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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 15 "Maple Project 9" }}
{PARA 0 "" 0 "" {TEXT -1 144 "Today we look at sequences and series.  \+
A sequence is just a list of numbers; as such we can just define them \+
as functions on Maple.  To define " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#
%\"nG" }{TEXT -1 4 "  = " }{XPPEDIT 18 0 "2*n^2/(n^2+1);" "6#*(\"\"#\"
\"\"*$)%\"nG\"\"#F%F%,&*$)%\"nG\"\"#\"\"\"F/\"\"\"F/!\"\"" }{TEXT -1 
36 " , we use typical function notation:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "a:=n->2*n^2/(n^2+1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 104 "Maple can list elements of the sequence.  For example, t
o list out the tenth thru twentieth terms, type:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "seq(a(n),n=10..20);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 77 "We can plot the sequence.  To plot the first 100 terms \+
of the above sequence:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "p
lot( [ seq([n,a(n)],n=1..100) ] , style=point, symbol=cross);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "(Other symbol options are diamond
, box, circle, and point.)  The above sequence appears to have a limit
, namely 2.  We evaluate it in the usual way." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "limit(a(n),n=infinity);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 79 "We can also implement sums and series using the \"sum
\" command.  Using the same " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG
" }{TEXT -1 25 "  as above, let's find   " }{XPPEDIT 18 0 "sum(a[n],n \+
= 1 .. 200);" "6#-%$sumG6$&%\"aG6#%\"nG/%\"nG;\"\"\"\"$+#" }{TEXT -1 
3 "  ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Sum(a(n),n=1..200
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 57 "Again, we don't have to use a capital \"s\" in our comm
and." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sum(ln(n),n=4..7);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 234 "We can sum infinite series.  If Maple re
turns the infinity symbol, then the series diverges.  You may recogniz
e the first series as a simple geometric series.  The second series di
verges because, as we saw, the terms don't go to zero." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sum(1/3^(n-1),n=1..infinity);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Sum(a(n),n=1..infinity);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 49 "I bet you're stoked now.  Let's do some problem
s." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "1)  a) " }{TEXT 273 37 "Plot
 a hundred terms of the sequence " }{TEXT -1 1 " " }{XPPEDIT 18 0 "a[n
];" "6#&%\"aG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(-1)^n;" "6#),
$\"\"\"!\"\"%\"nG" }{TEXT -1 6 " sin( " }{XPPEDIT 18 0 "1/n;" "6#*&\"
\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 4 " )  " }{TEXT 274 42 "and guess its \+
limit by looking at the plot" }{TEXT -1 64 ".  (It almost looks like w
e have 2 sequences in our plot.  The  " }{XPPEDIT 18 0 "(-1)^n" "6#),$
\"\"\"!\"\"%\"nG" }{TEXT -1 113 "  term makes the terms oscillate betw
een positive and negative values, i.e. values above and below the x-ax
is.)  " }{TEXT 272 38 "Then evaluate the limit using Maple.  " }}
{PARA 0 "" 0 "" {TEXT -1 29 "     b) Consider the series  " }{XPPEDIT 
18 0 "sum(1/(n^2),n = 1 .. infinity);" "6#-%$sumG6$*&\"\"\"\"\"\"*$)%
\"nG\"\"#F(!\"\"/%\"nG;\"\"\"%)infinityG" }{TEXT -1 4 "  . " }{TEXT 
275 232 " Plot the sequence of terms and the sequence of partial sums \+
on the same graph.  (Plot 50 of each; use the display command for this
, and also use different 'symbol' values in your plot commands so you \+
can see which plots are which.)" }{TEXT -1 2 "  " }{TEXT 276 4 "Then" 
}{TEXT -1 1 " " }{TEXT 277 29 "evaluate the sum using Maple." }{TEXT 
-1 153 "  Since the series converges, the terms must go to zero, and t
hat's what we saw in the plot.  (The value of this series is a famous \+
result due to Euler.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 52 "2) Remember the harmonic series is given by the sum " 
}{XPPEDIT 18 0 "sum(1/n,n = 1 .. infinity);" "6#-%$sumG6$*&\"\"\"\"\"
\"%\"nG!\"\"/%\"nG;\"\"\"%)infinityG" }{TEXT -1 338 "  .  We know it d
iverges (it gets infinitely large), but there is simply no way to see \+
this by just adding terms and seeing if the sequence of partial sums b
lows up or not.  To see what I mean, suppose we started with the first
 term of the harmonic series the day the universe was formed, 13 billi
on years ago, and added a new term every " }{TEXT 256 6 "second" }
{TEXT -1 3 ".  " }{TEXT 257 72 "About how large would the partial sum \+
be today, assuming a 365-day year?" }{TEXT -1 124 "  (Note: when you d
o this on Maple, it may give you a kooky-looking answer.  If it does, \+
just use \"evalf\" to get a decimal.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 88 "3) Remember the statement of the inte
gral test: if f(x) is positive and decreasing, and " }{XPPEDIT 18 0 "a
[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 15 " = f(n), then  " }{XPPEDIT 18 0 
"sum(a[n],n = 1 .. infinity);" "6#-%$sumG6$&%\"aG6#%\"nG/%\"nG;\"\"\"%
)infinityG" }{TEXT -1 29 "  converges  if and only if  " }{XPPEDIT 18 
0 "int(f(x),x = 1 .. infinity);" "6#-%$intG6$-%\"fG6#%\"xG/%\"xG;\"\"
\"%)infinityG" }{TEXT -1 130 "   converges (i.e. if the integral conve
rges, so does the sum.  If the integral diverges, so does the sum.)  C
onsider the series  " }{XPPEDIT 18 0 "sum(n*exp(-n),n = 1 .. infinity)
;" "6#-%$sumG6$*&%\"nG\"\"\"-%$expG6#,$F'!\"\"F(/%\"nG;\"\"\"%)infinit
yG" }{TEXT -1 5 "  .  " }{TEXT 279 16 "First check that" }{TEXT -1 1 "
 " }{TEXT 280 112 "the terms of this series do in fact decrease by tak
ing a derivative; what is the interval on which it decreases?" }{TEXT 
-1 2 "  " }{TEXT 283 69 "Then use the integral test to determine if it
 converges or diverges. " }{TEXT -1 121 " But note that this test does
 NOT give you the actual sum of the series; it only tells you if it co
nverges or diverges.  " }{TEXT 278 138 "To see this, sum the series.  \+
Use the \"evalf\" command to doublecheck that the sum of the series is
 not equal to the value of the integral." }{TEXT -1 0 "" }}}}{MARK "20
 3 2" 53 }{VIEWOPTS 1 1 0 1 1 1803 }