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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 
200 36 "Taylor Polynomials and Taylor Series" }}{PARA 0 "" 0 "" {TEXT 
251 0 "" }}{PARA 4 "" 0 "" {TEXT 252 29 "Syntax for Taylor Polynomials
" }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" {TEXT 251 126 "Ma
ple can easily find Taylor polynomials even for very complicated funct
ions. The example below illustrates the proper syntax." }}{PARA 0 "" 0
 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" {TEXT 203 14 "Example 1.    " }
{TEXT 251 22 "Consider the function " }{XPPEDIT 18 0 "f(x) = 1/(1+2*x)
;" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)*&\"\"#F)F'F)F)!\"\"" }{TEXT 251 
52 " .  Find and graph the first two Taylor polynomials " }{XPPEDIT 18
 0 "P[1](x);" "6#-&%\"PG6#\"\"\"6#%\"xG" }{TEXT 251 3 " , " }{XPPEDIT 
18 0 "P[2](x);" "6#-&%\"PG6#\"\"#6#%\"xG" }{TEXT 251 4 " of " }
{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 4 " at " }{XPPEDIT 
18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT 251 46 " .  How well do they appr
oximate the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT
 251 10 " at x=0.3?" }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 251 26 "Let's define the function " }{XPPEDIT 18 0 "f(
x);" "6#-%\"fG6#%\"xG" }{TEXT 251 1 "." }}{PARA 0 "" 0 "" {TEXT 251 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=x->1/(1+2*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(*&\"\"\"F-,&F-F-*&\"\"#F-F'F-F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 82 "To find
 the first and the second Taylor polynomials, we apply first the comma
nd \"" }{TEXT 204 6 "taylor" }{TEXT 251 47 "\". For convenience, we sh
all label outputs by " }{TEXT 205 2 "T1" }{TEXT 251 2 ", " }{TEXT 206 
2 "T2" }{TEXT 251 1 "." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "T1:=taylor(f(x),x=0,2); T2:=taylor(
f(x),x=0,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T1G+)%\"xG\"\"\"\"
\"!!\"#F'-%\"OG6#F'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T2G++%
\"xG\"\"\"\"\"!!\"#F'\"\"%\"\"#-%\"OG6#F'\"\"$" }}}{EXCHG {PARA 0 "> "
 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 39 "The fi
rst argument under the command \"" }{TEXT 207 6 "taylor" }{TEXT 251 
40 "\" is your function, the second a point " }{XPPEDIT 18 0 "x = a;" 
"6#/%\"xG%\"aG" }{TEXT 251 79 " at which you wish to expand the functi
on into its Taylor series, in this case " }{XPPEDIT 18 0 "x = 0;" "6#/
%\"xG\"\"!" }{TEXT 251 176 ", the third is responsible for the order o
f  the polynomial that you will see. Note that the polynomial that you
 obtain is one degree less than the third argument. The command " }
{TEXT 208 8 "\"taylor" }{TEXT 251 11 "\" returns " }{TEXT 209 16 "not \+
a polynomial" }{TEXT 251 16 " but the Taylor " }{TEXT 210 6 "series" }
{TEXT 251 5 "  at " }{XPPEDIT 18 0 "x = a;" "6#/%\"xG%\"aG" }{TEXT 
251 102 ", in which a  few first terms and a remainder are displayed. \+
The remainder is denoted, in general, by " }{XPPEDIT 18 0 "O((x-a)^n);
" "6#-%\"OG6#),&%\"xG\"\"\"%\"aG!\"\"%\"nG" }{TEXT 251 3 " . " }
{XPPEDIT 18 0 "O((x-a)^n);" "6#-%\"OG6#),&%\"xG\"\"\"%\"aG!\"\"%\"nG" 
}{TEXT 251 68 ", in this context, stands for a sum of terms of order a
t least n in " }{XPPEDIT 18 0 "x-a;" "6#,&%\"xG\"\"\"%\"aG!\"\"" }
{TEXT 251 28 ", that is, terms containing " }{XPPEDIT 18 0 "x-a;" "6#,
&%\"xG\"\"\"%\"aG!\"\"" }{TEXT 251 61 " in the power of at least n. It
 is important to realize that " }{TEXT 211 2 "T1" }{TEXT 251 2 ", " }
{TEXT 212 2 "T2" }{TEXT 251 65 " are series and not polynomials, as ma
ny commands, for example \"" }{TEXT 213 4 "plot" }{TEXT 251 47 "\", wi
ll not work with series. You can convert " }{TEXT 214 2 "T1" }{TEXT 
251 2 ", " }{TEXT 215 2 "T2" }{TEXT 251 60 " into corresponding polyno
mials using the following command:" }}{PARA 0 "" 0 "" {TEXT 251 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "convert(T1,polynom); conve
rt(T2,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&\"\"#F$
%\"xGF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&\"\"#F$%\"
xGF$!\"\"*&\"\"%F$)F'F&F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 66 "If you want to avoid anno
ying labeling or the use of the symbol \"" }{TEXT 216 1 "%" }{TEXT 
251 20 "\", you can apply \"" }{TEXT 217 6 "taylor" }{TEXT 251 9 "\" a
nd \"" }{TEXT 218 7 "convert" }{TEXT 251 78 "\" commands together. Let
's do it, and label the corresponding polynomials by " }{TEXT 219 2 "P
1" }{TEXT 251 2 ", " }{TEXT 220 2 "P2" }{TEXT 251 38 ".  Note the prop
er use of parentheses." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "P1:=convert(taylor(f(x),x=0,2),poly
nom); P2:=convert(taylor(f(x),x=0,3),polynom);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#P1G,&\"\"\"F&*&\"\"#F&%\"xGF&!\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#P2G,(\"\"\"F&*&\"\"#F&%\"xGF&!\"\"*&\"\"%F&)F)F(F
&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 251 18 "Now we shall plot " }{XPPEDIT 18 0 "f(x);" "6#-%
\"fG6#%\"xG" }{TEXT 251 240 " and the two polynomials. Usually when pl
otting functions we have to worry about the x-range only. Maple adjust
s the y-range automatically, so the whole graph is visible. It doesn't
 work very well if a function has a vertical asymptote, as " }
{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 8 " has at " }
{XPPEDIT 18 0 "x = -1/2;" "6#/%\"xG,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT 
251 111 ". In such cases it may be a good idea to specify the range fo
r y, as well.  The proper syntax looks as follows." }}{PARA 0 "" 0 "" 
{TEXT 251 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot([f(x
),P1,P2],x=-1..1,y=-5..10,color=[black,red, blue],thickness=[2,1,1]);"
 }}{PARA 13 "" 1 "" {TEXT 253 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 145 "We see the \+
thicker, black graph of the original function and the graphs of the fi
rst two polynomials. Maple displays also the vertical asymptote " }
{XPPEDIT 18 0 "x = -1/2;" "6#/%\"xG,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT 
251 86 ".  If you find it distracting, you can prevent the asymptote f
rom showing by adding \"" }{TEXT 221 12 "discont=true" }{TEXT 251 28 "
\" under the plot command.  " }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}
{PARA 0 "" 0 "" {TEXT 251 27 "Observe that the command \"" }{TEXT 222 
29 "convert(taylor(....),polynom)" }{TEXT 251 191 "\" returns the corr
esponding Taylor polynomial as an expression in terms of x and not as \+
a function of x. Hence, if you want to evaluate polynomials at a given
 x, you have to use the command " }{TEXT 223 6 "\"subs" }{TEXT 251 99 
"\" .  For example,  let's compare the values of the two polynomials a
nd the function f at x = .0.3." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f(.3); subs(x=.3,P1); subs(x
=.3,P2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++]i!#5" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"#w!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 78 "As we see, neither p
olynomial gives a very good approximation of the function " }{XPPEDIT 
18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 13 " at x = 0.3. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" 
{TEXT 252 24 "Intervals of Convergence" }}{PARA 0 "" 0 "" {TEXT 251 0 
"" }}{PARA 0 "" 0 "" {TEXT 251 169 "You can use Maple to graph Taylor \+
polynomials of higher order together with the original function. Such \+
graphs can illuminate the idea of the interval of convergence.   " }}
{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 224 12 
"Example 2.  " }{TEXT 251 11 "Let again  " }{XPPEDIT 18 0 "f(x) = 1/(1
+2*x);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)*&\"\"#F)F'F)F)!\"\"" }{TEXT 
251 51 "  .  Find a few higher-order Taylor polynomials of " }
{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 4 " at " }{XPPEDIT 
18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 251 28 ". Graph the polynomials \+
and " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 91 " in one \+
coordinate system. Can you guess the radius of convergence of the Tayl
or series of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 4 "
 at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 251 141 "?  Exam
ine your guess graphically and numerically. Confirm your guess analyti
cally by finding the exact value of the radius of convergence.  " }}
{PARA 0 "" 0 "" {TEXT 251 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 
63 "To save ourselves a lot of typing we shall use lists and the \"" }
{TEXT 225 3 "seq" }{TEXT 251 87 "\" command. We shall generate and the
n plot a list consisting of the original function " }{XPPEDIT 18 0 "f(
x);" "6#-%\"fG6#%\"xG" }{TEXT 251 154 " and consecutive polynomials of
 orders, say, 4 to 10. It will save us the effort of writing a separat
e command for each polynomial. Let's call our list \"" }{TEXT 226 1 "L
" }{TEXT 251 3 "\"." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 59 "L:=[f(x),seq(convert(taylor(f(x),x=1,i),p
olynom),i=5..11)];" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>%\"LG7**&\"\"
\"F',&F'F'*&\"\"#F'%\"xGF'F'!\"\",,#\"\"&\"\"*F'*&#F*F0F'F+F'F,*&#\"\"
%\"#FF'),&F+F'F'F,F*F'F'*&#\"\")\"#\")F')F8\"\"$F'F,*&#\"#;\"$V#F')F8F
5F'F',.F.F'F1F,F3F'F9F,F?F'*&#\"#K\"$H(F')F8F/F'F,,0F.F'F1F,F3F'F9F,F?
F'FEF,*&#\"#k\"%(=#F')F8\"\"'F'F',2F.F'F1F,F3F'F9F,F?F'FEF,FKF'*&#\"$G
\"\"%hlF')F8\"\"(F'F,,4F.F'F1F,F3F'F9F,F?F'FEF,FKF'FRF,*&#\"$c#\"&$o>F
')F8F;F'F',6F.F'F1F,F3F'F9F,F?F'FEF,FKF'FRF,FYF'*&#\"$7&\"&\\!fF')F8F0
F'F,,8F.F'F1F,F3F'F9F,F?F'FEF,FKF'FRF,FYF'FinF,*&#\"%C5\"'Zr<F')F8\"#5
F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 251 52 "As we see, Maple simplified the two initial term
s,  " }{XPPEDIT 200 0 "1/3-2/(9)(x-1);" "6#,&*&\"\"\"F%\"\"$!\"\"F%*&
\"\"#F%-\"\"*6#,&%\"xGF%F%F'F'F'" }{TEXT 200 32 ", in each Taylor poly
nomial to  " }{XPPEDIT 200 0 "5/9-2*x/9;" "6#,&*&\"\"&\"\"\"\"\"*!\"\"
F&*(\"\"#F&F'F(%\"xGF&F(" }{TEXT 251 10 ".   The \"" }{TEXT 227 4 "plo
t" }{TEXT 251 78 "\" command can be applied directly to a list of expr
essions. (All elements of " }{TEXT 228 1 "L" }{TEXT 251 89 " are expre
ssions. Technically, f is a function, but f(x) is an expression in ter
ms of x)." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 112 "plot(L,x=-1.5..3.5,y=-5..10,color=[black,blue,cyan
,green,yellow,brown,red,magenta],thickness=[2,1,1,1,1,1,1,1]);" }}
{PARA 13 "" 1 "" {TEXT 253 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 82 "Our best guess based \+
on the picture is that the radius of convergence R is about  " }
{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 251 23 " .  In
deed, for x > 1+ " }{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }
{TEXT 251 48 " , the polynomials seem to sharply diverge from " }
{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 100 ". As a matter \+
of fact, the higher order of the polynomial, the faster it diverges fr
om the graph of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 
13 " for x > 1 + " }{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }
{TEXT 251 191 " .  Look at the colors of the graphs: magenta and red c
orrespond to two highest-degree polynomials on the list. The situation
 with divergence is even more clear on the other side, for x < 1- " }
{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 251 57 " . The
 polynomials are all positive, while the values of " }{XPPEDIT 18 0 "f
(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 79 " are negative. We are guessing \+
then that the radius of convergence is at most  " }{XPPEDIT 18 0 "3/2;
" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 251 26 " . Within the interval (1
-" }{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 251 5 " , \+
1+" }{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 251 58 ")
 the polynomials seem to be getting closer and closer to " }{XPPEDIT 
18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 58 ". Hence, the radius of c
onvergence seems to be, indeed,   " }{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"
\"\"\"\"#!\"\"" }{TEXT 251 100 " .  Be warned, however, that the empha
sis in the reasoning presented above should be  on the word \"" }{TEXT
 229 5 "guess" }{TEXT 251 193 "\". As you know, some Taylor series con
verge very fast, some very slowly. You can never be sure that the poly
nomials that you are graphing are representative of the behavior of a \+
given series. " }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" 
{TEXT 251 48 "Let's try to confirm our guess that R  is about " }
{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 251 76 "  by g
raphing a couple of really long polynomials, say, of orders 20 and 25.
