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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" 
{TEXT 256 0 "" }{TEXT 257 14 "Taylor Series " }}{PARA 256 "" 0 "" 
{TEXT -1 0 "" }{TEXT 260 19 "Viewing Convergence" }{TEXT -1 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Notes to Student:" }}{PARA 0 "" 
0 "" {TEXT -1 290 "This lab is designed to help solidify the concepts \+
we have discussed in class by \"showing\" you convergence and divergen
ce. It also extends the fundamentals of series concepts to new areas o
f mathematical studies. Many of the examples are animated so you are e
ncouraged to replay them often." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 301 16 "Time Allocation:" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "50 minutes" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Intro
duction:" }}{PARA 0 "" 0 "" {TEXT -1 359 "The necessecity to approxima
te certain functions as sums of other functions has wide applications \+
in engineering. This arises because sometimes we cannot perform certai
n computations easily on a given function and must resort to expressin
g that function in terms of more elementary functions. For example, su
ppose we wish to evaluate the integral shown below:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {OLE 1 4628 1 "[xm]Br=WfoRrB:::wk;
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:::::::::::::::::::5:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 127 "There is no elementary antiderivative \+
for the integrand function but we can create an infinite power series \+
representation for " }}{PARA 0 "" 0 "" {TEXT -1 98 " this integral and
 then integrate term-by-term to obtain a series solution for the antid
erivative." }}{PARA 0 "" 0 "" {TEXT 267 119 "This lab is designed to h
elp you visualize the convergence of Taylor Polynomials and a few appl
ications of their usage." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Obje
ctives Defined:" }}{PARA 0 "" 0 "" {TEXT 258 293 "The goal of this lab
 is to establish a visual and conceptual understanding of the converge
nce/divergence of Taylor Polynomials. Although there are some signific
ant notational elements scattered throughout the module, the focus is \+
on the dynamics of convergence.  We shall achieve this goal by:" }}
{PARA 0 "" 0 "" {TEXT -1 63 "1. Deriving several approximating Taylor/
Maclaurin Polynomials." }}{PARA 0 "" 0 "" {TEXT -1 90 "2. Utilizing th
e power of animation to \"see\" the convergence of approximating polyn
omials." }}{PARA 0 "" 0 "" {TEXT -1 61 "3. Discussing \"interval of co
nvergence\" and overall behavior." }}{PARA 0 "" 0 "" {TEXT -1 47 "4. W
orking through specific numerical examples." }}{PARA 0 "" 0 "" {TEXT 
-1 71 "5. Reviewing all concepts presented through Student Practice Pr
oblems. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 60 "Defining Taylor/Macl
aurin Series and their nth-Polynomials:(" }{TEXT 299 8 "Animated" }
{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 72 "In class we developed th
e basic power series expansion about the point, " }{TEXT 261 1 "a" }
{TEXT -1 5 ", as:" }}{PARA 256 "" 0 "" {OLE 1 6180 1 "[xm]Br=WfoRrB:::
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:::::::::::1:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 265 25 "This series is called the" }{TEXT -1 1 " " }
{TEXT 266 39 "Taylor Series of the function f about (" }{TEXT 263 14 "
or centered at" }{TEXT 264 101 ") \"a\". If a = 0, then we get the spe
cial case which is called a Maclaurin Series. It looks as follow:" }}
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rTf:V<NvXf:^<v`\\f:V<Nl`fZ:j>DJ:vYE:cJtAjH<lDjwEJKxI;B:>l;B:;B:DJ:<Kd`
ppP@W\\Z:WM:<J<jPF:OZ:N`DZ:N:;B:_sFMw;sW:E:cJ@LjqTjDjw;<:[><Z<B>:::C:U
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:_;<j:Jq^QJD^;CU;e:qAB:^Q>:;b:^D::J<jPF:C:[Q;F;N@B:=:W=JDnOUv;e:qQ:[:J
XK:>Z<Jd:::C:UK;^:>X=jFJSZ:^[:Jm\\lNTPN`<JvDJXAjl]jY@JXAJ?Ijx@JXAjVLjW
AJXAJ:Hj<<:cJJ?Ke:qAB:^Q>:;b:^DVH::C:Uk:^:>X?JH>:_;<JAJm>ic>i`?_cBXkeE
WcE_Pi_\\;V[Cfh=Nt:^q;f^@Vr@>PUi:;Mjr:EB:^<Sps]jDjw;<:s?;J:DZJ^DVH::C:
UK;^:>X=jA>:_;<j<Jq@lJ@lJ<Ju_cF>s@fjAr:AU:<JMjWEj`@PtZ_gdOwo<R:BKa<@:B
UFF::]Jd::jBjpDpqR>H>_=t>;:fBFF_J>v?v?[Dw[j]BN_A[DweZ:^x?Kt;<[=SJvk?cD
r]<ZJBJcqZxQ^>\\IFbd:xo\\RJj:Hj:HZ:J;F:=B:AB:=J<>ZDFZ:B;>Z:>WZIF:HRmN:
?FDrZkE:?J;a\\<jXr:>h:F:Aj?R:DJ:HRmV:sZEF:N:?FDrZWM;NjcB:dJ;<J;]<\\RR<
J;Njcj:j;[<H>;B:;r<<:A>C:DZBL:@:CJIfNUZ@H<N;HRvV:Z;Z:bU<ZDB;p=Vj`c:DRN
B:K^<bZMK;rDhl;V;@:HBA::NjaJ:B:?FDB\\<J;NjcB:J>A=DrJJ;Vjrc:C:Q;F:bS::D
rMN:B:d:\\<;B\\[C:?J;a<J>E]<bZ=;?j;]a=::\\B;J?^:V?j:j;U@:Z<x>?::;:;JNJ
>=M:DZ<H?ZBX@?B:?VBb:HRv::>K<ZDB;rD?J;u\\<>Z<<B:<j?<Z:n>^;yayI:SL:<J:F
;<JwC:<:JK^:f?=JB<Z:VY[j=>:;JX?:<j:DJ;Dlc`qsLqlp@C:Uk;^:>x;NZ:V:;B:;::
::::::::::::::::::::::::::1:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Note: " }{TEXT 262 101 "If the f
unction f can be represented by a power series about \"a\" then the ab
ove equalities hold true." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 136 "Now, we can use partial sums to approximate f(x).
