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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 261 48 "MAT1154 - Equa\347\365e
s Diferenciais e de Diferen\347as " }}{PARA 256 "" 0 "" {TEXT 262 7 "S
\351ries." }}{PARA 256 "" 0 "" {TEXT -1 3 "por" }}{PARA 256 "" 0 "" 
{TEXT -1 17 "George Svetlichny" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Somando s\351ries" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 60 "O Maple conhece algumas s\351ries e \351 \+
capaz de caluclar a soma." }}{PARA 0 "" 0 "" {TEXT -1 47 "O seguinte p
rocedimento tomo um termo gen\351rico " }{TEXT 256 2 "T " }{TEXT -1 
13 "(por exemplo " }{XPPEDIT 257 0 "x^n/n;" "6#*&)%\"xG%\"nG\"\"\"F&!
\"\"" }{TEXT -1 85 ") , escreve a s\351rie com este termo gen\351rico,
 e depois a soma desta s\351rie a partir de " }{TEXT 258 3 "n=q" }
{TEXT -1 31 ", quando \351 capaz de calcul\341-la." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 23 "ser:=proc(T,q) local s:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "s:=Sum(T,n=q..infinity): print(s):" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "print(value(s)): end proc:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 22 "Agora alguns exemplos:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "ser(x^n,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "ser(x^n/n!,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "s
er(1/n^2,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ser(x^n/n,1
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "A pr\363xima \351 a s\351ri
e " }{TEXT 259 9 "harm\364nica" }{TEXT -1 71 " de reciprocas dos intei
ros sucessivos. Sabemos que esta s\351rie diverge." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 11 "ser(1/n,1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 95 "Se por\351m somarmos as reciprocas de inteiros sucessivos
 com sinais alternados, a s\351rie converge." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "ser(-(-1)^n/n,1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 64 "Algumas s\351ries s\363 se consegue somar utilizando fun
\347\365es especias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ser
(1/(1+n^2),0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Aqui aparece a \+
fun\347\343o especial " }{XPPEDIT 260 0 "Psi;" "6#%$PsiG" }{TEXT -1 
38 ". Vamos avaliar numeicamente a s\351rie. " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "evalf(sum(1/(1+n^2),n=0..infinity));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ser(1/n^3,1);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 48 "Aqui aparece a fun\347\343o especial zeta. O valor
 de " }{XPPEDIT 18 0 "Zeta(3);" "6#-%%ZetaG6#\"\"$" }{TEXT -1 4 " \351
: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(Zeta(3));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Algumas s\351ries n\343o podem ser
 somadas em forma expl\355cita:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "ser(sin(n)/n^2,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "O \+
Maple n\343o conseguiu avaliar esta somma apesar da s\351rie ser conve
rgente (Porque?). Podemos fazer uma estimativa num\351rica, somando os
 primeiros 100 termos por exemplo:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "evalf(sum(sin(n)/n,n=1..100));" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 25 "Ou, digamos, 1000 termos:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 31 "evalf(sum(sin(n)/n,n=1..1000));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 
"Expans\343o em s\351ries de Taylor" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 10 "O comando " }{TEXT 263 13 "taylor(f,a,b)" }{TEXT -1 9 " calcula
 " }{TEXT 266 1 "b" }{TEXT -1 40 " termos da s\351rie de Tylor de uma \+
fun\347\343o " }{TEXT 265 2 "f " }{TEXT -1 3 "em " }{TEXT 264 0 "" }
{TEXT -1 15 "torno do ponto " }{TEXT 267 2 "a." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 41 "O seguinte procedimento imprime a fun\347\343o " }
{TEXT 268 1 "f" }{TEXT -1 12 " seguido de " }{TEXT 269 14 "taylor(f,a,
b)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "tay:=proc(f,a,b)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(f):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "print(taylor(f,x=a,b)): end proc:" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 22 "Agora alguns exemplos:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 16 "tay(exp(x),0,5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "tay(exp(x),1,5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "tay(sin(x),0,9);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "tay(sin(x),2,9);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "tay(1/(1-x),0,8);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 139 "Se a fun\347\343o for um polin\364mio, ent\343o a s\351rie de \+
Taylor com o n\372mero de termos igual ao grau do polin\364mio mais um
, soma ao pr\363prio polin\364mio." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "tay(31+10*x-7*x^2+2*x^3+1*x^4,0,5);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 136 "A expans\343o em torno de um outro ponto com o
 mesmo n\372mero de termos reproduz o mesmo polin\364mio, s\363 agor e
scrito com soma de pot\352ncias de " }{TEXT 270 6 "(x-a)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "tay(31+10*x-7*x^2+2*x^3+1*x^4,-1,5)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "
" 0 "" {TEXT -1 35 "S\351ries de Taylor como aproxima\347\365es." }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Nesta se\347\343o vamos verificar
 at\351 que ponto as somas de um determinado n\372mero de termos da s
\351rie de Taylor aproxima a fun\347\343o dada:" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 41 "O seguinte procedimento expande a fun\347\343o " }
{TEXT 271 1 "f" }{TEXT -1 40 " numa s\351rie de Taylor em torno do pon
to " }{TEXT 272 1 "a" }{TEXT -1 94 " e depois plota a fun\347\343o jun
to com as aproxima\347\365es usando de um a 4 termos na s\351rie de Ta
ylor:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "paprox:=proc(f,a,c
,d,r,s)local f0,f1,f2,f3,f4:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f0:=
f;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f1:=convert(taylor(f,x=a,1),p
olynom);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f2:=convert(taylor(f,x=
a,2),polynom);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f3:=convert(taylo
r(f,x=a,3),polynom);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f4:=convert
(taylor(f,x=a,4),polynom);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "plot
([f0,f1,f2,f3,f4],x=c..d,y=r..s,color=[red,blue,green, yellow,brown],l
egend=[\"f(x)\",\"1 termo\",\"2 termos\", \"3termos\",\"4 termos\"]);
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 9 "A fun\347ao " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%
\"xG" }{TEXT -1 19 " em torno do ponto " }{TEXT 273 6 "x = 1." }{TEXT 
-1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "paprox(sin(x),1,
-2*Pi,2*Pi,-3,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "A fun\347ao \+
" }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 19 " em torno \+
do ponto " }{TEXT 274 6 "x = 0." }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "paprox(exp(x),0,-2,2,0,6);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 12 "O polin\364mio " }{XPPEDIT 18 0 "(1-x^2)^2;" "6#*
$,&\"\"\"F%*$%\"xG\"\"#!\"\"F(" }{TEXT -1 19 " em torno do ponto " }
{TEXT 275 7 "x =0.5." }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "paprox((1-x^2)^2,0.5,-2,2,-2,6);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 87 "Note que se forem incluidos cinco termos, a expans
\343o de Taylor repoduziria o polin\364mio." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 162 "O pr\363ximo procedimento \351 parecido, s\363 que ago
ra \351 incluida uma aproxima\347\343o com cinco termos, e cada aproxi
ma\347\343o \351 plotado separadamente. Usamos os memos exemplos." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "psaprox:=proc(f,a,c,d,r,s) l
ocal f0,f1,f2,f3,f4,f5: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f0:=f;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f1:=convert(taylor(f,x=a,1),polyn
om):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f2:=convert(taylor(f,x=a,2)
,polynom):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f3:=convert(taylor(f,
x=a,3),polynom):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f4:=convert(tay
lor(f,x=a,4),polynom): f5:=convert(taylor(f,x=a,5),polynom):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 89 "print(plot([f0,f1],x=c..d,y=r..s,color=[r
ed,blue],legend=[convert(f,string),\"1 termo\"])):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 373 "print(plot([f0,f2],x=c..d,y=r..s,color=[red,blue],
legend=[convert(f,string),\"2 termos\"])): print(plot([f0,f3],x=c..d,y
=r..s,color=[red,blue],legend=[convert(f,string),\"3 termos\"])): prin
t(plot([f0,f4],x=c..d,y=r..s,color=[red,blue],legend=[convert(f,string
),\"4 termos\"])): print(plot([f0,f5],x=c..d,y=r..s,color=[red,blue],l
egend=[convert(f,string),\"5 termos\"])): end proc:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 9 "A fun\347ao " }{XPPEDIT 18 0 "sin(x);" "6#-%$sin
G6#%\"xG" }{TEXT -1 19 " em torno do ponto " }{TEXT 276 6 "x = 1." }
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "psaprox(si
n(x),1,-2*Pi,2*Pi,-3,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "A fun
\347ao " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 19 " em
 torno do ponto " }{TEXT 277 6 "x = 0." }{TEXT -1 1 " " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "psaprox(exp(x),0,-2,2,0,6);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "O polin\364mio " }{XPPEDIT 18 0 "(
1-x^2)^2;" "6#*$,&\"\"\"F%*$%\"xG\"\"#!\"\"F(" }{TEXT -1 19 " em torno
 do ponto " }{TEXT 278 7 "x =0.5." }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 33 "psaprox((1-x^2)^2,0.5,-2,2,-2,6);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Na \372ltima plotagem a aproxima
\347\343o coincide com a pr\363pria fun\347\343o." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 197 "Longe do ponto de expan\347\343o a aproxima\347
\343o por s\351rie de Taylor de modo geral \351 bastante ruim. Veja os
 seguintes exemplos comparando uma fun\347\343o com a soma dos primeir
os seis termos da s\351rie de Taylor:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 9 "A fun\347\343o " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"x
G" }{TEXT -1 14 " no intervalo " }{TEXT 279 6 "(-6,6)" }{TEXT -1 1 ".
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "plot([sin(x),convert(t
aylor(sin(x),x=0,6),polynom)],x=-6..6,color=[red,blue],legend=[\"sin(x
)\",\"Taylor com 6 termos\"]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "
A fun\347\343o " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT 
-1 14 " no intervalo " }{TEXT 280 9 "(-15,10)." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 120 "plot([exp(x),convert(taylor(exp(x),x=0,6),pol
ynom)],x=-15..10,color=[red,blue],legend=[\"exp(x)\",\"Taylor com 6 te
rmos\"]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "A fun\347\343o " }
{XPPEDIT 281 0 "ln(x);" "6#-%#lnG6#%\"xG" }{TEXT -1 14 " no intervalo \+
" }{TEXT 282 5 "(0,4)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 114 "plot([ln(x),convert(taylor(ln(x),x=1,6),polynom)],x=
0..4,color=[red,blue],legend=[\"ln(x)\",\"Taylor com 6 termos\"]);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "Para conseguir uma aproxima\347
\343o por polin\364mios melhor em todo um intervalo n\343o se deve usa
r o polin\364mio de Taylor. Por exemplo para se aproximar " }{XPPEDIT 
283 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 14 " no intervalo " }
{TEXT 284 1 "(" }{XPPEDIT 285 0 "-Pi,Pi;" "6$,$%#PiG!\"\"F$" }{TEXT 
286 1 ")" }{TEXT -1 28 " o seguinte polin\364mio cubico" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-15*(Pi^2-21)*x/(2*Pi^4)+35*(Pi^
2-15)*x^3/(2*Pi^6)" "6#,&**\"#:\"\"\",&*$%#PiG\"\"#F&\"#@!\"\"F&%\"xGF
&*&F*F&*$F)\"\"%F&F,F,**\"#NF&,&*$F)F*F&F%F,F&F-\"\"$*&F*F&*$F)\"\"'F&
F,F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 35 "entre todos os  polin\364mios c\372bicos " }{TEXT 
287 3 "p, " }{TEXT -1 64 "\351 o que minimiza a integral do quadrado d
o erro, isto \351 minimiza" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "int((sin(x)-p(x))^2,x = -Pi .. Pi);" "6#-%$intG6$*$,&-%
$sinG6#%\"xG\"\"\"-%\"pG6#F+!\"\"\"\"#/F+;,$%#PiGF0F5" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "g:=(-15/2*(Pi^2-21)/(Pi^4))*x+(35/2*(Pi^2-15)/(Pi^6))
*x^3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "plot([sin(x),convert(tayl
or(sin(x),x=0,4),polynom),g],x=-Pi..Pi,color=[red,green,blue],legend=[
\"sin(x)\",\"Taylor com 4 termos\",\"Melhor c\372bica\"]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "4" 0 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }