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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 259 
"" 0 "" {TEXT -1 32 "Lesson 17: Convergence of Series" }}}{EXCHG 
{PARA 260 "" 0 "" {TEXT -1 14 "Definition: A " }{TEXT 256 6 "series" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[n],n=1..infinity" "6#-%$SumG6$&%
\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 41 " consists of two seque
nces: The sequence " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT 
-1 41 " of terms of the series and the sequence " }{XPPEDIT 18 0 "S[n]
=Sum(a[i],i=1..n" "6#/&%\"SG6#%\"nG-%$SumG6$&%\"aG6#%\"iG/F.;\"\"\"F'
" }{TEXT -1 142 " of partial sums of the series. If the sequence of pa
rtial sums converges to a limit L, then the series is said to converge
 to L and we write " }{XPPEDIT 18 0 "Sum(a[n],n=1..infinity)=L" "6#/-%
$SumG6$&%\"aG6#%\"nG/F*;\"\"\"%)infinityG%\"LG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 168 "Suppose you have established somehow, ei
ther directly or by some test, that a series converges to a number L. \+
How do we calculate this number to any specified accuracy?" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Not surprizingly,
 Maple can sum a lot of series already. For example, the sum of a " }
{TEXT 256 16 "geometric series" }{TEXT -1 30 " is easy for Maple to co
mpute." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Sum(a*r^n,n=0..infinity)=sum
(a*r^n,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&
%\"aG\"\"\")%\"rG%\"nGF)/F,;\"\"!%)infinityG,$*&F(F),&F+F)!\"\"F)F4F4
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Maple also knows that the har
monic series diverges to infinity. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "sum(1/n,n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "It also kno
ws how to compute the sums of the various convergent p-series. For exa
mple, the 3-series sums to" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "Sum(1/n^3,n=1..infinity)=sum(1/n^3,n=1..infinity);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*$)%\"nG\"\"$F(!\"\"/F+;F(%)in
finityG-%%ZetaG6#F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }
{TEXT 256 21 "Riemann Zeta function" }{TEXT -1 58 " is defined for all
 p > 1 to give the sum of the p-series." }}{PARA 0 "" 0 "" {TEXT -1 
43 "So, for example, the sum of the 3 series is" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "Zeta(3)=Zeta(3.);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%%ZetaG6#\"\"$$\"+.p0-7!\"*" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 111 "If the series converges fast enough you can look at th
e sequence of partial sums and get the desired accuracy. " }}{PARA 0 "
" 0 "" {TEXT -1 52 "Lets see how fast the 3-series converges to Zeta(3
)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "for n from 10 by 100 \+
to 300 do Sum(1/i^3,i=1..n)=evalf(sum(1/i^3,i=1..n)) od;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*$)%\"iG\"\"$F(!\"\"/F+;F(\"
#5$\"+')>`(>\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"
\"F(*$)%\"iG\"\"$F(!\"\"/F+;F(\"$5\"$\"+bf,-7!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*$)%\"iG\"\"$F(!\"\"/F+;F(\"$5#$\"+
>c/-7!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Well, the sum of the
 first 100 terms is accurate to 4 significant figures." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "You can't decide fo
r sure by looking at first few partial sums of a series that the serie
s converges. For example, look at a few partial sums of the harmonic s
eries." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "seq(evalf(sum(1/i
,i=1..100*n)),n=1..5); n:='n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"+
=vP(=&!\"*$\"+[4.yeF%$\"+!)Qm#G'F%$\"+\"pH*plF%$\"+IM#Gz'F%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Hmmm. You can't really tell by loo
king at these that the harmonic series doesn't converge." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 
-1 8 "Problems" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 261 "" 
0 "" {TEXT -1 321 "Exercises: In each of the problems below, determine
 whether the series converges or diverges. Give a reason in each case.
 For the convergent series, get an estimate correct to 2 decimal place
s of the sum of the series using psums or some other word of your own \+
devising. You can check with sum to see if Maple can sum it." }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum( 1/(3+2*n)^2,n=1..infinity);
 " "6#-%$SumG6$*&\"\"\"F'*$,&\"\"$F'*&\"\"#F'%\"nGF'F'F,!\"\"/F-;F'%)i
nfinityG" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "This \+
series converges by comparison with the p-series " }{XPPEDIT 18 0 "Sum
(1/n^2,n=1..infinity)" "6#-%$SumG6$*&\"\"\"F'*$%\"nG\"\"#!\"\"/F);F'%)
infinityG" }{TEXT -1 39 ". Each partial sum is bounded above by " }
{XPPEDIT 18 0 "Zeta(2)" "6#-%%ZetaG6#\"\"#" }{TEXT -1 148 " and the pa
rtial sums form an increasing sequence, so we know the sequence of par
tial sums converge. Checking to see what is programmed into Maple, " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "sum(1/(3+2*n)^2,n=1..infin
ity)=\nevalf(sum(1/(3+2*n)^2,n=1..infinity));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&#!#5\"\"*\"\"\"*$)%#PiG\"\"#F(#F(\"\")$\"*S%*eA\"!\"
*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "we get an exact sum. Check a
 few partial sums ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "seq(
evalf(sum(1/(3+2*i)^2,i=1..100*n)),\nn=1..5); n:='n':" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6'$\"+$y%Q,7!#5$\"+y\"=N@\"F%$\"+_ih<7F%$\"+)[v'>7F%
$\"+7V\"4A\"F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum(1/(n*(ln \+
(n))^2 ),n=2..infinity);" "6#-%$SumG6$*&\"\"\"F'*&%\"nGF'*$-%#lnG6#F)
\"\"#F'!\"\"/F);F.%)infinityG" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "1/(n*ln(n)^2)" "6#*&\"
\"\"F$*&%\"nGF$*$-%#lnG6#F&\"\"#F$!\"\"" }{TEXT -1 94 " is a decreasin
g for n > 1 (Take the derivative), so we can use the integral test on \+
this one." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "int(1/(n*(ln (
n))^2),n=1..infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Since the integral diverges, \+
the series diverges. " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum(1/
(2*n+1)^(1/3),n=0..infinity);" "6#-%$SumG6$*&\"\"\"F'),&*&\"\"#F'%\"nG
F'F'F'F'*&F'F'\"\"$!\"\"F//F,;\"\"!%)infinityG" }{TEXT -1 56 " This se
ries diverges, since it is a p-series with p < 1" }}}{EXCHG {PARA 262 
"" 0 "" {XPPEDIT 18 0 "Sum((1+2^n)/(1+3^n),n=0..infinity);" "6#-%$SumG
6$*&,&\"\"\"F()\"\"#%\"nGF(F(,&F(F()\"\"$F+F(!\"\"/F+;\"\"!%)infinityG
" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum(( n^5+4
*n^3+1)/(2*n^9+n^4+2 ),n=0..infinity);" "6#-%$SumG6$*&,(*$%\"nG\"\"&\"
\"\"*&\"\"%F+*$F)\"\"$F+F+F+F+F+,(*&\"\"#F+*$F)\"\"*F+F+*$F)F-F+F2F+!
\"\"/F);\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "This \+
series converges by comparison with " }{XPPEDIT 18 0 "Sum(n^6/n^9" "6#
-%$SumG6#*&%\"nG\"\"'*$F'\"\"*!\"\"" }{TEXT -1 2 ". " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 58 "evalf(int(( n^5+4*n^3+1)/\n (2*n^9+n^4+2 \+
),n=1..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6^g4V!#5" }}
}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum(sin (1/n^4 ),n=1..infinity)
;" "6#-%$SumG6$-%$sinG6#*&\"\"\"F**$%\"nG\"\"%!\"\"/F,;F*%)infinityG" 
}{TEXT -1 55 " This series converges by comparison with the 4-series.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalf(sum(sin(1/(n^4)),
n=1..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?@`P#*!#5" }}}
{EXCHG {PARA 263 "" 0 "" {XPPEDIT 18 0 "Sum( n*e^(-n^2),n=1..infinity)
; " "6#-%$SumG6$*&%\"nG\"\"\")%\"eG,$*$F'\"\"#!\"\"F(/F';F(%)infinityG
" }}}}}{MARK "17" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 
1 2 33 1 1 }