 These partial sums are called Taylor Polynomials of degree n. Let's l
ook at a few for " }{TEXT 268 7 "f(x) = " }{XPPEDIT 269 0 "exp(x);" "6
#-%$expG6#%\"xG" }{TEXT -1 37 ". We will expand this function about " 
}{TEXT 280 5 "a = 0" }{TEXT -1 89 ". The code below will compute the f
irst 4 Taylor polynomials.Place the cursor in the red " }{TEXT 270 7 "
restart" }{TEXT -1 27 " command and press \"Enter\"." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 142 "restart:with(plots):\nfor i from 1 to 4 do\na[i
]:=convert(taylor(exp(x),x=0,i),polynom):\nend do:\nMATRIX([[T0,T1,T2,
T3],[a[1],a[2],a[3],a[4]]]);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 
"The approximating Taylor Polynomials are shown above. " }}{PARA 0 "" 
0 "" {TEXT -1 45 "To see how well these functions approximate  " }
{TEXT 271 7 "f(x) = " }{XPPEDIT 257 0 "exp(x);" "6#-%$expG6#%\"xG" }
{TEXT -1 27 " near the expansion point \"" }{TEXT 272 5 "a = 0" }
{TEXT -1 68 "\", we shall superimpose them onto the same graph. It is \+
best to run " }{TEXT 273 24 "individual frame changes" }{TEXT -1 15 " \+
as opposed to " }{TEXT 274 23 "streaming frame changes" }{TEXT -1 11 "
 using the " }{TEXT 275 18 "Animation Toolbar " }{TEXT -1 80 "which ap
pears at the top when you click on the picture. Press \"Enter\" to beg
in. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 577 "restart: with(plot
s):\nf:=x->exp(x):\nc:=0:\nn:=4:\nT:=proc(n,x) convert(taylor(f(x),x=c
,n+1),polynom) end:\nGraph:=display(seq(plot(convert(taylor(f(x),x=c,i
),polynom),x=-2..1,y=-.1..3,thickness=3,title=\"Taylor Polynomial Conv
ergence\\nf(x) = exp(x)\\n a = 0\",color=magenta,titlefont=[TIMES,BOLD
,14]),i=1..n),insequence=true):\np:=display(seq(textplot([-0.9,2,cat(`
T`,i-1,` =`,convert(convert(taylor(f(x),x=c,i),polynom),string))],font
=[TIMES,BOLD,10],color=magenta),i=1..n),insequence=true):\nFunction:=p
lot(f(x),x=-2..1,y=-.1..3,color=blue,thickness=3):\ndisplay(Graph,Func
tion,p);\n\n\n" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 1 "\"" }{TEXT 
281 18 "Click on the graph" }{TEXT -1 221 "\" and then use the sequent
ial frame button ( ->| ) to see how the Taylor Polynomials \"fit\" the
 exponential graph near x = 0. Pay attention as to how accurate the ap
proximating polynoimials are as they increase in degree. " }}{PARA 
257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 290 
0 "" }{TEXT 291 28 "Example of sinx about a = 0." }}{PARA 257 "" 0 "" 
{TEXT -1 58 "The next example shows the first 6 Taylor Polynomials for
 " }{TEXT 276 33 "f(x) = sinx expanded about a = 0." }{TEXT 277 1 " " 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 573 "restart: with(plots):\nf:=x->sin
(x):\nc:=0:\nn:=6:\nT:=proc(n,x) convert(taylor(f(x),x=c,n+1),polynom)
 end:\nGraph:=display(seq(plot(convert(taylor(f(x),x=c,i),polynom),x=-
6..5,y=-3..5,thickness=3,title=\"Taylor Polynomial Convergence\\nf(x) \+
= sin(x)\\n a = 0\",color = magenta,titlefont=[TIMES,BOLD,14]),i=1..n)
,insequence=true):\np:=display(seq(textplot([-3,3,cat(`T`,i-1,` =`,con
vert(convert(taylor(f(x),x=c,i),polynom),string))],font=[TIMES,BOLD,10
],color=magenta),i=1..n),insequence=true):\nFunction:=plot(f(x),x=-6..
5,y=-3..5,color=blue,thickness=3):\ndisplay(Graph,Function,p);\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 283 1 "\"" }{TEXT 282 18 "Click on the gr
aph" }{TEXT 284 88 "\" and then use the sequential frame button agian \+
to see how the Taylor Polynomials \"fit\"" }{TEXT -1 189 ". Notice how
 the even numbered approximating polynomials are equivalent to the odd
 numbered. Also, as we increase the degree of the Taylor Polynomial th
e interval of convergence \"expands\". " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 292 0 "" }{TEXT 293 27 "Exam
ple of lnx about a = 2." }}{PARA 0 "" 0 "" {TEXT -1 60 "Our next examp
le creates the first 4 Taylor Polynomials for " }{TEXT 278 12 "f(x) = \+
ln(x)" }{TEXT -1 14 " centered at \"" }{TEXT 279 6 " a = 2" }{TEXT -1 
55 "\". Press \"Enter\" to see the first 4 Taylor Polynomials." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "res
tart:with(plots):\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 120 "for i from 1 to
 4 do\na[i]:=convert(taylor(ln(x),x=2,i),polynom):\nend do:\nMATRIX([[
T0,T1,T2,T3],[a[1],a[2],a[3],a[4]]]);\n" }}{PARA 0 "" 0 "" {TEXT -1 
89 "Notice the ascending powers in the approximating polynolmials. Pre
ss \"Enter\" to continue." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
572 "restart: with(plots):\nf:=x->ln(x):\nc:=2:\nn:=4:\nT:=proc(n,x) c
onvert(taylor(f(x),x=c,n+1),polynom) end:\nGraph:=display(seq(plot(con
vert(taylor(f(x),x=c,i),polynom),x=-5.5..5,y=-8..5,thickness=3,title=
\"Taylor Polynomial Convergence\\nf(x) = ln(x)\\na = 2\",color = magen
ta,titlefont=[TIMES,BOLD,14]),i=1..n),insequence=true):\np:=display(se
q(textplot([-1,5,cat(`T`,i-1,` =`,convert(convert(taylor(f(x),x=c,i),p
olynom),string))],font=[TIMES,BOLD,10],color=magenta),i=1..n),insequen
ce=true):\nFunction:=plot(f(x),x=-5..5,y=-8..5,color=blue,thickness=3)
:\ndisplay(Graph,Function,p);\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 286 1 "\"" }{TEXT 285 18 "Click on the graph" }{TEXT 287 138 "\"
 and then use the sequential frame button agian to see how the Taylor \+
Polynomials approximate the given function near the expansion point" }
{TEXT -1 56 ". Notice the restricted domain for our chosen function. \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 288 0 "" }{TEXT 289 63 "Example with limited interval of converg
ence centered at a = 0." }}{PARA 0 "" 0 "" {TEXT -1 200 "Our final exa
mple shows that many Taylor Series Polynomials only \"converge well\" \+
over a certain interval. Both in class and in your book we discussed  \+
\"interval of convergence\". Consider the function:" }}{PARA 256 "" 0 
"" {XPPEDIT 294 0 "f(x) = 1/(1-x);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)F
'!\"\"F+" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "we can create
 the Taylor Polynomial expansion for, say, the first 9 terms. Press \"
Enter\" to continue." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 608 "restart: w
ith(plots):\nf:=x->1/(1-x):\nc:=0:\nn:=9:\nT:=proc(n,x) convert(taylor
(f(x),x=c,n+1),polynom) end:\nGraph:=display(seq(plot(convert(taylor(f
(x),x=c,i),polynom),x=-5..1,y=-5..4,thickness=3,title=\"Taylor Polynom
ial Convergence\\nf(x) = 1/(1-x)\\na = 0\",color = magenta,titlefont=[
TIMES,BOLD,14]),i=1..n),insequence=true):\np:=display(seq(textplot([-1
.4,6,cat(`T`,i-1,` =`,convert(convert(taylor(f(x),x=c,i),polynom),stri
ng))],font=[TIMES,BOLD,10],color=magenta,view=[-5..1,-5..6]),i=1..n),i
nsequence=true):\nFunction:=plot(f(x),x=-5..1,y=-5..4,color=blue,thick
ness=3,discont=true):\ndisplay(Graph,Function,p);\n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 296 1 "\"" }{TEXT 295 18 "Click on the graph" }{TEXT 
297 127 "\" and use the sequential frame button to see how the Taylor \+
Polynomials approximate the given function near the expansion point" }
{TEXT -1 2 ". " }{TEXT 302 125 "Notice that the approximating polynomi
als do not extend for values less than negative 1. Replay the animatio
n to verify this." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 55 "Using Taylor/Maclaurin Series to Approxim
ate Integrals:" }}{PARA 0 "" 0 "" {TEXT -1 109 "We can use a Taylor Se
ries expansion to approximate the value of an integral. Consider the i
ntegral given by:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {OLE 1 4616 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyy
y:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::fyyyyya:nYf::G:I:wAyA:::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyJ;B
:Abm?:;JZC:bKB::>bmfoqIr]^r;V:r;B:F:YLpfF>:::::::::J?NZ;vyyyyy=J:B::::
:::c:;:=j[vGUMrvC?MoJ::::::::JCNZ;F:f:;jysy;Z::::::j:>:M=;:AB:Y<FCWMmv
G;J::::::::JCB:yyyxI:;Z::::::j;>:CJ:>:nyyM;:nyyyYE:G:IZ:>;F;nYV;^;f;;J
AjA>:[Z:F<N<;jCJDjDJE>:i:k:m:o:q:s:wAyA:::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::F:DZ:B::::::::::F:wyyAb<vYxy;tV;J:>R
<:TNEj`@Pt\\Pd`QrPPJLMQ<MJPnrPqjLqnxPq^;<:Z_SKGlAj\\@AJQLncpNdL]nIalHH
<J::::::j;J::::::::::JHB:?jD`WP@:;dGU<;jvUm]^wdAIER:LZ>WdG[;V\\QGghGbf
_g<GfmGg]Wbi_f_?d[wf_WH]R;cTTUUSaEBWTSiEB_tUUURWmBHjc`qpDpql`KfZ=wgoOh
cgfl?^HgF;R>CRMcDBEDXceV=MC>[[vg?wk>w_;?dnA\\IvbYGbJOxYn];?dni\\IFb:^x
?_x_^xIPbYGB<[=we;kAQNteli\\NtyrZ<ZJBJcy]=tYKDw[r==dDyDKyTKcyQE\\;w[FF
Z;FZ;r:?J;N:=j:V:=J<>ZD>\\>J:<JnIJ[lptP`IFZ;r:xjR<j:N:AVSHZVj?@j\\FHem
j^HMmqnG;KaFFJufF>::::::;C:?jysy:>:<::::::C:KT:<J:<:?ja^G>d:NHEms>@[C:
>Z::::::::kJ;@j;>:C:yayA:<::::::OB:;B:sF;<Z:::::::::::::yay=J:B:::::::
::::::::::::jysy:>:<::::::::=R:DJ<TRCDlPDZ=@i:V:sZFFZ;B:<J>E=;B:V;;b:>
@@J:jR<J:pJ;LJ=iLKDiDN:AFSDZ=@I;J::::::::::vYxy;<:C:=J:<jysy?:;:=C<<J@
M:?E;F:MZ=F`<B:?jJK:^:<J;@j:f:<j:Djyyyy=J<JQ@jFf:F`<F:;j<j?D:;B:E:Sb:;
C;;u;a:Gc;YJHvyyuy;B:kMBB:;xyNoyyyyA^AJ_I:QB:;JMJ@fc[KdRS;B:[o<B:Mb:vy
<f:^\\<nd;^g>f<Vy<F_yA:;B:>l;B:<Z<>ZJ^dcgg_;<<UMq>:C:US:F[:>Z:N`DZ:FZ:
B:_C:WW:E:cjX@jQ@jDjw?ux]:JBA:<Z<:::J<jPF:C:[Y:F;;JSd:<j:B:]C:[d:E:cje
Ijyg<;jw;<:[V:B:D::::C:UK:^:>X=j>>:_;<j:B:[C:[d:JDNu;^r=f<;jw;<:sg:=J:
>Z<Jd:::C:Uk:^:>X;j>JSZ:F:<Jv>d<:cJZEJqAjDjw?Zx]:JXK:>Z<Jd:::C:UK:^:>X
=J@>Z:N@B:AB:<jlLPvv\\=R:Lj<<:cJgOx<><<jw?UB:>l;B:;b:DZJVdscRYEU@J<<:U
k:^Z:Jr?:MJ:N`DZ:F:<jP<JbD:^\\<>i=>B[Z:VY;RyB:>l;Z<b:<;vGB:f_;>:C:[q:F
;;JSd:<j:B:_a@FNn>>a;fc[_H`a_gdO__;fZ;Z>WDR:Zf;rU:JdRS:JqF<a\\\\wG@k<H
>ZHK::E<=MC>;:LYBvB<OsKbLyDB:cYK:B>H^;kAQ^dZID:<[:_X=tYKLBxjZ:xo\\RJj:
@Z=N:?J;F:=j;F:CJ:dj:<JBL:;B:KAj:HZIB:=J;Vj_[VB:QR:=R:;b:CJC:kE:A:lj:Z
:>klB:;B:V;;b:>@;R:jR>ZGZeZDZ^[=@I;B:QB:n>^;yayI:S\\:>:[B:;B:qQBv:<J:^
Q;Z:FZ<NZ\\VdsWhngfgK<jPF:C:[Q:V:;B:;:::::::::::::::::::::::::::::::::
::::::::::1:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "Suppose we w
ish to approximate this integral to within 0.001. We first find the Ma
claurin (Taylor Series centered at a = 0) Series for " }{XPPEDIT 18 0 
"e^x;" "6#)%\"eG%\"xG" }{TEXT -1 61 ".If we examine the first 5 terms \+
an obvious pattern emerges;:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
127 "restart:with(plots):\nfor i from 1 to 5 do\nprint(convert(taylor(
exp(x),x=0,i),polynom)):\nend do:\nr:=Sum('x^k/k!',k=0..infinity);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 17 "We conclude that:" }}{PARA 256 "" 0 "" {OLE 1 5128 1 "[xm
]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:I:K
:wAyA:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=?Z:VZHQ:>:;`:Z@[::JZH
mPYyZIL\\Aj;J:`:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HY
LkNG>::::::::N<?R:=:EJ:vYxY:B::::::F:;jnJ:j;<ja:MLqnFYM:>::::::::N<<jy
yiy=J:B::::::V:;J<>:;:wyyN::wyyyq<J=j=B:K:M:wAQ:S:UJ:n;v;;JBB:]:_J:V<^
<f<n<;jEJFjFJGjGJHjHJIjIJJ<Z:F^:JKjKJyky;:::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::F:D:<::::::::::=Jyyy;d:yayYZHQ:>:;`:Z@o
<fc[_hb_ds?h_?^nn_j>^?GhoGfnGgioGSZ::RSPN]v;fbk;n_>WdG_dnBwmcf]]:>::::
::V:>::::::::::^=<J;f\\SAb;>Z:>bpfC>:miGILpeXvfZ;B;Lncp>@jCXontpZlPpDn
jHqnHp[xPlPPb@pmPpsF\\?^dcgg_WhZnc_whZNdigg[oG]r:aTXUeRYEU@k<HnqPQs`pl
DQJtnl>ZK^ZG_dZfbr_hlGF_J>@lqPnAMnQ@NbLYBxj]xoZ<OsyKI@NbLyDxjZ<JtQNtQL
tyR[xoZB>Hni@>YVKciDyDLcy]=D:<[:_xIHby?cnA\\IFbdve>we?_xYgB@nAlj:@j:@Z
=N:?J;F:=j;F:CJ:dj:<J;Vj_s:l;QR:=R:;b:>i:J?DJ<TrXfa=B>[m\\<jrfH=MtFGYM
q>>Wlj:gmlJ::::::>^:N:yay=J:B::::::^:<Jh@:;B:N:YLpJbNHEms>@[C:>Z::::::
::kJ;@j;>:C:yayA:<::::::OB:;B:sd;B::::::::::::jysy:>:<::::::::::::::::
:::vYxI:;Z::::::::j:N:inqIJ:h:=:A^=BA:joB:;b<=:b:DrJJ:hZ:N:A>Cb[b:\\RR
R:^:wbBEo@DJ<TrBDllC:;::::::vYxy;>:C:=J:vYxY;B:;:ed<V`@B:?c;<j:F;HjBH:
CjOK:^:;J;@j:f:FZ<vyyyy:^:n_;F=E:]c:=B:f:V[<Z:f:^[<>n?>fCV<<JMTjA^=yyy
xY:<Jv>\\:B:;xy>qyyyyA^CJfP:QB:;JMJ@fc[KdRS;B:;S:MZ:Ny<jFJ:b:?K<jP@:EZ
:^\\<>v<Nq@fZ:B:ab:;i:qR<EJ:F[<J:Dj<JDnb?vl?f<Vy<>uxE:<:[V:<J:<Z<>ZJ^d
cgg_KaBBJqJ<B:Uk:F;;JSd:;B:=B:>`:>d<f:^<kI;WD<e:qQ:tI<:sg:B:D::::C:UK;
^:>x;F:JSZ:F:F^:>Z>f:;JDv_;n^Bf<VYk@<:so:B:;b:^D::J<<:Uk:^:>X?B:M:_;<j
:jo>d<:cJwDJ_@jDjw;<:s?;J:D:c<::^:f??J<JrG:M:_;<j:Jv>d<:cJ]va>f\\:jw;<
:s?;J:D:c<::^:f?=J<JrO:M:_;<j:jo:^<eb:?r;eJ:VY;:<:s?;J:D:c<::^:f??J<Jr
G:M:_;<j:Jv>Z>f:^<;u:[kDjw;<:s?;J:D:c<::^Z:jPF:C:[Q;F;N@B:=:E]>f:<JD>Y
Gr;e:qAB:^Q>:;b:^D::J<<:UK;^:>X=j>JSZ:F:VG;B;E:cJf@J]MJBB:qQf[:JBA:DZ<
B>aTXDpql`;^Z:jPF:C:[Q;F;;JSd:<j:jL?d<:cb:GLrMj>>:_;<j:jX>d<:cJjnl<><;
B:qQ:Z:JBA:DZ<B>jqZ:jPN:C:[q:F;;JSd:<j:jX>Z>:cb:gx:?C;[Z:VYpI;B:>l;Z<b
:<;vGB:f?=J<JrO:MJ:N`DZ:F:fVTjOC:G;ST:UTRc=fdO?b;fZ;Z>WDR:Zf;::^DP@:nG
]jcbRY]KfZ=Ks:>:j\\jjN<Kjq:LYBvB<OsKbLyDxZ:^x?;ZJrJ@>YVKd<xb:B><NtGby?
C[r==<rQE\\;;=R:HJ;<:?j:<j:V:=J<>ZDB:<j;Q`=BA<j?@j:>Z<Jv<J:<J?D:u[=JrF
:?jE[Ej:j;s:;::VG;B:;b<=:b:;bZJ:hZ:j;K<Z<:@:CJIj@jBJ:<j?<:G;Sjysy=J`B:
;j><:oA<:JK<J<B:UK;>\\:B:qQBv:<J:^Q;Z:FZ<NZ\\VdsWhngfgK<<:Uk;^:>X=j;>Z
:>::::::::::::::::::::::::::::::::::::::::::::::::::::::1:" }{TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "If we substitute " }{OLE 1 4104 1 
"[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G
:jy;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=?Z:VZHQ:>:;`:Z@[::JZH
mPYyZIL\\Aj;J:T:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HY
LkNG>::::::::N<?R:=:EJ:vYxY:B::::::F:;jnJ:j;<ja:MLqnFYM:>::::::::N<<jy
yiy=J:B::::::V:;J<>:;:wyyN::wyyyq<J=j=B:K:wAO:Q:S:UJ:n;v;;JBB:]:_J:V<^
<f<n<;jEJyky;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::F:DZ:B::::::::::F:wyyAb<vYxy;tV;J:>R<:TNEj`@Pt\\Pd`QrPPJLMQ
<MJPnrPqjLqnxPq^;<:Z_SKGlAj\\@AJQLncpNdL]nIalHH<J::::::j;J::::::::::JH
B:?jD`W@@:;dGU<;jvUm]^wdAIER:LZ>WdG[;V\\QGghGbf_g<GfmGg]Wbi_f_?d[wf_WH
]R;cTTUUSaEBWTSiEB_tUUURWmBHjc`qpDpql`KfZ=wgoOhcgfl?^HgF;R>CRMcDBEDXce
V=MC>[[vg?wk>w_;?dnA\\IvbYGbJOxYn];?dni\\IFb:^x?_x_^xIPbYGB<[=we;kAQNt
eli\\NtyrZ<ZJBJcy]=tYKDw[r==dDyDKyTKcyQE\\;w[FFZ;FZ;r:?J;N:=j:V:=J<>ZD
FZ:NZ>njLN^gN:AFX;r:l;QR:=R:;b:;B\\_;Ob:@RJfb:FHemj^HMmqnG;KaFFJufF>::
::::;C:?jysy:>:<::::::CZ:>_;>Z:>Z:J;vCS=[LsfFaMR>`:J:<:::::::>=?R:AJ:^
:vYxY:B::::::F;>l=Z:::::::::::::yay=J:B:::::::::::::::::::jysy:>:<::::
::::?j:=\\<JmN:x`<JsN:<<:\\A<:::::yayYZ:B:C:=J:vYxY;B:;:ed<Fr<B:gt:F:M
J:HjOA:?JB?Z:J<J;@j:f:<j:Djyyyy=J<JQ@jFf:F`<FZ:B:E:Qb:>Z:f:^[<>j<>fCV<
<JMTjA^=yyyxY:<Jv><<JZyIUwyyQJ`:Uc:<j?<:G;Sj`@@c\\_?Z:>@EZ:F[<J:Dj<JDD
JW@JPEjDjwEJfui:B:>l;B:<Z<>ZJ^dcgg_;<<nG^:f_;j><J:<JSd:;B:=B:<JS<JbDj<
>:cJJDJ?IjD>:qQ:tI<:sg:;Z:b::::^:f?=J<JrA:aZ:N`DZ:>;;JJ<LmxPs<LJ\\Q:?J
[_;>v:>J;N:Lj<JDJ>AjDjw;<:so:>Z:>Z<B>c<::^Z:jP>:C:[Y:=j>JSZ:FZ:B:kM:Lj
<JDJV><<jw;<:[V:b:DZJVdscRYEU@J<<:Uk:^:>X;j>>:_c<B:=Z:f_:B;[N:<JMv;eR:
UTRc=fdOG[;fZ;>:LncZ;:l@::JdRS:JqF<a\\\\wG@k<H>:>:j\\jjN<Kjq:LYBvB<OsK
bLyDB:P>:<[=SJvk?cDr]<ZJBJcqZxQ^>\\IFBZYgB@>FZ;r:?J;N:=j:V:=J<>ZDFZ:B;
:AN:BA<j?@:;b:;B\\>:Ob:>>?j:;G=x@NHBB:BX>Z:V[:JMJ@vYxy:^C<J:>\\:B:qQBv
:>:sO:B:=b:?bBaTXaEWEUU^Z:B:US:;J<JrG:AJ:<J:::::::::::::5:" }{TEXT -1 
34 " we obtain the following integral:" }}{PARA 256 "" 0 "" {OLE 1 
7688 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya
:nYf::G:I:K:M:O:Q:S:UJ:nYvY:::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=A:
Abm?Z:>:;`:Z@[::JZHmPYyZIL\\Aj;ZMZ:j:vCSmlJ::::::::::OJ;@jyyyyyY;jysy;
Z:::::::^<>:F:AlqfG[maNFO=;::::::::_J;vyyuy:>:<::::::=J:FG>:VZ:vCj^nGG
mq>:;::::::::_Z;F:f:;jysy;Z::::::j;>:CJ:>:nyyM;:nyyyYE:G:IZ:>;F;N;;j?J
@j@JAjA>:[Z:F<nYV<^<f<n<;jEJFjFJGjGJHjHJIjIJJ<Z:F^:JKjKJLjLJMjMJNjNJOj
OJPjPJQjQJRB:];_;a;c;e;g;i;k;m;o;q;s;u;wK:vA>B<jZJ[j[>:C<EL:nBvB>CFC;J
_j_J`Jyky;:::::::::::::::::::::::::::::::::=Z<Z::::::::::j:nyyYZDjysyA
bm?:;JZC:bKg:UTRcETcTX[US;CGWCF;SK]UW=EWMuUWM@B::P`?oBY:ETV:WCKaTMcDGL
yVDurB:;::::::A:;::::::::::sZ:N:eRpSt:>bpfC>:miGILpeXvfZ;B;Lncp>@jCXon
tpZlPpDnjHqnHp[xPlPPb@pmPpsF\\?^dcgg_WhZnc_whZNdigg[oG]r:aTXUeRYEU@k<H
nqPQs`plDQJtnl>ZK^ZG_dZfbr_hlGF_J>@lqPnAMnQ@NbLYBxj]xoZ<OsyKI@NbLyDxjZ
<JtQNtQLtyR[xoZB>Hni@>YVKciDyDLcy]=D:<[:_xIHby?cnA\\IFbdve>we?_xYgB@nA
lj:@j:@Z=N:?B:?j:F:Aj:^:;b<=B:[B;>Z:>w=Nbfgh_s==r:xjR<j:N:AVSHZVj?@j\\
FHemj^HMmqnG;KaFFJufF>::::::;C:?jysy:>:<::::::C:Ku:>Z:J;vCS=[D:_mlVH[K
R<:;B:::::::JFNZ;vyyuyC:yayA:<::::::_J:<JSu:<:?:Jb:a=[;;B:::::::JF>:ya
y=J:B::::::nYyA<::::::::::::jysy:>:<::::::::=R:DJ<TRCDlPDZ=@i:V:sZFFZ;
B:<J>E=<:QJ:D:[[;>:F`:>ZGNZ>njeN^le<?j;M`<rZkM;LZwRIAjA@=HZJ>H=:QZ;Z;V
:=B:?B;D;Z=BAV;=R:>ZBP@>ZDN:gFDDJ:TsNr::Z=XG<J:dj:N:b:DrJB:;r<<J;VJ^:u
[<B\\[[;J<n]\\fNUb:F\\\\fVh:<J>;RF<j@jO:HRlj;]a<^:u[=:V:=r:>Z<B\\NZ>b>
V:QZ;J:HBAJHZ;:KfFB:O:F@NJ>Z<^:QC:Qs:h:j;;:B\\c;;b<Zlb:DBN:D;:rZ::>kpZ
:N;jSNZ:NJb:V?DZ<p??::Aj:H:;J<N<D:]K:DJ:TSNB:`<^=:<j;u@dj;V;:k=:J>E]<r
ZW=;b<=:b:DrJB:?Zhb::B:?JE]<F?<j?>Z<B\\[C:;R:jR>ZGZeN>Z^[=@I;J:::::yay
YZ:J<j:>Z:vYxY;J:J@p=iw;>iHj:F;HJXXZ:^:]W:;B:<J<>:?R:=j<B:=b:yyyyI:C:W
S:mj<jRDj:>:E:Qb:;B:<j<J@DZA>fqV<n^@v;sjyyiyAZ:>i:>\\:B:;xyvnyyyyA^q>:
Ig;<j?<:G;Sj`@@c\\_?Z:>b;f:<j>D:yi:E:cb:Ir:iT=[B:;B:qi:MHyYS:<Z:>l;B:D
J:DZJVdsgGl`;^Z:jP@:MB:;B:_c<>Z:FZ:B:K;E:cb:>lK>\\:>Z:VY:<JBA:DZ<B>jqZ
:jPF:CJ:>x;j>>:_c<B:=Z:F?;j>JwEZ:F=>:yQK^:f?;j<JDDJ:vbGfZ:jCDZ>N`Mf:^<
VgVf:;jCjTuk<JDjP=l<jCjuEl<JDjmMl<jCJPXl<>:cJNMJXeLBB:qQbpyQ@:;B:[V:b:
DZJ:Y=<:UK;^Z:JrG:MJ:N`DZ:F:>?f:^\\<JGqLBB:qAZ:>l;Z<b:<;vGB:f?=J<JrOZ:
F;;JSd:<j:jNj<JDDJ:nhifZ:>:a:seFE:cJEHJpLjDjwG_yA:;B:>l;Z:b:^dcSSnC<<U
MqJ<jPN:CB:>X=j>>:_;<j:JS<JQAj<JD>u;vmJf<VYkhyB:>l;Z:b::::^:f?=J<JrO:M
J:N`DZ:F:N`:>d<f:;JDvc<vtWf<VYZ:JBA:<Z<:::J<jPN:C:[q:f;;B:_c<B:C:_CGgc
F?D;eC;cf<[d:E:cjxEjQ@jDjw;<:[><Z<:::J<jPF:C:[Q;^;;B:_c<B:AB:;jR@mRfvB
f]H>d<f:^<iV;yqDjw;<:[><Z<:::J<jPN:C:[q:^;N@B:A:[CF[C:EI<:^<Ge;SD=e:qA
B:>LB:D::::C:Uk:^:>X?j>>:_;<j:JR<JbD:^<;F;qW@e:qi:;byB:>l;Z:b::::^:f??
J<JrG:O:_;<J;>Z:F@Si>Lj<<:cJFTJVmkDjw;<:[><Z<B>:::C:Uk:^:>X?J@JSZ:V:<j
RDmSNc>v`>B;E:cJ:_\\cf<VYZ:JXEZ:B:D::::C:UK;^Z:JrG:[B:<JSZ:n:<JQtLQt<;
N:?qB>J;N:Lj<JD>=;G>eB:VYZ:JX[:b::::^Z:jPF:C:[Q;v;;JSZ:f:;jJ@lJ@\\:>\\
GNh>v^>>\\EB;E:cJZ@jXDkDjw?UB:^Q=J:>Z<Jd:::C:UK;^:>x;F:MJ:<JSZ:F:;jo>d
<:cja^kkf<VYZ:JXK:>Z<Jd:::C:Uk:^:>X?j>JSZ:F:nG[d:JDfb=nchf\\:jw;<:s?;J
:D:c<::^:f??J<JrG:M:_;<j:Jq:^<ov:Cx:e:qAB:^Q>:;b:^D::J<jPF:C:[Q;F;N@B:
=:kMbD:^<GMtejD>:qAB:^Q>:;b:^D::J<jPN:C:[q:F;N@B:=:Q]<:cJYEjwxjDjw?:B:
^Q>:;b:^D::J<jPF:C:[Q;F;N@B:=:kM:Lj<JDf`=VpVf<VYZ:JXK:>Z<Jd:::CB:f??J<
JrG:UB:N@B:C:kEXkEX?D;oC;Yf<Lj<JD>JWY:e:qAB:^Q>:;b:^D::J<jPF:C:[Q;N<;J
SZ:v:<jlLPvLPvL`:v\\=R:uR=@jt=\\;B;E:cJF>eIf<VYZ:JXK:>Z<Jd:::C:US:?J<J
rG:O:_;<J;>:QeU=HB;B;E:cJvDjwhJBB:qQf[:JBAZ:b:DZJVDDpqZ:jPF:CB:>X?j>>:
_c<B:=:EObD:^\\<NM_i:MB:N@B:=:UC:[d:JDnFEw<M:_;<j:jX>DJDnQQc=[Z:VY;:<:
[V:b:DZJ:Y=<:US:?J<JrG:MJ:N`DZ:F:f_:>Z>:cb:kr:KI@O:_;<J;>:GiwYv@:cJ:gk
AN<N@B:IZ:fawf_ef_ef_:f]HFb?Na>Nb>f@ay:;B;E:cJvMJn]K?JSZ:N:vvqykU:^<kf
;KI@O:_;<J;Jn]yAi;JDnx>vtD><;B:qawM:<:[V:b:DZJ:Y=<:US:=J<JrO:MJ:N`DZ:F
:fv:>Z@fZ:B:cb:kS;;L@<:_;<j;B:_iv_ql]:;b;]W:;B:G;kt:UTRc=PpdPO`Hj<@:Ln
cBG@jCZfkj::a=c\\_;:F<a\\\\wG@k<H>bV;;;R>j\\^H=MC>;Y=Bw[r=ItQ<Osy;[Dwe
Z:^x?Kt[:B>H^;kAQ^dBLxb:B><NtGby?C[r==<rQE\\;;=R:HJ;N:?j:F:Aj:<J<>ZDB:
[B;>Z:>w=:xj:r=<j:N:AVSBA<j?@j:@J:DJ<N<rZkE:A:lj:Z:>klB:<:QJ:D:[[;:]K:
p:h<d:AFSrZkM;LZw:Y:HZJ>H=:QJ:@:@j;<j:<J;LZL:r:l;Qj:J:\\RSJ:dJ;nlbJ:Ts
Nr::Z=Xg:>Z:>ZDF:?:DJ:DrJ>Z:>ZEB:?j;K<fADZB@@@:CJIfNU:]:h:<J>;RF<j@jO:
HRlj;]a<^:u[=:V:=r:>Z<B\\NZ>b>V:QZ;J:HBAJHZ;:KfFB:O:F@NJ>Z<^:QC:Qs:h:j
;;:B\\c;;b<Zlb:DBN:D;:rZ::>kpJ:<J?:aK;<J;[<jOb:DRQN::V:=r:>:CJCb:F@;b:
;b_K[:RDJH:B:AfU<ZDV:Q:Jv::>klb:HRqJ:dj::DZ<H><J;bED::<J;nlb:M[:V;;b:
\\RR<J:@:F@;R=rD?;BSrZkM:<j?<:G;Sjysy=J`B:;j>JwC:<:JK^:f??JB<J:<jw?\\:
v:<J:^Q;Z:FZ<NZ\\VdsWhngfgK<<:Uk;^Z:JrG:AJ:<J:::::::::::::::::::::::::
:::::::::::::::1:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "We ca
n now integrate term by term to obtain:" }}{PARA 256 "" 0 "" {OLE 1 
9736 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyyq
yyyYJ<j<Jyk=J>j>J?j?J@j@>:W:YJ:><<jBJyky;:::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::
gj<vyyuyj;j;tV;B:;JZC:bKB::>bmfoqIr]^r;V:>r?B:<:ElrfH=MtFGYMq>>Wlj:gml
J::::::>>?jyyiy=J:B:::::::ST;B:F:YLpfF>:::::::::J?N:ry:>:<::::::gJ:^<>
:N:Y<>D_mlVH[KR<:;B:::::::JFNZ;N:<j:>:yayA:<::::::G:_f?B:F:N:;j;<:CJ:f
:<J=j=J>j>J?>:Q:SJ:f;n;v;><<jBJCjCJDjDJyyIwAyA::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::^=N:eRp[S;F:V;>:>R<j<@Z>B
KaTMcDG@jCXontpZlPpDnjHqnHp[xPlPPb@pmPpsF\\?^dcgg_WhZnc_whZNdigg[oG]r:
aTXUeRYEU@k<HnqPQs`plDQJtnl>ZK^ZG_dZfbr_hlGF_J>@lqPnAMnQ@NbLYBxj]xoZ<O
syKI@NbLyDxjZ<JtQNtQLtyR[xoZB>Hni@>YVKciDyDLcy]=D:<[:_xIHby?cnA\\IFbdv
e>we?_xYgB@nAlj:@j:@Z=N:?J;F:=j;F:CJ:dJ<FZ;FZ:>\\>J:<JnIJ[lptP`IFZ=r=]
C:=J;Vj_s:l;QR:=R:DJ<TRCDlPDZ=@IA:lj:@:<J>E=<:QJ:D:[[;:]K:pJ;LJ=iLKDiD
N:AFSDZ=@I?B;p]wV:Y:HZ;V:=B:?J>c:CJCr:h:=Z:VJHZVj?F:B\\_;;b<=B:?ZkZ<H>
<J;LZe>ZeV:Q:rZ::>knZ:N;JSN:?>D:_::VJ::\\RUJ:d:rFDJ:DBN:h<?J;u\\<:Z=R:
AZ:B;D;?J>A]:NZ:nlbb:;jNr:l;:;rZW=Or:h:Zr;:B\\_K;Vjh[:N[;:N:?VB>Z<^:Q;
DRQN::>Z<>ZP@?RD>ZEB:?J>=M:<j:ZkB:U:D<Jlj;]a<^:u[:FZ:V:Cb:@:@j;B:?:JCr
:j::HZV:=:\\rS>:;b<:DZ<P??J>A]<bZ=[:NZ>rD;rDAj?:HB::KVG<J:<J?:N:?fDB\\
_K;Nj[Z<JC::A>:ZBx@>ZD:b:DROJ>I]<bZ=;?ZeJ;Njhb::r:@j;B:LZLJ;RF<J;<JE]<
;jNr:l;:;rZWM:<J?H::\\A:ZBP@?j;u`<^:Q;\\rRB:;b<=j;FZ=J:B\\_K;VJ:B\\][<
b_K;`<?J;e<HB<J;A<;b:C:bZWK;Z:NZ:nlb:M[:ZB^WAjHr<<J;>kj>Z:F:RF<ZDNZ>nj
w::?B;D;A:@:;:;R:jS>:?B;h\\eV:Q:jR:J;BF;B:;b:N<:=:@]::L]<JS>:rDh<:F@:_
K;RFD:g;b>::]K::N:H]<JSZj>Z:>Z<jOV_<bZN:?>J:<J::::::::::=J:DJ:<Z::::::
::::jysyA:C:=Z:vYxY;J:jND=sT?>ePZ:::::yayY:^:F:vYxY;B:;:M;s<j:F;HJPh:C
jKEZ:>Z:^:;J;@j:f:<j:Djyyyy=J<JQ@jFf:F`<F:;j<<:Qb:<Z:f:^[<>fM>jgV<n^@v
;sjyyiyAZ:>I[B:<JZyIOwyyyy;TEN[MB:QB:<JMJ@fc[_hb_ds?h_C:;D;MZ:Ny<B:m:;
jyEJK^:;jP@:EZ:^\\<n\\=ncHf:V\\<n\\=>mJf:^<>lLf:V<NuNf:;JDJ_Uk<jCjA`k<
>:Mb:;jyg:^<ww:yFA[J:<jwEJbpyQ@:;B:[V:<Z<>Z<B>aTXU]fS:CB:f?=j>>:_c<B:=
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{TEXT -1 89 "We can let MAPLE compute the integral internally. Run the
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g limit:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {OLE 1 
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0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "You should  verify this result  by
 applying L'Hospital's Rule on the original problem." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 
"Student Problem Set:" }}{PARA 0 "" 0 "" {TEXT -1 117 "The following p
roblems are for you to work out by hand and are representative of the \+
examples discussed in this lab. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 38 "#1. Find a Maclaurin Series for cos x." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "#2. Use the Mac
laurin series for " }{XPPEDIT 18 0 "e^x;" "6#)%\"eG%\"xG" }{TEXT -1 
56 " and your answer in #1 to determine the following limit:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {OLE 1 5128 1 "[xm]Br=WfoR
rB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:I:K:wAyA:::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=?Z:VZHQ:>:;`:Z@[::JZHmPYyZIL
\\Aj;J:`:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HYLkNG>::
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::::::V:;J<>:;:wyyN::wyyyq<J=j=B:K:M:wAQ:S:UJ:n;v;;JBB:]:_J:V<^<f<n<;j
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::::::::::::::::::::::::F:D:<::::::::::=Jyyy;d:yayYZHQ:>:;`:Z@o<fc[_hb
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_J>vGY>LYBvB<OsKbLyDB:cYK:B>H^;kAQ^dZID:<[:_X=tYKLBxjZ:xo\\RJj:@Z=NZ:J
;F:=j;F:CJ:dj:<:iJ:@Z<N:S]:Njc;U]EFZ:B:AN:D:_VBW_B@`:R:=J?:=B:>kkJ<N<N
:AM;<J;y@VHVJ<:<J>K<V?Q[=L;b>B:P@l[:V;=J:DJ:HRv>Z:>:V[:B:G;Sjysy=J`>Z:
>:MB:NYB::?C:CZ:f_;N:[B:<jw?<IJ:>:sO:B:=b:?bBaTXaEWEUU^Z:jPV:CB:>x;F:A
J:<J::::::::::::::::::::::::::::::::::::::::::::::::3:" }{TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 52 "#3. Verify your result from #2 by L'Hospital's \+
Rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 45 "#4. Derive the Maclaurin Series for xcos(
2x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 55 "#5. Approximate the following integral to
 within 0.001." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {OLE 1 4616 1 "[xm]Br=WfoRrB:::wk;ny
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::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:I:wAyA:::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::1:" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 7 "#6. If " }{TEXT 300 4 "  f(" }{TEXT -1 5 "x) = " }{XPPEDIT 18 0 
"e^(x^2);" "6#)%\"eG*$%\"xG\"\"#" }{TEXT -1 14 ", show that   " }
{XPPEDIT 18 0 "f^(2*n);" "6#)%\"fG*&\"\"#\"\"\"%\"nGF'" }{TEXT -1 8 "(
0)  =  " }{XPPEDIT 18 0 "(2*n)!/n!;" "6#*&-%*factorialG6#*&\"\"#\"\"\"
%\"nGF)F)-F%6#F*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "
Conclusions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }{TEXT 259 99 "You have now completed the lab for Taylor Seri
es. From this lab you should have seen the following:" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "1. It is often possibl
e to represent a function by way of an infinite series." }}{PARA 0 "" 
0 "" {TEXT -1 177 "2. Near a specified point we can tell the value of \+
a function just by knowing the value of the function at the specified \+
point and     all its derivative's values at that point." }}{PARA 0 "
" 0 "" {TEXT -1 75 "3. Some approximating polynomials are only converg
ent in certain intervals." }}{PARA 0 "" 0 "" {TEXT -1 94 "4. We can us
e a series to approximate the value of a difficult integral to a desir
ed accuracy." }}{PARA 0 "" 0 "" {TEXT -1 77 "5. We can evaluate limits
 by substituting appropriate series representations." }}}}{MARK "0 2 1
" 19 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }