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{SECT 0 {PARA 200 "" 0 "" {TEXT 220 20 "Lab 11: Power Series" }}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 
202 "" 0 "" {TEXT 221 12 "Introduction" }}{PARA 203 "" 0 "" {TEXT 222 
49 "A power series is an infinite series of the form " }{XPPEDIT 18 0 
"Sum(c[n]*x^n, n=0..infinity)" "6#-%$SumG6$*&&%\"cG6#%\"nG\"\"\")%\"xG
F*F+/F*;\"\"!%)infinityG" }{TEXT 223 1 " " }{TEXT 222 12 ", where the \+
" }{XPPEDIT 18 0 "c[n]" "6#&%\"cG6#%\"nG" }{TEXT 223 1 " " }{TEXT 222 
13 " are certain " }{TEXT 224 12 "coefficients" }{TEXT 222 5 " and " }
{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 223 1 " " }{TEXT 222 4 " is " }
{TEXT 224 8 "variable" }{TEXT 222 1 "." }}{SECT 1 {PARA 202 "" 0 "" 
{TEXT 221 45 "A Geometric Series, Partial Sum Approximation" }}{PARA 
203 "" 0 "" {TEXT 222 41 "In class we already looked at the series " }
{XPPEDIT 18 0 "Sum(x^n, n=0..infinity)=1+x+x^2+x^3+x^4" "6#/-%$SumG6$)
%\"xG%\"nG/F);\"\"!%)infinityG,,\"\"\"F/F(F/*$F(\"\"#F/*$F(\"\"$F/*$F(
\"\"%F/" }{TEXT 223 1 " " }{TEXT 222 16 "+...  This is a " }{TEXT 224 
16 "geometric series" }{TEXT 222 2 ". " }}{PARA 203 "" 0 "" {TEXT 222 
6 "Here  " }{XPPEDIT 18 0 "c[n]=1" "6#/&%\"cG6#%\"nG\"\"\"" }{TEXT 
223 1 " " }{TEXT 222 2 ", " }{XPPEDIT 18 0 "n=0" "6#/%\"nG\"\"!" }
{TEXT 223 1 " " }{TEXT 222 67 ",1,2,..., and we know that this series \+
converges for all values of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 223 
1 " " }{TEXT 222 14 " that satisfy " }{XPPEDIT 18 0 "abs(x) <1" "6#2-%
$absG6#%\"xG\"\"\"" }{TEXT 223 1 " " }{TEXT 222 2 ". " }}{PARA 203 "" 
0 "" {TEXT 222 55 "We know that the sum of a geometric series of the f
orm " }{XPPEDIT 18 0 "sum(a*r^n, n=0..infinity)" "6#-%$sumG6$*&%\"aG\"
\"\")%\"rG%\"nGF(/F+;\"\"!%)infinityG" }{TEXT 223 1 " " }{TEXT 222 14 
"   (provided  " }{XPPEDIT 18 0 "abs(r) < 1" "6#2-%$absG6#%\"rG\"\"\"
" }{TEXT 223 1 " " }{TEXT 222 7 ")  is  " }{XPPEDIT 18 0 "a/(1-r)" "6#
*&%\"aG\"\"\",&F%F%%\"rG!\"\"F(" }{TEXT 223 1 " " }{TEXT 222 1 "." }}
{EXCHG {PARA 203 "" 0 "" {TEXT 222 50 "Therefore the sum of the geomet
ric series above is" }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 50 "Sum(x^n, n
=0..infinity) = sum(x^n, n=0..infinity);" }}}{EXCHG {PARA 203 "" 0 "" 
{TEXT 222 41 "Note that the sum of a power series is a " }{TEXT 224 
24 "function of the variable" }{TEXT 225 1 " " }{XPPEDIT 213 0 "x" "6#
%\"xG" }{TEXT 225 1 " " }{TEXT 222 54 " -- not just a number as we hav
e encountered thus far." }}{PARA 203 "" 0 "" {TEXT 222 21 "This means \+
that, for " }{XPPEDIT 18 0 "abs(x) <1" "6#2-%$absG6#%\"xG\"\"\"" }
{TEXT 223 1 " " }{TEXT 222 15 ", the function " }{XPPEDIT 18 0 "f(x) =
 1/(1-x)" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)F'!\"\"F+" }{TEXT 223 1 " \+
" }{TEXT 222 27 " and the sum of the series " }{XPPEDIT 18 0 "g(x) = S
um(x^n,n = 0 .. infinity)" "6#/-%\"gG6#%\"xG-%$SumG6$)F'%\"nG/F,;\"\"!
%)infinityG" }{TEXT 223 1 " " }{TEXT 222 1 " " }{TEXT 224 12 "are the \+
same" }{TEXT 222 1 "." }}{PARA 203 "" 0 "" {TEXT 222 34 "Let's illustr
ate this graphically." }}{PARA 203 "" 0 "" {TEXT 222 31 "Since the ser
ies converges to  " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 223 1 " " }
{TEXT 222 10 " only for " }{XPPEDIT 18 0 "abs(x) < 1" "6#2-%$absG6#%\"
xG\"\"\"" }{TEXT 223 1 " " }{TEXT 222 53 " we plot on an interval slig
htly smaller than [-1,1]." }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 151 "f :
= x -> 1/(1-x);\ng := x -> sum(x^n, n=0..infinity);\neps:=0.01:\nplot(
[f(x), g(x)], x=-1+eps..1-eps, color=[green,red], linestyle=[2,3], axe
s=framed); " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 222 32 "In practice one
 often takes the " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT 223 1 " " }
{TEXT 222 67 "-th partial sum of a power series (which is a polynomial
 of degree " }{XPPEDIT 2 0 "N-1" "6#,&%\"NG\"\"\"F%!\"\"" }{TEXT 223 
1 " " }{TEXT 222 19 ", and converges to " }{XPPEDIT 18 0 "f" "6#%\"fG
" }{TEXT 223 1 " " }{TEXT -1 1 " " }{TEXT 222 3 "as " }{XPPEDIT 2 0 "N
" "6#%\"NG" }{TEXT 223 1 " " }{TEXT 222 35 " approaches infinity) to r
epresent " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 223 1 " " }{TEXT 222 
39 " instead of the full (infinite) series." }}{PARA 203 "" 0 "" 
{TEXT 222 36 "This, of course, results only in an " }{TEXT 224 13 "app
roximation" }{TEXT 222 4 " to " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 
223 1 " " }{TEXT 222 1 "." }}{PARA 203 "" 0 "" {TEXT 222 19 "Here is a
 plot of  " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 223 1 " " }{TEXT 222 
38 " together with the 10-th partial sum, " }{XPPEDIT 18 0 "s[10]" "6#
&%\"sG6#\"#5" }{TEXT 223 1 " " }{TEXT 222 5 ", of " }{XPPEDIT 18 0 "g
" "6#%\"gG" }{TEXT 223 1 " " }{TEXT 222 23 " (the infinite series)." }
}{PARA 203 "" 0 "" {TEXT 222 14 "Note that the " }{XPPEDIT 18 0 "N" "6
#%\"NG" }{TEXT 223 1 " " }{TEXT 222 38 "-th partial sum consists of th
e first " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT 223 1 " " }{TEXT 222 31 
" terms of the power series, so " }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 
104 "s := (x,N) -> sum(x^n, n=0..N-1);\nplot([f(x),s(x,10)], x=-1+eps.
.1-eps, color=[green,red], axes=framed);" }}}{PARA 203 "" 0 "" {TEXT 
222 38 "This plot shows that (especially near " }{XPPEDIT 18 0 "x=0" "
6#/%\"xG\"\"!" }{TEXT 223 1 " " }{TEXT 222 51 ", the center of the int
erval) the approximation of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 223 
1 " " }{TEXT 222 4 " by " }{XPPEDIT 18 0 "s[10]" "6#&%\"sG6#\"#5" }
{TEXT 223 1 " " }{TEXT 222 15 " is very good. " }}{PARA 203 "" 0 "" 
{TEXT 222 39 "Towards the endpoints of the interval [" }{XPPEDIT 18 0 
"-1,1" "6$,$\"\"\"!\"\"F$" }{TEXT 223 1 " " }{TEXT 222 44 "], however,
 the quality is noticably poorer." }}{EXCHG {PARA 203 "" 0 "" {TEXT 
222 72 "Let's also take a look at the terms in one of these partials s
ums, e.g. " }{XPPEDIT 18 0 "s[4]" "6#&%\"sG6#\"\"%" }{TEXT 223 1 " " }
{TEXT 222 12 " is given by" }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 7 "s(x,
4);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 222 43 "Note that this is a pol
ynomial of degree 3." }}{PARA 203 "" 0 "" {TEXT 222 16 "In general, th
e " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT 223 1 " " }{TEXT 222 39 "-th \+
partial sum of a power series is a " }{TEXT 224 10 "polynomial" }
{TEXT 222 11 " of degree " }{XPPEDIT 18 0 "N-1;" "6#,&%\"NG\"\"\"F%!\"
\"" }{TEXT 223 1 " " }{TEXT 222 2 ". " }}{PARA 203 "" 0 "" {TEXT 222 
19 "This means that an " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT 223 1 " \+
" }{TEXT 222 50 "-th partial sum approximation of a function is an " }
{TEXT 224 13 "approximation" }{TEXT 222 43 " (of a possibly very compl
icated function) " }{TEXT 224 15 "by a polynomial" }{TEXT 222 1 "." }}
}}{SECT 1 {PARA 202 "" 0 "" {TEXT 221 20 "Test for Convergence" }}
{EXCHG {PARA 203 "" 0 "" {TEXT 222 60 "Here are examples of two more (
very important) power series." }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 86 "
Sum((-1)^n/(2*n)!*x^(2*n), n=0..infinity) = sum((-1)^n/(2*n)!*x^(2*n),
 n=0..infinity);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 222 3 "and" }}
{PARA 201 "> " 0 "" {MPLTEXT 1 0 94 "Sum((-1)^n/(2*n+1)!*x^(2*n+1), n=
0..infinity) = sum((-1)^n/(2*n+1)!*x^(2*n+1), n=0..infinity);" }}}
{EXCHG {PARA 203 "" 0 "" {TEXT 222 45 "Our goal is to determine for wh
ich values of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 223 1 " " }{TEXT 
222 23 " these identities hold." }}{PARA 203 "" 0 "" {TEXT 222 31 "In \+
order to do this we use the " }{TEXT 224 10 "Ratio Test" }{TEXT 222 1 
"." }}}{EXCHG {PARA 203 "" 0 "" {TEXT 222 82 "First we define the term
s in the series as functions so we can easily evaluate at " }{XPPEDIT 
18 0 "n" "6#%\"nG" }{TEXT 223 1 " " }{TEXT 222 5 " and " }{XPPEDIT 18 
0 "n+1" "6#,&%\"nG\"\"\"F%F%" }{TEXT 223 1 " " }}{PARA 201 "> " 0 "" 
{MPLTEXT 1 0 32 "a := n -> (-1)^n/(2*n)!*x^(2*n);" }}}{EXCHG {PARA 
203 "" 0 "" {TEXT 222 38 "The Ratio Test states that the series " }
{XPPEDIT 18 0 "Sum(a[n],n = 0 .. infinity);" "6#-%$SumG6$&%\"aG6#%\"nG
/F);\"\"!%)infinityG" }{TEXT 223 1 " " }{TEXT 222 15 " converges if  \+
" }{XPPEDIT 18 0 "limit(abs(a[n+1])/abs(a[n]),n = infinity) = L;" "6#/
-%&limitG6$*&-%$absG6#&%\"aG6#,&%\"nG\"\"\"F0F0F0-F)6#&F,6#F/!\"\"/F/%
)infinityG%\"LG" }{TEXT 223 1 " " }{TEXT 222 6 "  < 1." }}{PARA 203 "
" 0 "" {TEXT 222 9 "Therefore" }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 46 "
abs(a[n+1])/abs(a[n]) = abs(a(n+1))/abs(a(n));" }}{PARA 203 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 12 "simplify(
%);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 222 3 "and" }}{PARA 201 "> " 0 
"" {MPLTEXT 1 0 84 "Limit(abs(a[n+1])/abs(a[n]), n=infinity) = limit(a
bs(a(n+1))/abs(a(n)), n=infinity);" }}{PARA 203 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 203 "" 0 "" {TEXT 222 2 "or" }}{PARA 201 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 222 
67 "This tells us that the series will converge to the cosine function
 " }{TEXT 224 8 "for any " }{XPPEDIT 204 0 "x" "6#%\"xG" }{TEXT 224 1 
" " }{TEXT 222 41 ", since 0 < 1 regardless of the value of " }
{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 223 1 " " }{TEXT 222 1 "." }}}}
{SECT 1 {PARA 202 "" 0 "" {TEXT 221 46 "An Animation of the Partial Su
m Approximations" }}{EXCHG {PARA 203 "" 0 "" {TEXT 222 48 "Let's creat
e an animation to illustrate how the " }{XPPEDIT 18 0 "N" "6#%\"NG" }
{TEXT 223 1 " " }{TEXT 222 81 "-th partial sums approximate the limiti
ng function increasingly more accurate as " }{XPPEDIT 18 0 "N" "6#%\"N
G" }{TEXT 223 1 " " }{TEXT 222 11 " increases." }}{PARA 203 "" 0 "" 
{TEXT 222 21 "We plot the function " }{XPPEDIT 18 0 "f(x) = cos(x)" "6
#/-%\"fG6#%\"xG-%$cosG6#F'" }{TEXT 223 1 " " }{TEXT 222 18 " on the in
terval [" }{XPPEDIT 18 0 "-4*Pi" "6#,$*&\"\"%\"\"\"%#PiGF&!\"\"" }
{TEXT 223 1 " " }{TEXT 222 2 ", " }{XPPEDIT 18 0 "4*Pi" "6#*&\"\"%\"\"
\"%#PiGF%" }{TEXT 223 1 " " }{TEXT 222 82 "] together with the partial
 sum approximation (for which we increase the value of " }{XPPEDIT 18 
0 "N" "6#%\"NG" }{TEXT 223 1 " " }{TEXT 222 40 " by one in each frame \+
of the animation)." }}{PARA 203 "" 0 "" {TEXT 222 100 "For best unders
tanding step through the animation one frame at a time by clicking on \+
the ->| button." }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 139 "N:=15:\ns := \+
(x,N) -> sum((-1)^n/(2*n)!*x^(2*n), n=0..N-1);\nwith(plots):\ntrue_plo
t := plot(cos(x), x=-4*Pi..4*Pi, y=-1.2..1.2, color=green):" }
{MPLTEXT 1 0 159 "\napproximations := animate(s(x,nu), x=-4*Pi..4*Pi, \+
nu=1..N, frames=N, view=[-4*Pi..4*Pi,-1..1], color=red, numpoints=200)
:\ndisplay(\{true_plot,approximations\});" }}}{EXCHG {PARA 203 "" 0 "
" {TEXT 222 52 "And what are the first few polynomials we used here?" 
}}{PARA 201 "> " 0 "" {MPLTEXT 1 0 49 "N:=4:\nfor nu from 1 to N do\n \+
  s[nu]=s(x,nu);\nod;" }}}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 221 13 "As
signment 12" }}{SECT 1 {PARA 202 "" 0 "" {TEXT 221 5 "Ex.1:" }}{PARA 
203 "" 0 "" {TEXT 222 88 "Modify the discussion of the series for cosi
ne for the one representing sine, i,e., for " }{XPPEDIT 18 0 "f(x) = s
in(x)" "6#/-%\"fG6#%\"xG-%$sinG6#F'" }{TEXT 223 1 " " }{TEXT 222 2 ": \+
" }}{PARA 203 "" 0 "" {TEXT -1 22 "a) Using the identity " }{XPPEDIT 
18 0 "sin(x) = cos(x-Pi/2);" "6#/-%$sinG6#%\"xG-%$cosG6#,&F'\"\"\"*&%#
PiGF,\"\"#!\"\"F0" }{TEXT -1 76 ", start with the power series for cos
ine and obtain a power series for sine." }}{PARA 203 "" 0 "" {TEXT 
222 80 "b) Find the center and interval of convergence of your power s
eries found in a)." }}{PARA 203 "" 0 "" {TEXT 222 62 "c) Create an ani
mation of the partial sum approximations (use " }{XPPEDIT 18 0 "N=15" 
"6#/%\"NG\"#:" }{TEXT 223 1 " " }{TEXT 222 11 " as above)." }}{PARA 
203 "" 0 "" {TEXT 222 47 "d) Print the first few partial sum polynomia
ls." }}}{SECT 1 {PARA 202 "" 0 "" {TEXT 221 5 "Ex.2:" }}{PARA 203 "" 
0 "" {TEXT 222 26 "Consider the power series " }{XPPEDIT 18 0 "Sum(x^n
/(n*3^n),n = 1 .. infinity)" "6#-%$SumG6$*&)%\"xG%\"nG\"\"\"*&F)F*)\"
\"$F)F*!\"\"/F);F*%)infinityG" }{TEXT 223 1 " " }{TEXT 222 1 "." }}
{PARA 203 "" 0 "" {TEXT 222 48 "a) Print the first 5 terms of this pow
er series." }}{PARA 203 "" 0 "" {TEXT 222 22 "b) For what values of " 
}{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 223 1 " " }{TEXT 222 98 " does it \+
converge? (Use the Ratio Test, but also decide what happens at the end
s of the interval)." }}{PARA 203 "" 0 "" {TEXT 222 34 "c) Print the fi
rst 5 partial sums." }}{PARA 203 "" 0 "" {TEXT 222 33 "d) What is the \+
limiting function?" }}{PARA 203 "" 0 "" {TEXT 222 64 "e) Create an ani
mation of the partial sum approximations (up to " }{XPPEDIT 18 0 "N=15
" "6#/%\"NG\"#:" }{TEXT 223 1 " " }{TEXT 222 39 " terms) on the interv
al of convergence." }}}}{PARA 204 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 0 \+
0" 13 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }