)F)%(~.~.~.~GF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 68 "There is a sign change whenever the deno
minator is a perfect square." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 86 "We shall show that this series converges by inv
estigating a suitable bracketed series." }}{PARA 0 "" 0 "" {TEXT -1 
62 "Group together the consecutive terms which have the same sign." }}
{PARA 0 "" 0 "" {TEXT -1 15 "In detail, let " }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "b[1]=1+1/2+1/3" "6#/&%\"bG6#\"\"\",(F'F'*&F'F
'\"\"#!\"\"F'*&F'F'\"\"$F+F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=1/4+1/5+1/6+1/7+1/8" "6#/&%\"bG6#
\"\"#,,*&\"\"\"F*\"\"%!\"\"F**&F*F*\"\"&F,F**&F*F*\"\"'F,F**&F*F*\"\"(
F,F**&F*F*\"\")F,F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "b[3]=1/9+1/10+1/11+1/12+1/13+1/14+1/15" "6#/&%\"bG6#
\"\"$,0*&\"\"\"F*\"\"*!\"\"F**&F*F*\"#5F,F**&F*F*\"#6F,F**&F*F*\"#7F,F
**&F*F*\"#8F,F**&F*F*\"#9F,F**&F*F*\"#:F,F*" }{TEXT -1 2 ", " }}{PARA 
0 "" 0 "" {TEXT -1 4 "   :" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "b[k] = 1/(k^2)+1/(k^2+1)+1/(k^2+2)+` . . . `+1/(k^2+2*k
);" "6#/&%\"bG6#%\"kG,,*&\"\"\"F**$F'\"\"#!\"\"F**&F*F*,&*$F'F,F*F*F*F
-F**&F*F*,&*$F'F,F*F,F*F-F*%(~.~.~.~GF**&F*F*,&*$F'F,F**&F,F*F'F*F*F-F
*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(1/(k^2+i),i = 0 .. 2*k);" 
"6#/%!G-%$SumG6$*&\"\"\"F),&*$%\"kG\"\"#F)%\"iGF)!\"\"/F.;\"\"!*&F-F)F
,F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 23 "and consider the \+
series" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^(k
-1)*b[k],k=1..infinity)" "6#-%$SumG6$*&),$\"\"\"!\"\",&%\"kGF)F)F*F)&%
\"bG6#F,F)/F,;F)%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 49 "We can show that this bracketed series converges." }}{PARA 0 "
" 0 "" {TEXT -1 19 "Split the terms of " }{XPPEDIT 18 0 "b[k]" "6#&%\"
bG6#%\"kG" }{TEXT -1 28 " into two groups by letting " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[k] = 1/(k^2)+` . . . `+1/(k^2+k-
1);" "6#/&%\"cG6#%\"kG,(*&\"\"\"F**$F'\"\"#!\"\"F*%(~.~.~.~GF**&F*F*,(
*$F'F,F*F'F*F*F-F-F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "an
d" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d[k] = 1/(k^2+k)
+` . . . `+1/(k^2+2*k);" "6#/&%\"dG6#%\"kG,(*&\"\"\"F*,&*$F'\"\"#F*F'F
*!\"\"F*%(~.~.~.~GF**&F*F*,&*$F'F-F**&F-F*F'F*F*F.F*" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "b[k]=c[k]+d[k]
" "6#/&%\"bG6#%\"kG,&&%\"cG6#F'\"\"\"&%\"dG6#F'F," }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 14 "For each term " }{TEXT 269 1 "t" }{TEXT 
-1 4 " of " }{XPPEDIT 18 0 "c[k]" "6#&%\"cG6#%\"kG" }{TEXT -1 9 " we h
ave " }{XPPEDIT 18 0 "1/(k^2+k) < t" "6#2*&\"\"\"F%,&*$%\"kG\"\"#F%F(F
%!\"\"%\"tG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``<= 1/k^2" "6#1%!G*&\"\"
\"F&*$%\"kG\"\"#!\"\"" }{TEXT -1 12 " and, since " }{XPPEDIT 18 0 "c[k
]" "6#&%\"cG6#%\"kG" }{TEXT -1 5 " has " }{TEXT 270 1 "k" }{TEXT -1 
19 " terms, we see that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "1/(k+1) < c[k]" "6#2*&\"\"\"F%,&%\"kGF%F%F%!\"\"&%\"cG6#F'" }
{XPPEDIT 18 0 "``<= 1/k" "6#1%!G*&\"\"\"F&%\"kG!\"\"" }{TEXT -1 13 "  \+
------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 14 "For each term " }{TEXT 271 1 "t" }{TEXT -1 4 " of " }{XPPEDIT 
18 0 "d[k]" "6#&%\"dG6#%\"kG" }{TEXT -1 9 " we have " }{XPPEDIT 18 0 "
1/(k^2+2*k+1) < t" "6#2*&\"\"\"F%,(*$%\"kG\"\"#F%*&F)F%F(F%F%F%F%!\"\"
%\"tG" }{XPPEDIT 18 0 " ``<= 1/(k^2+k)" "6#1%!G*&\"\"\"F&,&*$%\"kG\"\"
#F&F)F&!\"\"" }{TEXT -1 12 " and, since " }{XPPEDIT 18 0 "d[k]" "6#&%
\"dG6#%\"kG" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "k+1" "6#,&%\"kG\"\"\"
F%F%" }{TEXT -1 19 " terms, we see that" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "1/(k+1) <d[k]" "6#2*&\"\"\"F%,&%\"kGF%F%F%!\"\"&
%\"dG6#F'" }{XPPEDIT 18 0 "``<= 1/k" "6#1%!G*&\"\"\"F&%\"kG!\"\"" }
{TEXT -1 15 "  ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 55 "By adding
 the inequalities (i) and (ii) it follows that" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "2/(k+1) <b[k]" "6#2*&\"\"#\"\"\",&%\"kG
F&F&F&!\"\"&%\"bG6#F(" }{XPPEDIT 18 0 "``<=2/k" "6#1%!G*&\"\"#\"\"\"%
\"kG!\"\"" }{TEXT -1 15 "  ------- (iii)" }}{PARA 0 "" 0 "" {TEXT -1 
6 "Since " }{XPPEDIT 18 0 "2/(k+1) <b[k]" "6#2*&\"\"#\"\"\",&%\"kGF&F&
F&!\"\"&%\"bG6#F(" }{XPPEDIT 18 0 "``<=2/k" "6#1%!G*&\"\"#\"\"\"%\"kG!
\"\"" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "2/(k+2) <b[k+1]" "6#2*&\"\"
#\"\"\",&%\"kGF&F%F&!\"\"&%\"bG6#,&F(F&F&F&" }{XPPEDIT 18 0 "``<=2/(k+
1)" "6#1%!G*&\"\"#\"\"\",&%\"kGF'F'F'!\"\"" }{TEXT -1 10 ", we have " 
}{XPPEDIT 18 0 "b[k+1] < b[k];" "6#2&%\"bG6#,&%\"kG\"\"\"F)F)&F%6#F(" 
}{TEXT -1 11 "  for eack " }{TEXT 272 1 "k" }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 17 "Also, since both " }{XPPEDIT 18 0 "2/(k+1)" "6#*&
\"\"#\"\"\",&%\"kGF%F%F%!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2/k
" "6#*&\"\"#\"\"\"%\"kG!\"\"" }{TEXT -1 14 " tend to 0 as " }{XPPEDIT 
18 0 "k->infinity" "6#f*6#%\"kG7\"6$%)operatorG%&arrowG6\"%)infinityGF
*F*F*" }{TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Limit(b[k],k = infinity) = 0;" "6#/-%&LimitG6$&%\"bG6#%
\"kG/F*%)infinityG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
41 "We can now see that the bracketed series " }{XPPEDIT 18 0 "Sum((-1
)^(k-1)*b[k],k=1..infinity)" "6#-%$SumG6$*&),$\"\"\"!\"\",&%\"kGF)F)F*
F)&%\"bG6#F,F)/F,;F)%)infinityG" }{TEXT -1 42 " converges by the alter
nating series test." }}{PARA 0 "" 0 "" {TEXT -1 104 "Since all the ter
ms grouped together have the same sign, it follows that the original u
nbracketed series" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Su
m(a[k],k=1..infinity)" "6#-%$SumG6$&%\"aG6#%\"kG/F);\"\"\"%)infinityG
" }{TEXT -1 16 " also converges." }}{PARA 0 "" 0 "" {TEXT -1 97 "Furth
ermore, the sum of the original series can be calculated as the sum of
 the bracketed series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "b := k -> Sum(1/(k^2+j),j=0..2*k);
\nSum((-1)^(k-1)*b(k),k=1..infinity);\nevalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"bGf*6#%\"kG6\"6$%)operatorG%&arrowGF(-%$SumG6$*&\"
\"\"F0,&*$)9$\"\"#F0F0%\"jGF0!\"\"/F6;\"\"!,$F4F5F(F(F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\",&%\"kG\"\"\"F(F+F+-F$6$*&F+F+,
&*$)F*\"\"#F+F+%\"jGF+F(/F3;\"\"!,$F*F2F+/F*;F+%)infinityG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+A73%H\"!\"*" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The following picture shows pa
rtial sums of both the original series ( grey diamonds) and the bracke
ted series (red circles)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 411 "a := n -> (-1)^(floor(sqrt(n))-1)/
n:\nptsA := [seq([n,sum(evalf(a(i)),i=1..n)],n=1..65)]:\nb := n -> sum
(1/(n^2+i),i=0..2*n):\nptsB := [seq([(n+1)^2-1,sum(evalf((-1)^(j-1)*b(
j)),j=1..n)],n=1..7)]:\nn := 'n':\nplot([ptsA,1.294081222,ptsA,ptsB],x
=0..65,style=[line$2,point$2],\n   color=[grey,COLOR(RGB,0,.7,0),COLOR
(RGB,.3,.3,.3),red],\n   linestyle=[2,3],symbol=[diamond,circle],view=
[0..65,0..2],\n   labels=[`n`,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 548 
271 271 {PLOTDATA 2 "6(-%'CURVESG6'7]o7$$\"\"\"\"\"!F(7$$\"\"#F*$\"3++
++++++:!#<7$$\"\"$F*$\"3!******HLLL$=F07$$\"\"%F*$\"3!******HLLLe\"F07
$$\"\"&F*$\"3%******HLLLQ\"F07$$\"\"'F*$\"3#******pmmm@\"F07$$\"\"(F*$
\"33+++C&4Q2\"F07$$\"\")F*$\"3g******R_4)[*!#=7$$\"\"*F*$\"37+++N1#*f5
F07$$\"#5F*$\"3)******\\j?*f6F07$$\"#6F*$\"35+++E(H3D\"F07$$\"#7F*$\"3
)*******eI;M8F07$$\"#8F*$\"3*******f8'369F07$$\"#9F*$\"3++++2Z^#[\"F07
$$\"#:F*$\"3++++u8=\\:F07$$\"#;F*$\"3++++u8o'[\"F07$$\"#<F*$\"3!******
\\%y&yU\"F07$$\"#=F*$\"31+++*G-BP\"F07$$\"#>F*$\"3.+++52n>8F07$$\"#?F*
$\"3)*******42np7F07$$\"#@F*$\"3,+++j;0A7F07$$\"#AF*$\"3-+++<rfw6F07$$
\"#BF*$\"31+++c)=J8\"F07$$\"#CF*$\"3/+++!>_94\"F07$$\"#DF*$\"33+++!>_9
8\"F07$$\"#EF*$\"3/+++GP\"*p6F07$$\"#FF*$\"3)******\\w]p?\"F07$$\"#GF*
$\"3\"******40lEC\"F07$$\"#HF*$\"3#*******3y9x7F07$$\"#IF*$\"32+++V6[5
8F07$$\"#JF*$\"3$******p?RFM\"F07$$\"#KF*$\"3%******p?*)RP\"F07$$\"#LF
*$\"3.+++QAH/9F07$$\"#MF*$\"3.+++-SqL9F07$$\"#NF*$\"3!******4VvAY\"F07
$$\"#OF*$\"3)******Hl(\\M9F07$$\"#PF*$\"31+++E1Z29F07$$\"#QF*$\"3))***
**f$[:\"Q\"F07$$\"#RF*$\"3-+++6Q^b8F07$$\"#SF*$\"35+++6Q^I8F07$$\"#TF*
$\"3*******pcBhI\"F07$$\"#UF*$\"3'******H/9BG\"F07$$\"#VF*$\"33+++H#e!
f7F07$$\"#WF*$\"3)******f&4LO7F07$$\"#XF*$\"33+++M(3T@\"F07$$\"#YF*$\"
3-+++/'pB>\"F07$$\"#ZF*$\"3#******z+$4r6F07$$\"#[F*$\"3/+++v'f-:\"F07$
$\"#\\F*$\"3'******z$ymq6F07$$\"#]F*$\"3'******z$ym!>\"F07$$\"#^F*$\"3
%******4ov-@\"F07$$\"#_F*$\"3#*******\\k]H7F07$$\"#`F*$\"3'******RPu$[
7F07$$\"#aF*$\"33+++$*G*oE\"F07$$\"#bF*$\"3/+++6Z2&G\"F07$$\"#cF*$\"3*
)*****R&=$HI\"F07$$\"#dF*$\"33+++9dZ?8F07$$\"#eF*$\"3(******H4<xL\"F07
$$\"#fF*$\"3)******\\CmYN\"F07$$\"#gF*$\"3'******>\"HLr8F07$$\"#hF*$\"
33+++bjs(Q\"F07$$\"#iF*$\"3!******pQbQS\"F07$$\"#jF*$\"3#******HSG(>9F
07$$\"#kF*$\"3#******HS.TS\"F07$$\"#lF*$\"3/+++)y=()Q\"F0-%'COLOURG6&%
$RGBG$\")=THv!\")F[`lF[`l-%&STYLEG6#%%LINEG-%*LINESTYLEG6#F--%'SYMBOLG
6#%(DIAMONDG-F$6'7S7$$F*F*$\"3)******>A\"3%H\"F07$$\"3WLL$3-:oT\"F0F^a
l7$$\"3ym;z%>y&\\EF0F^al7$$\"3<LLe9$\\f.%F0F^al7$$\"3NML3--^JaF0F^al7$
$\"3,n;HK#Q/#oF0F^al7$$\"3PL$e9J_\"3\")F0F^al7$$\"3c+]7`c]T%*F0F^al7$$
\"3ULekeh/#3\"!#;F^al7$$\"32+D1>&*\\>7FhblF^al7$$\"3#om;a>()3O\"FhblF^
al7$$\"3_L$3x`@a[\"FhblF^al7$$\"39++D6xhD;FhblF^al7$$\"3!****\\Pa*Qm<F
hblF^al7$$\"39++v)G\\?!>FhblF^al7$$\"3dLekL8CD?FhblF^al7$$\"3#pm;/fG<<
#FhblF^al7$$\"3!omm;^AeH#FhblF^al7$$\"37+D\"GCt,W#FhblF^al7$$\"3_mm;*4
[zc#FhblF^al7$$\"3V+D\"yOP\"3FFhblF^al7$$\"3A+vV.2jTGFhblF^al7$$\"3An;
z%*p\"4)HFhblF^al7$$\"3=n\"H2HD)3JFhblF^al7$$\"3wL$3xm\"zYKFhblF^al7$$
\"3KLeke**4!R$FhblF^al7$$\"3$)*\\(=Z-&[^$FhblF^al7$$\"3sL$ek(Re\\OFhbl
F^al7$$\"3%*****\\-rx)y$FhblF^al7$$\"3>+]i?/&\\#RFhblF^al7$$\"3W+D\"y5
0n0%FhblF^al7$$\"3\"***\\PCi*H?%FhblF^al7$$\"3Rmm;*HXWL%FhblF^al7$$\"3
S++vV_zuWFhblF^al7$$\"3/Lek`J(>g%FhblF^al7$$\"3T++D,A,TZFhblF^al7$$\"3
Qm\"z%4r$=([FhblF^al7$$\"3_+D1Moe3]FhblF^al7$$\"3WmmT&o%GU^FhblF^al7$$
\"3K+DJXRD#G&FhblF^al7$$\"3-MLLy41<aFhblF^al7$$\"3aL$3-k?\\b&FhblF^al7
$$\"3_n\"zWvQ;p&FhblF^al7$$\"3p++]-&os\"eFhblF^al7$$\"3On;/rVDhfFhblF^
al7$$\"3-MLL[q.!4'FhblF^al7$$\"3y+D\"GCYtA'FhblF^al7$$\"3#3](=i'o(ejFh
blF^al7$Fc_lF^al-%&COLORG6&Fj_lF*$FG!\"\"F*F^`l-Fc`l6#F3-Ff`l6#%'CIRCL
EG-F$6'F&-F`jl6&Fj_l$F3FcjlF][mF][m-F_`l6#%&POINTGFb`lFe`l-F$6'7)F17$F
K$\"3a+++Q_4)[*FOFhoFerF\\vF]z7$Fi^l$\"3++++/%G(>9F0-Fh_l6&Fj_l$\"*+++
+\"F]`lF]alF]alF^[mFdjlFfjl-%+AXESLABELSG6$%\"nG%!G-%%VIEWG6$;F]alFc_l
;F]alF," 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv
e 1" "Curve 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 4" }}{PARA 0 "" 0 "" {TEXT -1 69 "See: Mathemat
ica in Action by Stan Wagon, Springer-Verlag,  page 440." }}{PARA 0 "
" 0 "" {TEXT -1 49 "The following is a rapidly convergent series for \+
" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "Sum((-1/4)^n*(2/(4*n+1)+2/(4*n+2)+1/(4
*n+3)),n = 0 .. infinity) = Pi;" "6#/-%$SumG6$*&),$*&\"\"\"F+\"\"%!\"
\"F-%\"nGF+,(*&\"\"#F+,&*&F,F+F.F+F+F+F+F-F+*&F1F+,&*&F,F+F.F+F+F1F+F-
F+*&F+F+,&*&F,F+F.F+F+\"\"$F+F-F+F+/F.;\"\"!%)infinityG%#PiG" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "evalf(Sum((-1/4)^n*(2/(4*n+1)+2/(4*n+2)+1/(4*n+3)),\n
  n=0..infinity),50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S^P*Rpr>%)
G]zKQVEYQKz*e`EfTJ!#\\" }}}{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 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }