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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT 266 10 "Objective:" }}{PARA 0 "" 0 "" {TEXT -1 64 " To d
eepen the understanding of convergence of  infinite series." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Using a compute
r algebra system will help you  evaluate and plot partial sums of a se
ries." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 265 14 "Solved Example" }
{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 21 "Consider the series  " }
{XPPEDIT 18 0 "sum(1/(n^1.5),n = 1 .. infinity);" "6#-%$sumG6$*&\"\"\"
F')%\"nG-%&FloatG6$\"#:!\"\"F./F);F'%)infinityG" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 62 "   a) Using Maple, graph the first twenty
 terms of the series." }}{PARA 0 "" 0 "" {TEXT -1 55 "   b) Using Mapl
e, graph the first twenty partial sums." }}{PARA 0 "" 0 "" {TEXT -1 
178 "   c) Based on the graphs, make conjectures about the convergence
s of the sequences of terms and partial sums. Make a conjecture, if yo
u can, about the convergence of the series." }}{PARA 0 "" 0 "" {TEXT 
-1 55 "   d) Evaluate the 50th, 100th, 200th,.. partial sums ." }}
{PARA 0 "" 0 "" {TEXT -1 123 "   e) Using calculus, determine the conv
ergence of the series. In a case of a convergent series estimate the s
um using (d)." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 0 "" }{TEXT 256 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT 
-1 3 "   " }}{SECT 0 {PARA 5 "" 0 "" {TEXT 267 43 "Graph the first twe
nty terms of the series." }}{PARA 5 "" 0 "" {TEXT 268 35 " Define the \+
nth term of the series:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a
:=n->1/n^1.5;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6
$%)operatorG%&arrowGF(*&\"\"\"F-*$)9$$\"#:!\"\"F-F3F(F(F(" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }{TEXT 269 46 "Define the sequence of the first \+
twenty terms:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p:=seq([n,a
(n)],n=1..20);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pG667$\"\"\"$F
'\"\"!7$\"\"#$\"+1R`NN!#57$\"\"$$\"+(*3]C>F.7$\"\"%$\"++++]7F.7$\"\"&$
\"+5>FW*)!#67$\"\"'$\"+u\"QT!oF;7$\"\"($\"+sC\\*R&F;7$\"\")$\"+#Q<%>WF
;7$\"\"*$\"+/Pq.PF;7$\"#5$\"+gwFiJF;7$\"#6$\"+BA,TFF;7$\"#7$\"+Ahi0CF;
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7$\"#<$\"+Z,oE9F;7$\"#=$\"+-qX48F;7$\"#>$\"+J7X27F;7$\"#?$\"+*)R.=6F;
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 27 "Plot the sequence of
 terms:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([p], n=0..20
,style=point,symbol=circle);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 255 144 
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" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 
271 0 "" }{TEXT -1 14 "b) Define the " }{TEXT 272 1 "n" }{TEXT -1 15 "
th partial sum:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "s:=n->sum(1/i^1.5,i=1..n);\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"sGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$sumG6$*&
\"\"\"F0*$)%\"iG$\"#:!\"\"F0F6/F3;F09$F(F(F(" }}}{PARA 0 "" 0 "" 
{TEXT -1 53 "Define the sequence of the first twenty partial sums:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "q:=seq([n,s(n)],n=1..20);\n
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+!R`NN\"!\"*7$\"\"$$\"+![.ga\"F.7$\"\"%$\"+![.5n\"F.7$\"\"&$\"+*>Y/w\"
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\"+\\ELM@F.7$\"#=$\"+>sUZ@F.7$\"#>$\"+J<]f@F.7$\"#?$\"+r?oq@F." }}}
{PARA 0 "" 0 "" {TEXT -1 34 "Plot the sequence of partial sums:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot([q],n=0..20,0..3,style=
point,symbol=circle);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 289 167 167 
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{TEXT -1 1 " " }}{SECT 0 {PARA 5 "" 0 "" {TEXT 274 2 "c)" }{TEXT -1 1 
" " }{TEXT 273 72 "Conjectures about the convergence of the terms, par
tial sums and series:" }}{PARA 0 "" 0 "" {TEXT -1 436 " It appears fro
m the graph in (a) that the sequence of terms is convergent to zero. B
y looking at the graph in (b), you may be tempted to say that the sequ
ence of partial sums is convergent to about 2.2. In fact , you cannot \+
draw any conclusions about the convergence because the sequence is inc
reasing and for all you know it may be increasing to infinity however \+
slowly. No conclusions can be made about the convergence of the series
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 257 1 " " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 275 3 "
d) " }{TEXT 276 1 " " }{TEXT 277 27 "Evaluate the partial sums: " }}
{PARA 259 "" 0 "" {TEXT 278 61 "(use copy and paste with some editing \+
to expedite the typing)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e
valf(subs(n=50,s(n)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z(R4L#
!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(subs(n=100,s(
n)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*4uGT#!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(subs(n=200,s(n)));\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[08rC!\"*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "evalf(subs(n=400,s(n)));\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+5yV7D!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "evalf(subs(n=800,s(n)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+hnoTD!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(subs(
n=1600,s(n)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+gJQiD!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(subs(n=3200,s(n)));\n
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+sF-xD!\"*" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 3 "   " }}
{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 279 30 "e) Conclusions u
sing Calculus:" }}{PARA 0 "" 0 "" {TEXT -1 10 "This is a " }{TEXT 280 
1 "p" }{TEXT -1 15 "-series where  " }{TEXT 282 1 "p" }{TEXT -1 13 "=1
.5.  Since " }{TEXT 281 1 "p" }{TEXT -1 58 ">1 the series is convergen
t. The limit (sum) is about 2.6." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
283 147 "                                                             \+
                                                                      \+
                " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 258 0 "" }
{TEXT 259 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 
{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" }{TEXT 261 0 "" }
{TEXT 262 0 "" }{TEXT 263 10 "Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 
59 " For each of the series (i)-(iii) answer questions (a)-(e)." }}
{PARA 0 "" 0 "" {TEXT -1 4 "(i) " }{XPPEDIT 18 0 "sum(((-3)/Pi)^n,n = \+
1 .. infinity);" "6#-%$sumG6$)*&,$\"\"$!\"\"\"\"\"%#PiGF*%\"nG/F-;F+%)
infinityG" }{TEXT -1 16 "            (ii)" }{XPPEDIT 18 0 "sum(n*e^(-n
^2),n = 1 .. infinity);" "6#-%$sumG6$*&%\"nG\"\"\")%\"eG,$*$F'\"\"#!\"
\"F(/F';F(%)infinityG" }{TEXT -1 18 "            (iii) " }{XPPEDIT 18 
0 "sum(n/sqrt(2*n^3-1),n = 1 .. infinity);" "6#-%$sumG6$*&%\"nG\"\"\"-
%%sqrtG6#,&*&\"\"#F(*$F'\"\"$F(F(F(!\"\"F1/F';F(%)infinityG" }{TEXT 
-1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
62 "   a) Using Maple, graph the first twenty terms of the series." }}
{PARA 0 "" 0 "" {TEXT -1 55 "   b) Using Maple, graph the first twenty
 partial sums." }}{PARA 0 "" 0 "" {TEXT -1 169 "   c) Using the graphs
 in (a) and (b), comment on the convergence of the sequence of terms, \+
on the convergence of the partial sums, and on the convergence of the \+
series." }}{PARA 0 "" 0 "" {TEXT -1 68 "   d) Using Maple, evaluate th
e 50th, 100th, 200th,... partial sums." }}{PARA 0 "" 0 "" {TEXT -1 
226 "   e) Using calculus, determine the convergence of the series -- \+
present a detailed argument. In the case of a convergent geometric ser
ies find the sum algebraically; for all other convergent series estima
te the sum using (d)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 264 10 "Problem 2:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 256 1 " " }{TEXT -1 329 "Consider the harmon
ic series. Plot its first twenty partial sums, plot its first thirty p
artial sums. Are you convinced the series is divergent? - Probably not
. Look at s(4), s(8), s(16),..,s(128) and present an argument for the \+
divergence of the series. Look at the differences between each two con
secutive sums you've computed." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 70 "_________________________________________
_____________________________" }}{PARA 0 "" 0 "" {TEXT -1 161 "MSIP Gr
ant #P120A80089-98:  \"Three Urban Calculus Reform programs: Adopting \+
the Best\" 1998-2001                                                  \+
                " }}}{MARK "4 8 8 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }