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{SECT 0 {PARA 256 "" 0 "" {TEXT 261 0 "" }{TEXT 262 0 "" }{TEXT -1 0 "
" }{TEXT 267 0 "" }{TEXT 268 13 "TAYLOR SERIES" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 263 21 "Calculus  III Project" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 10 "Objective:" }}{PARA 3 "" 0 "" {TEXT 269 
79 "To obtain graphical evidence for the interval of convergence of a \+
power series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "For any given " }{TEXT 278 2 "n " }{TEXT -1 4 "and " }
{TEXT 277 1 "a" }{TEXT -1 31 ", Maple will help you find the " }{TEXT 
280 1 "n" }{TEXT -1 40 "th degree Taylor polynomial centered at " }
{TEXT 279 1 "a" }{TEXT -1 62 ". It will also help you graph the functi
on and the polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 264 
15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 29 " Consider the Taylor series  " }{XPPEDIT 18 0 "sum((-1)
^n*(x-1)^(n+1)/(n+1),n = 0 .. infinity);" "6#-%$sumG6$*(),$\"\"\"!\"\"
%\"nGF)),&%\"xGF)F)F*,&F+F)F)F)F),&F+F)F)F)F*/F+;\"\"!%)infinityG" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 258 1 " " }{TEXT -1 85 "   a) Using calculus, find the \+
radius of convergence and the interval of convergence." }}{PARA 0 "" 
0 "" {TEXT -1 60 "    b) The series is the Taylor series  centered  at
 1 for  " }{TEXT 283 4 "lnx " }{TEXT -1 21 ".  Using Maple, plot " }
{TEXT 284 3 "lnx" }{TEXT -1 13 " and the 16th" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "degree Taylor polynomial in one window." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 256 4 "    " }{TEXT -1 2 "c)" }{TEXT 266 1 " " 
}{TEXT -1 9 "For what " }{TEXT 285 1 "x" }{TEXT -1 6 " does " }
{XPPEDIT 18 0 "T[16];" "6#&%\"TG6#\"#;" }{TEXT -1 112 " seem to agree \+
with the function?  Comment on what this interval and the interval of \+
convergence have in common." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 296 0 "" }{TEXT 297 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 
115 "   a) Applying the ratio test, you find that the radius of conver
gence is 1; the interval of convergence is (0,2]. " }{TEXT 265 6 "Note
: " }{TEXT -1 53 "in your answers you must present a detailed argument
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "   b
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"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=x->ln(
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(x), x=a,n) computes the Taylor expansion of " }{TEXT 286 4 "f(x)" }
{TEXT -1 7 " about " }{TEXT 287 1 "a" }{TEXT -1 13 " up to order " }
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0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Plot " }{XPPEDIT 18 
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45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 1 " " }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" 
{TEXT -1 3 "c) " }{XPPEDIT 18 0 "T[16];" "6#&%\"TG6#\"#;" }{TEXT -1 
20 " seems to represent " }{TEXT 281 3 "lnx" }{TEXT -1 16 "  well for \+
 0.1<" }{TEXT 295 1 "x" }{TEXT -1 159 "<1.9,  which is roughly the int
erval of convergence.   The function and the polynomial are very diffe
rent from each other outside the interval of convergence. " }}}{PARA 
259 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 298 
150 "                                                                 \+
                                                                      \+
               " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 0 "" }{TEXT 258 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 259 11 "Problem 1: \+
" }}{PARA 3 "" 0 "" {TEXT 273 15 "Consider the se" }{TEXT 275 0 "" }
{TEXT 270 0 "" }{TEXT 271 0 "" }{TEXT 272 6 "ries  " }{XPPEDIT 274 0 "
sum((-1)^n*x^(2*n+1)/(2*n+1),n = 0 .. infinity);" "6#-%$sumG6$*(),$\"
\"\"!\"\"%\"nGF))%\"xG,&*&\"\"#F)F+F)F)F)F)F),&*&F0F)F+F)F)F)F)F*/F+;
\"\"!%)infinityG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "   a) \+
Using calculus, find the radius of convergence and interval of converg
ence." }}{PARA 0 "" 0 "" {TEXT -1 58 "   b) The series is the Taylor s
eries centered  at 0 for  " }{TEXT 289 7 "arctanx" }{TEXT -1 22 " .  U
sing Maple, plot " }{TEXT 290 7 "arctanx" }{TEXT -1 54 "  and its 15th
 degree Taylor polynomial in one window." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 256 3 "   " }{TEXT -1 2 "c)" }{TEXT 257 1 " " }{TEXT -1 
33 "Highlight the interval  on which " }{XPPEDIT 18 0 "T[15];" "6#&%\"
TG6#\"#:" }{TEXT -1 112 " seems to agree with the function. Comment on
 what this interval and the interval of convergence have in common." }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "
" }{TEXT 276 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 19 "In one win
dow plot " }{TEXT 291 7 "arctanx" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "
T[15];" "6#&%\"TG6#\"#:" }{TEXT -1 51 " about 1- - try to guess the ra
dius of convergence;" }}{PARA 0 "" 0 "" {TEXT -1 5 "plot " }{TEXT 292 
7 "arctanx" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "T[15];" "6#&%\"TG6#\"#
:" }{TEXT -1 58 " about -1 -- try to guess the radius of convergence; \+
plot " }{TEXT 293 7 "arctanx" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "T[15
];" "6#&%\"TG6#\"#:" }{TEXT -1 168 " about 2 -- try to guess the radiu
s of convergence. Try other points as the centers of the Taylor expans
ion -- how does the radius of convergence depend on the center? " }
{TEXT 260 9 "Remember:" }{TEXT -1 40 " when plotting, your choice of r
ange of " }{TEXT 294 1 "x" }{TEXT -1 35 " will depend on the center of
 the  " }{XPPEDIT 18 0 "T[n];" "6#&%\"TG6#%\"nG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 73 "________
_________________________________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "MSIP Gra
nt #P120A80089-98:  \"Three Urban Calculus Reform programs: Adopting t
he Best\" 1998-2001                                                   \+
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