" }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 153 "plot([f(x),convert(taylor(f(x),x=1,21),polynom),conv
ert(taylor(f(x),x=1,26),polynom)],x=-1.5..3.5,y=-5..10,color=[black,bl
ue,magenta],thickness=[2,1,1]);" }}{PARA 13 "" 1 "" {TEXT 253 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 251 123 "The graph seems to confirm our conjecture. You can, of \+
course, experiment on your own by changing the values or ranges for " 
}{TEXT 230 1 "i" }{TEXT 251 156 " in the commands above and graphing l
onger and longer polynomials. Be aware, however, that calculating and \+
graphing very long polynomials may take a while. " }{TEXT 231 82 "If y
ou start calculations that take longer than your patience allows, clic
k on the" }{TEXT 251 3 " \"" }{TEXT 232 4 "STOP" }{TEXT 251 4 "\"  " }
{TEXT 233 84 "button on the tool bar. It will stop calculations, altho
ugh usually not right away. " }}{PARA 0 "" 0 "" {TEXT 251 114 "We can \+
try to examine our guess as to the value of R numerically, by looking \+
at values of Taylor polynomials near " }{XPPEDIT 18 0 "1-3/2;" "6#,&\"
\"\"F$*&\"\"$F$\"\"#!\"\"F(" }{TEXT 251 5 " and " }{XPPEDIT 18 0 "1+3/
2;" "6#,&\"\"\"F$*&\"\"$F$\"\"#!\"\"F$" }{TEXT 251 37 "  versus the co
rresponding values of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }
{TEXT 251 45 ". Let's create a list of numerical values of " }
{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 59 " and polynomial
s of orders, say, 50 through 60 at  x = 2.3." }}{PARA 0 "" 0 "" {TEXT 
251 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "[evalf(f(2.3)),e
valf(seq(subs(x=2.3,convert(taylor(f(x),x=1,i),polynom)),i=51..61))];"
 }}{PARA 207 "" 1 "" {XPPMATH 20 "6#7.$\"+'G9dy\"!#5$\"+^G#py\"F&$\"+q
om%y\"F&$\"+a?i'y\"F&$\"+hv#\\y\"F&$\"+AhR'y\"F&$\"+pL7&y\"F&$\"+@kA'y
\"F&$\"+V/F&y\"F&$\"+e*)4'y\"F&$\"+74Q&y\"F&$\"+>K+'y\"F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
251 54 "Remember that the first number on the list represents " }
{XPPEDIT 18 0 "f(Float(23, -1));" "6#-%\"fG6#-%&FloatG6$\"#B!\"\"" }
{TEXT 251 79 ", the following numbers represent the values of consecut
ive Taylor polynomials " }{XPPEDIT 18 0 "P[50](Float(23, -1));" "6#-&%
\"PG6#\"#]6#-%&FloatG6$\"#B!\"\"" }{TEXT 251 9 " through " }{XPPEDIT 
18 0 "P[60](Float(23, -1));" "6#-&%\"PG6#\"#g6#-%&FloatG6$\"#B!\"\"" }
{TEXT 251 56 " . The values of the polynomials do seem to converge to \+
" }{XPPEDIT 18 0 "f(Float(23, -1));" "6#-%\"fG6#-%&FloatG6$\"#B!\"\"" 
}{TEXT 251 127 ". Hence, x = 2.3 is within the interval of convergence
, which  suggests  that R is at least 2.3-1= 1.3. Let's check the valu
es " }{XPPEDIT 18 0 "f(Float(27, -1));" "6#-%\"fG6#-%&FloatG6$\"#F!\"
\"" }{TEXT 251 9 "  versus " }{XPPEDIT 18 0 "P[50](Float(27, -1));" "6
#-&%\"PG6#\"#]6#-%&FloatG6$\"#F!\"\"" }{TEXT 251 9 " through " }
{XPPEDIT 18 0 "P[60](Float(27, -1));" "6#-&%\"PG6#\"#g6#-%&FloatG6$\"#
F!\"\"" }{TEXT 251 3 ".  " }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "[evalf(f(2.7)),evalf(seq(subs(x=2.7
,convert(taylor(f(x),x=1,i),polynom)),i=51..61))];" }}{PARA 208 "" 1 "
" {XPPMATH 20 "6#7.$\"+++]i:!#5$\"+$[,RE*!\")$!++bdY5!\"($\"+NAX*=\"F,
$!+JDrW8F,$\"+r3MF:F,$!+&)HlF<F,$\"+_+Mh>F,$!+B(=&>AF,$\"+))yy=DF,$!+Q
HH^GF,$\"+k')zMKF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 251 446 "The values  oscillate wildly, wh
ich suggests  that x = 2.7 is outside the interval of convergence, or \+
equivlantly, that R is less or equal to 2.7-1=1.7.   By modifying the \+
commands above, you can explore numerically convergence and divergence
 at points near the other end of the interval of convergence, that is,
 near x = -0.5. Note that when your test points get closer to the endp
oints -0.5 or 2.5 convergence or divergence became much slower. " }}
{PARA 0 "" 0 "" {TEXT 251 102 "In our example, both numerical and grap
hical experiments seem to confirm our guess that R is equal to " }
{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 251 3 " . " }}
{PARA 209 "" 0 "" {TEXT 200 345 "Warning: Graphical and numerical expe
riments may lead to good guesses as to the value of  the radius of con
vergence, but they can also be very misleading. The appearance of grap
hs depends heavily on the range for x and y, which can be very confusi
ng. Convergence or divergence may be so slow that your numerical exper
iments will not pick it up. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 81 "The only way to be sure
 that the radius of convergence in our example is indeed  " }{XPPEDIT 
18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 251 98 "  is to prove i
t analytically. Fortunately, it is quite easy. We can obtain the Taylo
r series for " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 4 "
 at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 251 24 " from th
e expansion of  " }{XPPEDIT 18 0 "1/(1-x);" "6#*&\"\"\"F$,&F$F$%\"xG!
\"\"F'" }{TEXT 251 5 "  at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }
{TEXT 251 32 ", which is the geometric series:" }}{PARA 210 "" 0 "" 
{XPPEDIT 18 0 "1/(1-x) = sum(x^n, n = 0 .. infinity);" "6#/*&\"\"\"F%,
&F%F%%\"xG!\"\"F(-%$sumG6$)F'%\"nG/F-;\"\"!%)infinityG" }{TEXT 200 2 "
 ." }}{PARA 0 "" 0 "" {TEXT 251 37 "To get a series involving powers o
f  " }{XPPEDIT 18 0 "x-1;" "6#,&%\"xG\"\"\"F%!\"\"" }{TEXT 251 43 " we
 so some algebraic juggling to  rewrite " }{XPPEDIT 18 0 "f(x);" "6#-%
\"fG6#%\"xG" }{TEXT 251 12 " as follows:" }}{PARA 211 "" 0 "" 
{XPPEDIT 18 0 "1/(1+2*x) = 1/(3+2*(x-1));" "6#/*&\"\"\"F%,&F%F%*&\"\"#
F%%\"xGF%F%!\"\"*&F%F%,&\"\"$F%*&F(F%,&F)F%F%F*F%F%F*" }{TEXT 200 3 " \+
= " }{XPPEDIT 18 0 "1/(3*(1+2/3*(x-1)));" "6#*&\"\"\"F$*&\"\"$F$,&F$F$
*&,$*&\"\"#F$F&!\"\"F$F$,&%\"xGF$F$F,F$F$F$F," }{TEXT 200 3 " = " }
{XPPEDIT 18 0 "(sum((-2/3*(x-1))^n, n = 0 .. infinity))/3;" "6#*(-%$su
mG6$),$*&,$*&\"\"#\"\"\"\"\"$!\"\"F-F-,&%\"xGF-F-F/F-F/%\"nG/F2;\"\"!%
)infinityGF-F-F-F.F/" }{TEXT 200 3 " = " }{XPPEDIT 18 0 "sum((-1)^n*2^
n*(x-1)^n/3^(n+1), n = 0 .. infinity);" "6#-%$sumG6$**),$\"\"\"!\"\"%
\"nGF))\"\"#F+F))\"\"$,&F+F)F)F)F*),&%\"xGF)F)F*F+F)/F+;\"\"!%)infinit
yG" }{TEXT 200 1 "." }}{PARA 0 "" 0 "" {TEXT 251 90 "From the uniquene
ss of power series expansion, the latter series is the Taylor series o
f  " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 4 " at " }
{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 251 34 ". Indeed, it is
 a power series at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 
251 21 " and it converges to " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG
" }{TEXT 251 22 " in a neighborhood of " }{XPPEDIT 18 0 "x = 1;" "6#/%
\"xG\"\"\"" }{TEXT 251 41 ". Hence, it must be the Taylor series of " 
}{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 4 " at " }
{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 251 75 ". On the other \+
hand, for any given x, the series is equal to the constant  " }
{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 251 40 "  times t
he geometric series with ratio " }{XPPEDIT 18 0 "r = -2/3*(x-1);" "6#/
%\"rG,$*&,$*&\"\"#\"\"\"\"\"$!\"\"F*F*,&%\"xGF*F*F,F*F," }{TEXT 251 
45 ". Hence, the series converges if and only if " }{XPPEDIT 18 0 "abs
(r) < 1;" "6#2-%$absG6#%\"rG\"\"\"" }{TEXT 251 15 ", that is when " }
{XPPEDIT 18 0 "abs(x-1) < 3/2;" "6#2-%$absG6#,&%\"xG\"\"\"F)!\"\"*&\"
\"$F)\"\"#F*" }{TEXT 251 47 " .  Thus, the radius of convergence is in
deed  " }{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 251 
3 " . " }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" {TEXT 251 
50 "Yet another way to see this is to use the formula " }{XPPEDIT 18 0
 "R = limit(abs(C[n])/abs(C[n+1]), n = infinity);" "6#/%\"RG-%&limitG6
$*&-%$absG6#&%\"CG6#%\"nG\"\"\"-F*6#&F-6#,&F/F0F0F0!\"\"/F/%)infinityG
" }{TEXT 251 7 "  with " }{XPPEDIT 18 0 "abs(C[n]) = 2^n/3^(n+1);" "6#
/-%$absG6#&%\"CG6#%\"nG*&)\"\"#F*\"\"\")\"\"$,&F*F.F.F.!\"\"" }{TEXT 
251 18 ", as in the text. " }}{PARA 0 "" 0 "" {TEXT 251 35 "Similarly,
 using the expansion of  " }{XPPEDIT 18 0 "1/(1-x);" "6#*&\"\"\"F$,&F$
F$%\"xG!\"\"F'" }{TEXT 251 46 "  , we can easily obtain the Taylor ser
ies of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 4 " at " 
}{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT 251 1 ":" }}{PARA 212 "
" 0 "" {XPPEDIT 18 0 "1/(1+2*x) = sum((-2)^n*x^n, n = 0 .. infinity);"
 "6#/*&\"\"\"F%,&F%F%*&\"\"#F%%\"xGF%F%!\"\"-%$sumG6$*&),$F(F*%\"nGF%)
F)F1F%/F1;\"\"!%)infinityG" }{TEXT 200 2 " ," }}{PARA 0 "" 0 "" {TEXT 
251 32 " whose radius of convergence is " }{XPPEDIT 18 0 "R = 1/2;" "6
#/%\"RG*&\"\"\"F&\"\"#!\"\"" }{TEXT 251 65 " .  A conjecture that come
s to mind is that the Taylor series at " }{XPPEDIT 18 0 "x = a;" "6#/%
\"xG%\"aG" }{TEXT 251 62 " of any function is convergent in a symmetri
c interval around " }{XPPEDIT 18 0 "x = a;" "6#/%\"xG%\"aG" }{TEXT 
251 612 " which can be extended until we hit the first \"singularity\"
 of the function. A \"singularity\" is a point at which the behavior o
f the function is \"irregular\", like, for example, a point of a verti
cal asymptote. Is this conjecture true? Yes, but only provided we look
 at singularities in the complex plane as well. If we consider only si
ngularities located on the real line, the conjecture is false. It is d
emonstrated by the example in Problem 1 of your homework. The function
 that you have there has no singularities on the real line, yet the ra
dius of convergence of its Taylor series at  x = 0 is finite." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" 
{TEXT 252 59 "Estimating the Error of Approximation by Taylor Polynomi
als" }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" {TEXT 251 4 "L
et " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT 251 22 " be a giv
en function, " }{XPPEDIT 18 0 "x = a;" "6#/%\"xG%\"aG" }{TEXT 251 16 "
 a given point, " }{XPPEDIT 18 0 "P[n](x);" "6#-&%\"PG6#%\"nG6#%\"xG" 
}{TEXT 251 31 " the n-th Taylor polynomial of " }{XPPEDIT 18 0 "h(x);"
 "6#-%\"hG6#%\"xG" }{TEXT 251 4 " at " }{XPPEDIT 18 0 "x = a;" "6#/%\"
xG%\"aG" }{TEXT 251 8 " .  Let " }{XPPEDIT 18 0 "E[n](x) = h(x)-P[n](x
);" "6#/-&%\"EG6#%\"nG6#%\"xG,&-%\"hGF)\"\"\"-&%\"PGF'F)!\"\"" }{TEXT 
251 78 " be the n-th error. We know  an error bound for Taylor approxi
mations. Namely:" }}{PARA 213 "" 0 "" {XPPEDIT 18 0 "abs(E[n](x)) <= M
*abs(x-a)^(n+1)/factorial(n+1);" "6#1-%$absG6#-&%\"EG6#%\"nG6#%\"xG*(%
\"MG\"\"\")-F%6#,&F-F0%\"aG!\"\",&F+F0F0F0F0-%*factorialG6#F7F6" }
{TEXT 200 42 "    for all x in [ a-d, a+d ] ,       (EB)" }}{PARA 0 ""
 0 "" {TEXT 251 6 "where " }{XPPEDIT 18 0 "M;" "6#%\"MG" }{TEXT 251 
24 " is such a constant that" }}{PARA 214 "" 0 "" {XPPEDIT 18 0 "abs(`
@@`(D, n+1)*h(x)) <= M;" "6#1-%$absG6#*&-%#@@G6$%\"DG,&%\"nG\"\"\"F.F.
F.-%\"hG6#%\"xGF.%\"MG" }{TEXT 200 35 "    for  all  x  in  [ a-d, a+d
 ] ." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0
 "D^(n+1);" "6#)%\"DG,&%\"nG\"\"\"F'F'" }{TEXT 251 601 "  denotes the \+
(n+1)-th derivative. We denote the error bound by (EB) for future refe
rence. This bound is of great importance for proving convergence of Ta
ylor series, for obtaining estimates of the error for arbitrarily long
 Taylor polynomials, and for many other tasks. You have to remember th
at the formula (EB) gives an upper bound for an error and not its exac
t value. The actual error is often much smaller than the estimate (EB)
. If we work with specific functions and specific polynomials, we don'
t always have to use (EB). We can simply graph the error and obtain th
e estimate from the graph. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 234 12 "Example 3.  " }{TEXT 251 
22 "Consider the function " }{XPPEDIT 18 0 "h(x) = exp(x)*sin(x);" "6#
/-%\"hG6#%\"xG*&-%$expGF&\"\"\"-%$sinGF&F+" }{TEXT 251 49 " . Find the
 smallest-degree Taylor polynomial at " }{XPPEDIT 18 0 "x = 0;" "6#/%
\"xG\"\"!" }{TEXT 251 39 " that stays within 0.1 of the function " }
{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT 251 179 " for all x in \+
the interval [-2,2]. ( Observe that in this problem the interval is fi
xed and we are looking for a polynomial that gives the desired approxi
mation in this interval.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 50 "We shall solve the problem \+
by graphing the errors " }{XPPEDIT 18 0 "E[n](x);" "6#-&%\"EG6#%\"nG6#
%\"xG" }{TEXT 251 153 " corresponding to a few first polynomials. Let'
s generate a list of errors corresponding to consecutive polynomials a
nd plot them in the interval [-2,2]." }}{PARA 0 "" 0 "" {TEXT 251 0 ""
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "h:=x->exp(x)*sin(x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(*&-%$expGF&\"\"\"-%$sinGF&F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "E:=[seq
(h(x)-convert(taylor(h(x),x=0,i),polynom),i=2..11)];" }}{PARA 215 "" 1
 "" {XPPMATH 20 "6#>%\"EG7,,&*&-%$expG6#%\"xG\"\"\"-%$sinGF*F,F,F+!\"
\",(F'F,F+F/*$)F+\"\"#F,F/,*F'F,F+F/F1F/*&#F,\"\"$F,)F+F7F,F/F4,,F'F,F
+F/F1F/F5F/*&#F,\"#IF,)F+\"\"&F,F,,.F'F,F+F/F1F/F5F/F:F,*&#F,\"#!*F,)F
+\"\"'F,F,,0F'F,F+F/F1F/F5F/F:F,F@F,*&#F,\"$I'F,)F+\"\"(F,F,FE,2F'F,F+
F/F1F/F5F/F:F,F@F,FFF,*&#F,\"&!oAF,)F+\"\"*F,F/,4F'F,F+F/F1F/F5F/F:F,F
@F,FFF,FLF/*&#F,\"'+M6F,)F+\"#5F,F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 324 "We shall pl
ot a few error functions at a time, so we don't get confused which is \+
which. We shall also plot the horizontal lines y= -0.1, y = 0.1 to see
 which errors stay within the bounds. Remember that E[k] denotes the k
-th element of the list E. We observe that E[3]=E[4] and E[7]=E[8], so
 we shall not plot E[4] and E[8]." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plot([-.1,.1,E[1],E[2],E[3]
,E[5]],x=-2..2,color=[magenta,magenta,red,blue,green,yellow]);" }}
{PARA 13 "" 1 "" {TEXT 253 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 118 "Clearly, none of the
 errors plotted above stays within bounds. Hence, the first five polyn
omials are not good enough. " }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "plot([-.1,.1,E[6],E[7],E[9],
E[10]],x=-2..2,color=[magenta,magenta,red,blue,green,yellow]);" }}
{PARA 13 "" 1 "" {TEXT 253 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 128 "The first error func
tion which stays within bounds is E[7]. Hence, the seventh Taylor poly
nomial is the first that approximates " }{XPPEDIT 18 0 "h(x);" "6#-%\"
hG6#%\"xG" }{TEXT 251 22 " in [-2,2] within 0.1." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 235 12 "Exam
ple 4.  " }{TEXT 251 4 "Let " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG"
 }{TEXT 251 51 " be as above. Find the thirties Taylor polynomial  " }
{XPPEDIT 18 0 "P[30](x);" "6#-&%\"PG6#\"#I6#%\"xG" }{TEXT 251 4 " of "
 }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT 251 68 " at x =0. Fin
d the longest interval around zero, [-d,d],  such that " }{XPPEDIT 18 
0 "P[30](x);" "6#-&%\"PG6#\"#I6#%\"xG" }{TEXT 251 22 " stays within 0.
01 of " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT 251 164 " for \+
all x in [-d,d]. (Observe that in this problem the Taylor polynomial i
s fixed and we are looking for an interval in which the desired approx
imation is valid.) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 251 26 "Let's find the polynomial " }
{XPPEDIT 18 0 "P[30](x);" "6#-&%\"PG6#\"#I6#%\"xG" }{TEXT 251 18 " and
 label it P30:" }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 42 "P30:=convert(taylor(h(x),x=0,31),polynom);" }}
{PARA 216 "" 1 "" {XPPMATH 20 "6#>%$P30G,P%\"xG\"\"\"*$)F&\"\"#F'F'*&#
F'\"\"$F')F&F-F'F'*&#F'\"#IF')F&\"\"&F'!\"\"*&#F'\"#!*F')F&\"\"'F'F4*&
#F'\"$I'F')F&\"\"(F'F4*&#F'\"&!oAF')F&\"\"*F'F'*&#F'\"'+M6F')F&\"#5F'F
'*&#F'\"(+uC\"F')F&\"#6F'F'*&#F'\")+sH(*F')F&\"#8F'F4*&#F'\"*+/3\"oF')
F&\"#9F'F4*&#F'\",+g?;-\"F')F&\"#:F'F4*&#F'\".+g,/%*Q\"F')F&\"#<F'F'*&
#F'\"/+S9OY]7F')F&\"#=F'F'*&#F'\"0+gt'3)eP#F')F&\"#>F'F'*&#F'\"2++c9#)
\\$*)\\F')F&\"#@F'F4*&#F'\"3++;g.[G)[&F')F&\"#AF'F4*&#F'\"5++o$G[]0BE
\"F')F&\"#BF'F4*&#F'\"7+++/^[9l\"py$F')F&\"#DF'F'*&#F'\"8+++_jI)o9*H#
\\F')F&\"#EF'F'*&#F'\":+++/:F%e'p2#H8F')F&\"#FF'F'*&#F'\"<+++C1B9K\"[K
e'R&F')F&\"#HF'F4*&#F'\"=+++g$fM@)>s[([4)F')F&F1F'F4" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 53 "
Lets define the error of the 30-th degree polynomial:" }}{PARA 0 "" 0 
"" {TEXT 251 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "E30:=h(
x)-P30;" }}{PARA 217 "" 1 "" {XPPMATH 20 "6#>%$E30G,R*&-%$expG6#%\"xG
\"\"\"-%$sinGF)F+F+F*!\"\"*$)F*\"\"#F+F.*&#F+\"\"$F+)F*F4F+F.*&#F+\"#I
F+)F*\"\"&F+F+*&#F+\"#!*F+)F*\"\"'F+F+*&#F+\"$I'F+)F*\"\"(F+F+*&#F+\"&
!oAF+)F*\"\"*F+F.*&#F+\"'+M6F+)F*\"#5F+F.*&#F+\"(+uC\"F+)F*\"#6F+F.*&#
F+\")+sH(*F+)F*\"#8F+F+*&#F+\"*+/3\"oF+)F*\"#9F+F+*&#F+\",+g?;-\"F+)F*
\"#:F+F+*&#F+\".+g,/%*Q\"F+)F*\"#<F+F.*&#F+\"/+S9OY]7F+)F*\"#=F+F.*&#F
+\"0+gt'3)eP#F+)F*\"#>F+F.*&#F+\"2++c9#)\\$*)\\F+)F*\"#@F+F+*&#F+\"3++
;g.[G)[&F+)F*\"#AF+F+*&#F+\"5++o$G[]0BE\"F+)F*\"#BF+F+*&#F+\"7+++/^[9l
\"py$F+)F*\"#DF+F.*&#F+\"8+++_jI)o9*H#\\F+)F*\"#EF+F.*&#F+\":+++/:F%e'
p2#H8F+)F*\"#FF+F.*&#F+\"<+++C1B9K\"[Ke'R&F+)F*\"#HF+F+*&#F+\"=+++g$fM
@)>s[([4)F+)F*F8F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 251 116 "Let's graph the error E30 toget
her with the lines y = -0.01, y= 0.01 to see in what interval it stays
 within bounds." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 58 "plot([.01,-.01,E30],x=-8..8,color=[magenta,ma
genta,blue]);" }}{PARA 13 "" 1 "" {TEXT 253 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 215 "For \+
your information, it took a few trials and errors as to what the range
 for x should be to obtain the above graph. To pinpoint the interval f
or which the error stays within bounds, we solve a couple of equations
." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "a:=fsolve(E30=0.01,x,-8..-6); b:=fsolve(E30=-0.01,x,6
..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$!+\\bzrw!\"*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+*3d/q(!\"*" }}}{EXCHG {PARA 0 "> "
 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 273 "(Be a
ware that it takes a little while for Maple to execute the seond comma
nd and find \"b\". Don't panic.)  The error stays within bounds in the
 interval [a,b]. If we want a symmetric interval around zero in which \+
the error stays within bounds, it will have to be [-a,a]. " }}{PARA 0 
"" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 236 11 "Example
 5. " }{TEXT 251 55 " Find a polynomial whose value at x = 0.7 approxi
mates " }{XPPEDIT 18 0 "sin(Float(7, -1));" "6#-%$sinG6#-%&FloatG6$\"
\"(!\"\"" }{TEXT 251 70 " to thirty decimal places.   Check the accura
cy of your approximation " }{TEXT 237 8 "without " }{TEXT 251 33 "usin
g Maple's value of sin(.7)!  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 460 "Of course, we are goin
g to use Taylor series of sin(x) at x = 0 and look for a Taylor polyno
mial that gives a desired approximation. Since we are not asked to fin
d the polynomial of smallest degree, we shall use the estimate (EB), r
ather than check consecutive polynomials one-by-one. Since any derivat
ive of sin(x) is either plus or minus sin(x) or cos(x), the absolute v
alue of  any derivative is bounded by 1 in any interval. Hence, the bo
und for the error " }{XPPEDIT 18 0 "E[n](x);" "6#-&%\"EG6#%\"nG6#%\"xG
" }{TEXT 251 24 " for any x  is given by:" }}{PARA 218 "" 0 "" 
{XPPEDIT 18 0 "abs(E[n](x)) <= abs(x)^(n+1)/factorial(n+1);" "6#1-%$ab
sG6#-&%\"EG6#%\"nG6#%\"xG*&)-F%F,,&F+\"\"\"F2F2F2-%*factorialG6#F1!\"
\"" }{TEXT 200 3 "  ." }}{PARA 0 "" 0 "" {TEXT 251 104 "Hence, to find
 a polynomial that approximates sin(0.7) as well as we want, we have t
o find n such that  " }{XPPEDIT 18 0 "abs(E[n](Float(7, -1))) < 10^(-3
0);" "6#2-%$absG6#-&%\"EG6#%\"nG6#-%&FloatG6$\"\"(!\"\")\"#5,$\"#IF1" 
}{TEXT 251 45 ". In other words, we have to find n such that" }}{PARA 
219 "" 0 "" {XPPEDIT 18 0 "Float(7, -1)^(n+1)/factorial(n+1) < 10^(-30
);" "6#2*&)-%&FloatG6$\"\"(!\"\",&%\"nG\"\"\"F-F-F--%*factorialG6#F+F*
)\"#5,$\"#IF*" }{TEXT 200 2 " ," }}{PARA 0 "" 0 "" {TEXT 251 17 "or, e
quivalently:" }}{PARA 220 "" 0 "" {XPPEDIT 18 0 "Float(7, -1)^(n+1)/fa
ctorial(n+1)-10^(-30) < 0;" "6#2,&*&)-%&FloatG6$\"\"(!\"\",&%\"nG\"\"
\"F.F.F.-%*factorialG6#F,F+F.)\"#5,$\"#IF+F+\"\"!" }{TEXT 200 2 " ." }
}{PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" {TEXT 251 152 "Maple
 can solve many inequalities, but the one above is too tricky to be so
lved with simple commands. Let's find a suitable n by trial and error.
 Define" }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "S:=.7^(n+1)/factorial(n+1)-10^(-30);" }}{PARA 11 "" 1
 "" {XPPMATH 20 "6#>%\"SG,&*&)$\"\"(!\"\",&%\"nG\"\"\"F-F-F--%*factori
alG6#F+F*F-#F-\"@+++++++++++++++\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
251 0 "" }}{PARA 0 "" 0 "" {TEXT 251 232 "Let's find a positive intege
r n for which S is less than zero. As n increases, S decreases, we try
 n = 20, S is positive. We try n= 30, S is negative. By trial and erro
r we find that the smallest n for which S is negative is n =25. " }}
{PARA 0 "" 0 "" {TEXT 251 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "subs(n=25,S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+$)QGsw!#S" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 ""
 {TEXT 251 234 "Hence, Taylor polynomial of degree 25 approximates sin
(.7) with desired accuracy. Let's find the polynomial and compare it's
 value at x = .7 with  Maple's value for sin(.7), which presumably is \+
accurate to the number of digits shown. " }}{PARA 0 "" 0 "" {TEXT 251 
1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "P25:=convert(taylor
(sin(x),x=0,26),polynom);" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>%$P25G,
<%\"xG\"\"\"*&#F'\"\"'F')F&\"\"$F'!\"\"*&#F'\"$?\"F')F&\"\"&F'F'*&#F'
\"%S]F')F&\"\"(F'F-*&#F'\"'!)GOF')F&\"\"*F'F'*&#F'\")+o\"*RF')F&\"#6F'
F-*&#F'\"++3-FiF')F&\"#8F'F'*&#F'\".+!oVn28F')F&\"#:F'F-*&#F'\"0+g4Guo
b$F')F&\"#<F'F'*&#F'\"3+?$)3/5X;7F')F&\"#>F'F-*&#F'\"5++W4<<U44^F')F&
\"#@F'F'*&#F'\"8++kw\\))Qn,_e#F')F&\"#BF'F-*&#F'\";+++%)f)4LV+@6b\"F')
F&\"#DF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 251 254 "Since Maple's default setting shows onl
y ten decimal places when evaluating numerically, we have to tell Mapl
e that we want  expansion up to 30, or better yet 31 digits, for value
s of sin(x) and P25 at x =.7. We accomplish that by the following comm
and: " }}{PARA 0 "" 0 "" {TEXT 251 1 " " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "evalf(sin(.7),31); evalf(subs(x=.7,P25),31);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"@()R^VhsO0\"pPso<Uk!#J" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#$\"@')R^VhsO0\"pPso<Uk!#J" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 251 32 "The \+
second argument under the \"" }{TEXT 238 5 "evalf" }{TEXT 251 161 "\" \+
command tells Maple how many digits you want to see.  Keep in mind tha
t Maple uses techniques similar to Taylor polynomials to compute its v
alues for sin(.7)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 4 "" 0 "" {TEXT 252 17 "
Homework Problems" }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" 
{TEXT 251 72 "After opening your homework worksheet type in and execut
e the command \"" }{TEXT 239 8 "restart;" }{TEXT 251 76 "\" to clear a
ny variable assingment from this worksheet like f, g, P20 etc. " }}
{PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" {TEXT 240 11 "Problem
 1. " }{TEXT 251 23 " Consider the function " }{XPPEDIT 18 0 "f(x) = 3
/(4+x^2);" "6#/-%\"fG6#%\"xG*&\"\"$\"\"\",&\"\"%F**$)F'\"\"#F*F*!\"\""
 }{TEXT 251 3 "  ." }}{PARA 0 "" 0 "" {TEXT 251 67 "(a)  Analytically,
 that is, \"by hand\", find the Taylor series of " }{XPPEDIT 18 0 "f(x
);" "6#-%\"fG6#%\"xG" }{TEXT 251 76 " at x = 0. (You can do it by subs
tituting into the well-known expansion for " }{XPPEDIT 18 0 "1/(1-x);"
 "6#*&\"\"\"F$,&F$F$%\"xG!\"\"F'" }{TEXT 251 412 "   at x=0.)  Find th
e exact value of the radius of convergence of the series. (Since no od
d powers appear in the expansion, you can't use the ratio test. Use th
e properties of the geometric series that you know.)  Explain how you \+
arrived at your answer. Use the math equation mode of Maple to type in
 your series and other mathematical formulas, if you need them. ( Inst
ructions for doing this are found below.) " }}{PARA 0 "" 0 "" {TEXT 
251 0 "" }}{PARA 0 "" 0 "" {TEXT 251 84 "(b) As in Example 2, illustra
te graphically the interval of convergence by plotting " }{XPPEDIT 18 
0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 251 158 " and a few higher-order po
lynomials in one coordinate system. Find polynomials and ranges for x \+
and y that allow you to generate a really convincing picture. " }{TEXT
 241 116 "Note that the center of the series is x=0 and not x=1, as in
 Example 2. Make appropriate adjustments in the syntax! " }}{PARA 0 ""
 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" {TEXT 251 322 "(c) At a couple \+
of points near one of the endpoints of the interval of convergence, on
e point inside the interval, one point outside the interval, demonstra
te numerically convergence or divergence of the series by evaluating h
igher-order Taylor polynomials at those points and comparing with the \+
corresponding values of f." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 0
 "" 0 "" {TEXT 242 43 "Instructions for Using the Equation Mode:  " }
{TEXT 251 147 "When you are in the text mode and want to enter a math \+
equation, positon the cursor where you want the formula to appear and \+
click on the button \"" }{XPPEDIT 18 0 "Sigma;" "6#%&SigmaG" }{TEXT 
251 184 "\" on the toolbar, right next to the \"T\" button. The cursor
 changes to a question mark. Start typing your formula using the stand
ard Maple syntax. For example, if you want to enter: " }{XPPEDIT 18 0 
"7*(sum((n+1)*(x-1)^n/2^n, n = 0 .. infinity));" "6#*&\"\"(\"\"\"-%$su
mG6$*(,&%\"nGF%F%F%F%),&%\"xGF%F%!\"\"F+F%)\"\"#F+F//F+;\"\"!%)infinit
yGF%" }{TEXT 251 10 "  , type: " }{TEXT 243 40 "7*sum((n+1)/(2^n)*(x-1
)^n,n=0..infinity)" }{TEXT 251 605 " . The text that you type will app
ear in a small strip at the bottom of toolbar. After you finish typing
 your formula, press \"ENTER\", and then click on \"T\". The formula w
ill appear where you wanted it and the cursor after it. Be prepared to
 see Maple  rearrange  the formula a bit, as well as what you typed in
 the strip, to its own liking. As long as it is mathematically equival
ent to what you wanted, live with it. If you want to edit  formula you
 typed already, highlight it, click on the small strip where the synta
x is to place cursor there, edit the equation, press \"ENTER\", click \+
on \"T\".  " }}{PARA 0 "" 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" {TEXT 
251 168 "There are things that Maple will not allow you to do. For exa
mple, you cannot enter a formula of the form \"a=b=c\". You have to en
ter \"a=b\" and \"b=c\", separately. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT
 251 0 "" }}{PARA 0 "" 0 "" {TEXT 244 12 "Problem 2.  " }{TEXT 251 22 
"Consider the function " }{XPPEDIT 18 0 "g(x) = exp(x)*cos(x);" "6#/-%
\"gG6#%\"xG*&-%$expGF&\"\"\"-%$cosGF&F+" }{TEXT 251 2 ". " }}{PARA 0 "
" 0 "" {TEXT 251 0 "" }}{PARA 0 "" 0 "" {TEXT 251 131 " (a)  Find the \+
smallest-degree Taylor polynomial at x =0 which approximates the funct
ion within 0.2 throughout the interval [-1,1]." }}{PARA 0 "" 0 "" 
{TEXT 251 0 "" }}{PARA 0 "" 0 "" {TEXT 251 46 " (b)  Find the 20-th de
gree Taylor polynomial " }{XPPEDIT 18 0 "P[20](x);" "6#-&%\"PG6#\"#?6#
%\"xG" }{TEXT 251 100 " at x = 0. Find the longest interval [-d,d] abo
ut x = 0 in which the polynomial stays within 0.1 of " }{XPPEDIT 18 0 
"g(x);" "6#-%\"gG6#%\"xG" }{TEXT 251 1 "." }}{PARA 0 "" 0 "" {TEXT 
251 0 "" }}{PARA 0 "" 0 "" {TEXT 245 12 "Problem 3.  " }{TEXT 251 107 
"Find a polynomial which approximates cos (0.4) to twenty decimal plac
es. Compare the two values at x = 0.4." }}{PARA 0 "" 0 "" {TEXT 251 0 
"" }}{PARA 0 "" 0 "" {TEXT 246 11 "Problem 4. " }{TEXT 251 35 "Using t
he ratio test and Maple's \"" }{TEXT 247 5 "limit" }{TEXT 251 64 "\" c
ommand to find the radius of convergence of the power series" }}{PARA 
222 "" 0 "" {XPPEDIT 18 0 "sum((x-3)^n/(n^2*4^n), n = 1 .. infinity);"
 "6#-%$sumG6$*&),&%\"xG\"\"\"\"\"$!\"\"%\"nGF**&)F-\"\"#F*)\"\"%F-F*F,
/F-;F*%)infinityG" }{TEXT 200 2 " ." }}{PARA 0 "" 0 "" {TEXT 251 60 "H
int: The proper syntax for finding limits of sequences is: " }{TEXT 
248 6 "limit(" }{TEXT 249 26 "expression in terms of n, " }{TEXT 250 
13 "n=infinity); " }{TEXT 251 1 "." }}{PARA 0 "" 0 "" {TEXT 251 0 "" }
}}{EXCHG {PARA 223 "" 0 "" {TEXT 200 77 "MTH 142 Maple Worksheets writ
ten by B. Kaskosz and L. Pakula, Copyright 1999." }}}{EXCHG {PARA 224 
"" 0 "" {TEXT 200 26 "Last modified August 1999." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }