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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart;with(plots):
with(DETools):" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 256 22 "Equa\347\365
es de Diferen\347as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 10 "Somat\363rios" }}{PARA 0 "" 0 "" {TEXT -1 79 "S
omas de termos de uma sequ\352ncia s\343o escritos de forma compacta c
om nota\347\343o de " }{TEXT 257 10 "somat\363rios" }{TEXT -1 35 ". A \+
soma de termos de m at\351 n, com " }{XPPEDIT 18 0 "m <= n;" "6#1%\"mG
%\"nG" }{TEXT -1 49 " , a(m)+a(m+1), ..., a(n), se escreve ent\343o co
mo:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(a(i),i=m..n)" "6#-%$sumG6$
-%\"aG6#%\"iG/F);%\"mG%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 133 "Da mesma forma que a vari\341vel de integra\347\343o de uma in
tegral \351 irrelevante, tanto faz que letra usamos para o \355ndice d
e soma. Ou seja," }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(a(i),i=m..n)=
sum(a(k),k=m..n)" "6#/-%$sumG6$-%\"aG6#%\"iG/F*;%\"mG%\"nG-F%6$-F(6#%
\"kG/F3;F-F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 101 "S\363 va
mos evitar usar a mesma letra para o \355ndice e um de seus extremos, \+
como faz\355amos para integrais." }}{PARA 0 "" 0 "" {TEXT -1 123 "V
\341rias propriedades que voc\352s conhecem para integrais valem para \+
somat\363rios. Algumas com pequenas modifica\347\365es, entretanto." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "
Propriedades de somat\363rios" }}{PARA 0 "" 0 "" {TEXT 260 12 "Lineari
dade:" }}{PARA 0 "" 0 "" {TEXT -1 80 "Somat\363rios s\343o lineares. O
u seja, se c \351 uma constante e a e b sequ\352ncias, temos" }}{PARA 
256 "" 0 "" {XPPEDIT 18 0 "sum(c*a(i),i=m..n)=c*sum(a(i),i=m..n)" "6#/
-%$sumG6$*&%\"cG\"\"\"-%\"aG6#%\"iGF)/F-;%\"mG%\"nG*&F(F)-F%6$-F+6#F-/
F-;F0F1F)" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(a(i)+
b(i),i=m..n)=sum(a(i),i=m..n)+sum(b(i),i=m..n)" "6#/-%$sumG6$,&-%\"aG6
#%\"iG\"\"\"-%\"bG6#F+F,/F+;%\"mG%\"nG,&-F%6$-F)6#F+/F+;F2F3F,-F%6$-F.
6#F+/F+;F2F3F," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Convenci
onamos que o somat\363rio vazio " }{XPPEDIT 18 0 "sum(a(i),i = m+1 .. \+
m) = 0;" "6#/-%$sumG6$-%\"aG6#%\"iG/F*;,&%\"mG\"\"\"F/F/F.\"\"!" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "Com essa conven\347\343
o, tamb\351m vale que" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(a(i),i =
 m .. n) = sum(a(i),i = m .. p)+sum(a(i),i = p+1 .. n);" "6#/-%$sumG6$
-%\"aG6#%\"iG/F*;%\"mG%\"nG,&-F%6$-F(6#F*/F*;F-%\"pG\"\"\"-F%6$-F(6#F*
/F*;,&F6F7F7F7F.F7" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 12 "ao \+
menos se " }{XPPEDIT 18 0 "m<=p" "6#1%\"mG%\"pG" }{TEXT -1 3 " e " }
{XPPEDIT 18 0 "p<=n" "6#1%\"pG%\"nG" }{TEXT -1 189 ". Note que a essa \+
\351 uma varia\347\343o ligeira da regra para integrais, sendo que os \+
limites nas duas somas do lado direito, diferentemente do caso das int
egrais, n\343o s\343o iguais, mas sim p e p+1." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 18 "Mudan\347a de \355ndice:
" }}{PARA 0 "" 0 "" {TEXT -1 185 "Podemos tamb\351m fazer algumas pouc
as mudan\347as de \355ndice nos somat\363rios. Como n\343o podemos alt
erar o n\372mero de parcelas, estamos limitados a shifts ou mudan\347a
 na ordem dos termos. Ou seja:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(a(i),i=0..n)=sum(a(j-r),j=r..n+r)" 
"6#/-%$sumG6$-%\"aG6#%\"iG/F*;\"\"!%\"nG-F%6$-F(6#,&%\"jG\"\"\"%\"rG!
\"\"/F4;F6,&F.F5F6F5" }{TEXT -1 19 ", onde a mudan\347a \351 " }
{XPPEDIT 18 0 "i=j-r" "6#/%\"iG,&%\"jG\"\"\"%\"rG!\"\"" }{TEXT -1 1 ";
" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(a(i),i = 0 .. n) = sum(a(n-j),j
 = 0 .. n);" "6#/-%$sumG6$-%\"aG6#%\"iG/F*;\"\"!%\"nG-F%6$-F(6#,&F.\"
\"\"%\"jG!\"\"/F5;F-F." }{TEXT -1 13 ", onde temos " }{XPPEDIT 18 0 "i
=n-j" "6#/%\"iG,&%\"nG\"\"\"%\"jG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Com algumas adapta
\347\365es isso vale tamb\351m para somat\341rios com limites diferent
es de 0 e n (quais?)." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 34 "Alguns exemplos que sabemos somar." }}
{PARA 0 "" 0 "" {TEXT -1 43 "Usando mudan\347a de \355ndice, podemos n
otar que" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(a(i+1)-a(i),i=0..n)=sum
(a(i+1),i=0..n)-sum(a(i),i=0..n)" "6#/-%$sumG6$,&-%\"aG6#,&%\"iG\"\"\"
F-F-F--F)6#F,!\"\"/F,;\"\"!%\"nG,&-F%6$-F)6#,&F,F-F-F-/F,;F3F4F--F%6$-
F)6#F,/F,;F3F4F0" }{TEXT -1 5 "=\n = " }{XPPEDIT 18 0 "sum(a(i),i = 1 \+
.. n+1)-sum(a(i),i = 0 .. n);" "6#,&-%$sumG6$-%\"aG6#%\"iG/F*;\"\"\",&
%\"nGF-F-F-F--F%6$-F(6#F*/F*;\"\"!F/!\"\"" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "a(n+1)+sum(a(i),i = 1 .. n)-(sum(a(i),i = 1 .. n)+a(0))
;" "6#,(-%\"aG6#,&%\"nG\"\"\"F)F)F)-%$sumG6$-F%6#%\"iG/F/;F)F(F),&-F+6
$-F%6#F//F/;F)F(F)-F%6#\"\"!F)!\"\"" }{TEXT -1 6 " =\n=  " }{XPPEDIT 
18 0 "a(n+1)-a(0)" "6#,&-%\"aG6#,&%\"nG\"\"\"F)F)F)-F%6#\"\"!!\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 78 "Essa soma \351 chamada de soma telesc\363pica, pois todos
 os termos intermedi\341rios (" }{XPPEDIT 18 0 "1<=i, i<=n" "6$1\"\"\"
%\"iG1F%%\"nG" }{TEXT -1 14 ") se cancelam." }}{PARA 0 "" 0 "" {TEXT 
-1 53 "Usando isso, podemos calcular v\341rias somas cl\341ssicas." }}
{PARA 0 "" 0 "" {TEXT -1 12 "Por exemplo:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Para " }{XPPEDIT 262 0 "a(i)=i^2" "
6#/-%\"aG6#%\"iG*$F'\"\"#" }{TEXT 263 1 "," }{TEXT -1 39 " recuperamos
 a soma de uma p.a., j\341 que" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum((
i+1)^2-i^2,i=0..n-1)=n^2" "6#/-%$sumG6$,&*$,&%\"iG\"\"\"F+F+\"\"#F+*$F
*F,!\"\"/F*;\"\"!,&%\"nGF+F+F.*$F3F," }{TEXT -1 51 ", mas tamb\351m, e
xpandindo o somando e simplificando:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 
"sum((i+1)^2-i^2,i=0..n-1)=sum(2*i+1,i=0..n-1)" "6#/-%$sumG6$,&*$,&%\"
iG\"\"\"F+F+\"\"#F+*$F*F,!\"\"/F*;\"\"!,&%\"nGF+F+F.-F%6$,&*&F,F+F*F+F
+F+F+/F*;F1,&F3F+F+F." }{TEXT -1 1 "=" }{XPPEDIT 18 0 "2*sum(i,i=0..n-
1)+sum(1,i=0..n-1)" "6#,&*&\"\"#\"\"\"-%$sumG6$%\"iG/F*;\"\"!,&%\"nGF&
F&!\"\"F&F&-F(6$F&/F*;F-,&F/F&F&F0F&" }{TEXT -1 30 ", que d\341, como \+
era de esperar:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(i,i=0..n-1)=(n^2
-n)/2" "6#/-%$sumG6$%\"iG/F';\"\"!,&%\"nG\"\"\"F-!\"\"*&,&*$F,\"\"#F-F
,F.F-F2F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 7 "Usando " }{XPPEDIT 264 0 "a(i)=i^3" "6#/-%\"aG6#%\"
iG*$F'\"\"$" }{TEXT -1 66 " e aplicando o mesmo truque podemos calcula
r a soma dos quadrados:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum((i+1)^3-
i^3,i = 0 .. n-1) = n^3;" "6#/-%$sumG6$,&*$,&%\"iG\"\"\"F+F+\"\"$F+*$F
*F,!\"\"/F*;\"\"!,&%\"nGF+F+F.*$F3F," }{TEXT -1 10 " e tamb\351m " }
{XPPEDIT 18 0 "sum((i+1)^3-i^3,i = 0 .. n-1) = sum(3*i^2+3*i+1,i = 0 .
. n-1);" "6#/-%$sumG6$,&*$,&%\"iG\"\"\"F+F+\"\"$F+*$F*F,!\"\"/F*;\"\"!
,&%\"nGF+F+F.-F%6$,(*&F,F+*$F*\"\"#F+F+*&F,F+F*F+F+F+F+/F*;F1,&F3F+F+F
." }{TEXT -1 10 ", ou seja," }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(i^2,
i=0..n-1)=n^3/3-sum(i,i=0..n-1)-n/3" "6#/-%$sumG6$*$%\"iG\"\"#/F(;\"\"
!,&%\"nG\"\"\"F/!\"\",(*&F.\"\"$F3F0F/-F%6$F(/F(;F,,&F.F/F/F0F0*&F.F/F
3F0F0" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "n^3/3-n^2/2+n/2-n/3" "6#,**&%
\"nG\"\"$F&!\"\"\"\"\"*&F%\"\"#F*F'F'*&F%F(F*F'F(*&F%F(F&F'F'" }{TEXT 
-1 3 " = " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sum(i^2,i = 0 .
. n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"$F&*$)%\"nGF
'F&F&F&*&#F&\"\"#F&*$)F*F-F&F&!\"\"*&#F&\"\"'F&F*F&F&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 95 "Podemos prosseguir assim e calcular uma a
 uma soma de cubos, pot\352ncias quartas e a\355 por diante." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Outro exemplo, \+
que tamb\351m podemos calcular usando somas telesc\363picas \351" }}
{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(k*2^k,k=0..n-1)" "6#-%$sumG6$*&%\"k
G\"\"\")\"\"#F'F(/F';\"\"!,&%\"nGF(F(!\"\"" }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 24 "Mais uma vez, agora com " }{XPPEDIT 265 0 "a(k)=k
*2^k" "6#/-%\"aG6#%\"kG*&F'\"\"\")\"\"#F'F)" }{TEXT -1 7 ", temos" }}
{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum((k+1)*2^(k+1)-k*2^k,k=0..n-1)=n*2^n
" "6#/-%$sumG6$,&*&,&%\"kG\"\"\"F+F+F+)\"\"#,&F*F+F+F+F+F+*&F*F+)F-F*F
+!\"\"/F*;\"\"!,&%\"nGF+F+F1*&F6F+)F-F6F+" }{TEXT -1 10 ", e tamb\351m
" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum((k+1)*2^(k+1)-k*2^k,k=0..n-1)=s
um((k)*2^(k),k=0..n-1)-sum(2^(k+1),k=0..n-1)" "6#/-%$sumG6$,&*&,&%\"kG
\"\"\"F+F+F+)\"\"#,&F*F+F+F+F+F+*&F*F+)F-F*F+!\"\"/F*;\"\"!,&%\"nGF+F+
F1,&-F%6$*&F*F+)F-F*F+/F*;F4,&F6F+F+F1F+-F%6$)F-,&F*F+F+F+/F*;F4,&F6F+
F+F1F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 63 "A \372ltima som
a acima \351 a soma de uma p.g., que conhecemos, donde" }}{PARA 0 "" 
0 "" {XPPEDIT 18 0 "sum(k*2^k,k = 0 .. n-1) = n*2^n-(2^(n+1)-2);" "6#/
-%$sumG6$*&%\"kG\"\"\")\"\"#F(F)/F(;\"\"!,&%\"nGF)F)!\"\",&*&F0F))F+F0
F)F),&)F+,&F0F)F)F)F)F+F1F1" }{TEXT -1 1 "=" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 20 "sum(k*2^k,k=0..n-1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,(*&%\"nG\"\"\")\"\"#F%F&F&*&F(F&F'F&!\"\"F(F&" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 133 "H\341 um outro truque que podemos usar para ca
lcular essa soma que \351 \372til em outras situa\347\365es, e que tam
b\351m usa c\341lculo mais fortemente:" }}{PARA 0 "" 0 "" {TEXT -1 95 
"Em vez de considerar o problema original, vamos trocar 2 por x, um pa
r\342metro e definir a fun\347\343o" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "
g(x)=sum(x^k,k=0..n-1)" "6#/-%\"gG6#%\"xG-%$sumG6$)F'%\"kG/F,;\"\"!,&%
\"nG\"\"\"F2!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 91 "Como
 temos uma forma fechada para a soma de uma p.g., podemos calcular g'(
x) de dois modos:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(g(x),x)=sum(k
*x^(k-1),k=0..n-1)" "6#/-%%diffG6$-%\"gG6#%\"xGF*-%$sumG6$*&%\"kG\"\"
\")F*,&F/F0F0!\"\"F0/F/;\"\"!,&%\"nGF0F0F3" }{TEXT -1 2 " =" }
{XPPEDIT 18 0 " x^(-1)*sum(k*x^(k),k=0..n-1)" "6#*&)%\"xG,$\"\"\"!\"\"
F'-%$sumG6$*&%\"kGF')F%F-F'/F-;\"\"!,&%\"nGF'F'F(F'" }}{PARA 0 "" 0 "
" {TEXT -1 1 "e" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(g(x),x)=diff((x
^n-1)/(x-1),x)" "6#/-%%diffG6$-%\"gG6#%\"xGF*-F%6$*&,&)F*%\"nG\"\"\"F1
!\"\"F1,&F*F1F1F2F2F*" }{TEXT -1 1 "=" }{XPPEDIT 18 0 "x^n*n/x/(x-1)-(
x^n-1)/(x-1)^2;" "6#,&**)%\"xG%\"nG\"\"\"F'F(F&!\"\",&F&F(F(F)F)F(*&,&
)F&F'F(F(F)F(*$,&F&F(F(F)\"\"#F)F)" }}{PARA 0 "" 0 "" {TEXT -1 31 "e i
gualando, obtemos finalmente" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(k*x
^k,k=0..n-1)=x*(x^n*n/x/(x-1)-(x^n-1)/(x-1)^2)" "6#/-%$sumG6$*&%\"kG\"
\"\")%\"xGF(F)/F(;\"\"!,&%\"nGF)F)!\"\"*&F+F),&**)F+F0F)F0F)F+F1,&F+F)
F)F1F1F)*&,&)F+F0F)F)F1F)*$,&F+F)F)F1\"\"#F1F1F)" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 101 "Esse truque pode ser iterado v\341rias v
ezes, para obter, depois de alguma \341lgebra, por exemplo soma de " }
{XPPEDIT 18 0 "k^2*2^k, k^3*2^k" "6$*&%\"kG\"\"#)F%F$\"\"\"*&F$\"\"$)F
%F$F'" }{TEXT -1 14 " e por a\355 vai." }}}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Usando somas telesc\363pic
as para resolver uma ED" }}{PARA 0 "" 0 "" {TEXT -1 71 "usando o que v
imos sobre somas telesc\363picas podemos facilmente resolver" }}{PARA 
0 "" 0 "" {XPPEDIT 18 0 "y(k+1)-y(k)=b(k)" "6#/,&-%\"yG6#,&%\"kG\"\"\"
F*F*F*-F&6#F)!\"\"-%\"bG6#F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 35 "De fato, somando para k de 0 a n-1:" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "sum(y(k+1)-y(k),k=0..n-1)=sum(b(k),k=0..n-1)" "6#/-%$su
mG6$,&-%\"yG6#,&%\"kG\"\"\"F-F-F--F)6#F,!\"\"/F,;\"\"!,&%\"nGF-F-F0-F%
6$-%\"bG6#F,/F,;F3,&F5F-F-F0" }{TEXT -1 9 ", ou seja" }}{PARA 0 "" 0 "
" {XPPEDIT 18 0 "y(n)-y(0)=sum(b(k),k=0..n-1)" "6#/,&-%\"yG6#%\"nG\"\"
\"-F&6#\"\"!!\"\"-%$sumG6$-%\"bG6#%\"kG/F4;F,,&F(F)F)F-" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 36 "Se em vez de saber y(0) soub\351ssem
os " }{XPPEDIT 18 0 "y(n[0])" "6#-%\"yG6#&%\"nG6#\"\"!" }{TEXT -1 32 "
, bastaria somar para k indo de " }{XPPEDIT 18 0 "n[0]" "6#&%\"nG6#\"
\"!" }{TEXT -1 3 " a " }{XPPEDIT 18 0 "n-1" "6#,&%\"nG\"\"\"F%!\"\"" }
{TEXT -1 11 " e ter\355amos" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "y(n)-y(n
[0]) = sum(b(k),k = n[0] .. n-1);" "6#/,&-%\"yG6#%\"nG\"\"\"-F&6#&F(6#
\"\"!!\"\"-%$sumG6$-%\"bG6#%\"kG/F6;&F(6#F.,&F(F)F)F/" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
15 "Soma por partes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 14 "Se definirmos " }{XPPEDIT 18 0 "dz(k)=z(k+1)-z(k)" "6#/
-%#dzG6#%\"kG,&-%\"zG6#,&F'\"\"\"F-F-F--F*6#F'!\"\"" }{TEXT -1 31 ", a
 soma telesc\363pica se reduz a" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(
dz(k),k=0..n-1)=z(n)-z(0)" "6#/-%$sumG6$-%#dzG6#%\"kG/F*;\"\"!,&%\"nG
\"\"\"F0!\"\",&-%\"zG6#F/F0-F46#F-F1" }{TEXT -1 22 ", que lembra basta
nte " }{XPPEDIT 18 0 "int(1,z=0..n)=z(n)-z(0)" "6#/-%$intG6$\"\"\"/%\"
zG;\"\"!%\"nG,&-F)6#F,F'-F)6#F+!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 61 "Se z for dada pelo produto termo a termo de duas sequ
\352ncias: " }{XPPEDIT 18 0 "z(n)=x(n)*y(n)" "6#/-%\"zG6#%\"nG*&-%\"xG
6#F'\"\"\"-%\"yG6#F'F," }{TEXT -1 17 ", temos ent\343o que" }}{PARA 0 
"" 0 "" {XPPEDIT 18 0 "dxy(n)=x(n+1)*y(n+1)-x(n)*y(n)" "6#/-%$dxyG6#%
\"nG,&*&-%\"xG6#,&F'\"\"\"F.F.F.-%\"yG6#,&F'F.F.F.F.F.*&-F+6#F'F.-F06#
F'F.!\"\"" }{TEXT -1 2 " =" }{XPPEDIT 18 0 "x(n+1)*(y(n+1)-y(n))+y(n)*
x(n+1)-x(n)*y(n)=x(n+1)*(y(n+1)-y(n))+y(n)*(x(n+1)-x(n))" "6#/,(*&-%\"
xG6#,&%\"nG\"\"\"F+F+F+,&-%\"yG6#,&F*F+F+F+F+-F.6#F*!\"\"F+F+*&-F.6#F*
F+-F'6#,&F*F+F+F+F+F+*&-F'6#F*F+-F.6#F*F+F3,&*&-F'6#,&F*F+F+F+F+,&-F.6
#,&F*F+F+F+F+-F.6#F*F3F+F+*&-F.6#F*F+,&-F'6#,&F*F+F+F+F+-F'6#F*F3F+F+
" }{TEXT -1 4 " =\n=" }{XPPEDIT 18 0 "x(n+1)*dy(n)+y(n)*dx(n);" "6#,&*
&-%\"xG6#,&%\"nG\"\"\"F*F*F*-%#dyG6#F)F*F**&-%\"yG6#F)F*-%#dxG6#F)F*F*
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "Ou seja, para cada k:
" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "y(k)*dx(k)=dxy(k)-x(k+1)*dy(k)" "6#
/*&-%\"yG6#%\"kG\"\"\"-%#dxG6#F(F),&-%$dxyG6#F(F)*&-%\"xG6#,&F(F)F)F)F
)-%#dyG6#F(F)!\"\"" }{TEXT -1 48 ", que podemos agora somar com k indo
 de 0 a n-1:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(y(k)*dx(k),k = 0 ..
 n-1) = x(n)*y(n)-x(0)*y(0)-sum(x(k+1)*dy(k),k = 0 .. n-1);" "6#/-%$su
mG6$*&-%\"yG6#%\"kG\"\"\"-%#dxG6#F+F,/F+;\"\"!,&%\"nGF,F,!\"\",(*&-%\"
xG6#%\"nGF,-%\"yG6#F;F,F,*&-F96#F2F,-F=6#F2F,F5-%$sumG6$*&-F96#,&%\"kG
F,F,F,F,-%#dyG6#FKF,/FK;F2,&F;F,F,F5F5" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 12 "Compare com " }{XPPEDIT 18 0 "int(u,v = 0 .. n) = u(
n)*v(n)-u(0)*v(0)-int(v,u = 0 .. n-1);" "6#/-%$intG6$%\"uG/%\"vG;\"\"!
%\"nG,(*&-F'6#F,\"\"\"-F)6#F,F1F1*&-F'6#F+F1-F)6#F+F1!\"\"-F%6$F)/F';F
+,&F,F1F1F9F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 97 "Essa t
\351cnica serve de motiva\347\343o para o uso de somas telesc\363picas
 ao resolver soma de termos como " }{XPPEDIT 18 0 "k*2^k" "6#*&%\"kG\"
\"\")\"\"#F$F%" }{TEXT -1 14 ", por exemplo." }}}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 11 "Produt\363rios" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 65 "Produtos de termos de uma sequ\352ncia a(
m) a(m+1) ... a(n), quando " }{XPPEDIT 18 0 "m<=n" "6#1%\"mG%\"nG" }
{TEXT -1 50 ", podem ser escritos em nota\347\343o de produt\363rio co
mo" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "product(a(i),i=m..n)" "6#-%(produ
ctG6$-%\"aG6#%\"iG/F);%\"mG%\"nG" }{TEXT -1 47 ", sendo que convencion
amos que o produto vazio " }{XPPEDIT 18 0 "product(a(i),i=m+1..m)=1" "
6#/-%(productG6$-%\"aG6#%\"iG/F*;,&%\"mG\"\"\"F/F/F.F/" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 42 "Propriedades b\341sicas dos produt
\363rios s\343o (" }{XPPEDIT 18 0 "m<=p" "6#1%\"mG%\"pG" }{TEXT -1 2 "
, " }{XPPEDIT 18 0 "p<=n" "6#1%\"pG%\"nG" }{TEXT -1 2 "):" }}{PARA 0 "
" 0 "" {XPPEDIT 18 0 "product(a(i),i=m..n)=product(a(k),k=m..n)" "6#/-
%(productG6$-%\"aG6#%\"iG/F*;%\"mG%\"nG-F%6$-F(6#%\"kG/F3;F-F." }
{TEXT -1 59 "  (resultado n\343o depende do nome dado ao \355ndice do \+
produto)" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "product(a(i)*b(i),i=m..n)=p
roduct(a(i),i=m..n)*product(b(i),i=m..n)" "6#/-%(productG6$*&-%\"aG6#%
\"iG\"\"\"-%\"bG6#F+F,/F+;%\"mG%\"nG*&-F%6$-F)6#F+/F+;F2F3F,-F%6$-F.6#
F+/F+;F2F3F," }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "product(a(i),i=m..n)=pr
oduct(a(i),i=m..p)*product(a(i),i=p+1..n)" "6#/-%(productG6$-%\"aG6#%
\"iG/F*;%\"mG%\"nG*&-F%6$-F(6#F*/F*;F-%\"pG\"\"\"-F%6$-F(6#F*/F*;,&F6F
7F7F7F.F7" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "product(alpha,i=m..n)=alph
a^(n-m+1)" "6#/-%(productG6$%&alphaG/%\"iG;%\"mG%\"nG)F',(F,\"\"\"F+!
\"\"F/F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
49 "Essa nota\347\343o \351 \372til por exemplo ao resolver a ED:" }}
{PARA 256 "" 0 "" {XPPEDIT 18 0 "y(i+1)=a(i)*y(i)" "6#/-%\"yG6#,&%\"iG
\"\"\"F)F)*&-%\"aG6#F(F)-F%6#F(F)" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "i
>=0" "6#1\"\"!%\"iG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 81 "De
 fato, tomando o produt\363rio de ambos lados, com i variando de 0 at
\351 n-1 obtemos" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "product(y(i+1),i=
0..n-1)=product(a(i),i=0..n-1)*product(y(i),i=0..n-1)" "6#/-%(productG
6$-%\"yG6#,&%\"iG\"\"\"F,F,/F+;\"\"!,&%\"nGF,F,!\"\"*&-F%6$-%\"aG6#F+/
F+;F/,&F1F,F,F2F,-F%6$-F(6#F+/F+;F/,&F1F,F,F2F," }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 99 "Mas o produto do lado esquerdo \351 o pro
duto dos termos y(0), y(1), ..., y(n) e pode ser escrito como" }
{XPPEDIT 18 0 " product(y(i),i=1..n)" "6#-%(productG6$-%\"yG6#%\"iG/F)
;\"\"\"%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "A equa
\347\343o fica portanto" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "product(y(i)
,i = 1 .. n) = product(a(i),i = 0 .. n-1)*product(y(i),i = 0 .. n-1);
" "6#/-%(productG6$-%\"yG6#%\"iG/F*;\"\"\"%\"nG*&-F%6$-%\"aG6#F*/F*;\"
\"!,&F.F-F-!\"\"F--F%6$-F(6#F*/F*;F7,&F.F-F-F9F-" }{TEXT -1 50 ", ou s
eja, simplificando os dois produt\363rios em y:" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "y(n) = product(a(i),i = 0 .. n-1)*y(0);" "6#/-%\"yG6#%
\"nG*&-%(productG6$-%\"aG6#%\"iG/F/;\"\"!,&F'\"\"\"F4!\"\"F4-F%6#F2F4
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Note que esse argumen
to " }{TEXT 266 15 "s\363 \351 v\341lido se " }{XPPEDIT 267 0 "a(i)<>0
" "6#0-%\"aG6#%\"iG\"\"!" }{TEXT -1 15 ", para todo i, " }{XPPEDIT 18 
0 "i>=0" "6#1\"\"!%\"iG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
52 "Al\351m do produt\363rio de uma constante, sabemos tamb\351m:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Product(alpha^k,k=0..n-1)=a
lpha^Sum(k,k=0..n-1),Product(alpha^k,k=0..n-1)=product(alpha^k,k=0..n-
1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%(ProductG6$)%&alphaG%\"kG/F)
;\"\"!,&%\"nG\"\"\"F/!\"\")F(-%$SumG6$F)F*/F$)F(,&*&#F/\"\"#F/F.F/F0*&
#F/F:F/*$)F.F:F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Pro
duct(k,k=1..n)=convert(product(k,k=1..n),factorial);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(ProductG6
$%\"kG/F';\"\"\"%\"nG-%*factorialG6#F+" }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 13 "F\363rmula para " }{XPPEDIT 18 0 "y(n+1)=a(n)*y(n)+b(n)" 
"6#/-%\"yG6#,&%\"nG\"\"\"F)F),&*&-%\"aG6#F(F)-F%6#F(F)F)-%\"bG6#F(F)" 
}{TEXT -1 1 "." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Uma vez que j
\341 sabemos usar somat\363rios e produt\363rios, bem como sabemos som
ar somas telesc\363picas, podemos falar finalmente da f\363rmula para \+
resolver a ED dada por" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {XPPEDIT 18 0 "y(n+1)-a(n)*y(n)=b(n)" "6#/,&-%\"yG6#,&%\"nG\"\"
\"F*F*F**&-%\"aG6#F)F*-F&6#F)F*!\"\"-%\"bG6#F)" }{TEXT -1 6 ", com " }
{XPPEDIT 18 0 "y(0)" "6#-%\"yG6#\"\"!" }{TEXT -1 6 " dado." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Multiplicando a E
D por um " }{TEXT 258 13 "fator somante" }{TEXT -1 15 " a determinar, \+
" }{XPPEDIT 18 0 "m(n+1)" "6#-%\"mG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 26 "
, obtemos a ED equivalente" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {XPPEDIT 18 0 "m(n+1)*y(n+1)-a(n)*m(n+1)*y(n)=m(n+1)*b(n)
" "6#/,&*&-%\"mG6#,&%\"nG\"\"\"F+F+F+-%\"yG6#,&F*F+F+F+F+F+*(-%\"aG6#F
*F+-F'6#,&F*F+F+F+F+-F-6#F*F+!\"\"*&-F'6#,&F*F+F+F+F+-%\"bG6#F*F+" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 15 "Ora, definindo " }{XPPEDIT 18 0 "x(n)=m(n)*b(n)" "6#/-%\"
xG6#%\"nG*&-%\"mG6#F'\"\"\"-%\"bG6#F'F," }{TEXT -1 3 " e " }{XPPEDIT 
18 0 "beta(n)=m(n+1)*b(n)" "6#/-%%betaG6#%\"nG*&-%\"mG6#,&F'\"\"\"F-F-
F--%\"bG6#F'F-" }{TEXT -1 29 ", vemos que a equa\347\343o acima \351" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x(n
+1)-x(n)=beta(n)" "6#/,&-%\"xG6#,&%\"nG\"\"\"F*F*F*-F&6#F)!\"\"-%%beta
G6#F)" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT 259 9 "desde que" }{TEXT -1 42 " m, por sua vez, satisfa\347a
 a seguinte ED :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "m(n+1) = m(n)/a(n);" "6#/-%\"mG6#,&%\"nG\"\"\"F)F)*&-F%
6#F(F)-%\"aG6#F(!\"\"" }{TEXT -1 27 ", cuja solu\347\343o sabemos ser \+
" }{XPPEDIT 18 0 "m(n)=product(1/a(i),i=0..n-1)*m(0)" "6#/-%\"mG6#%\"n
G*&-%(productG6$*&\"\"\"F--%\"aG6#%\"iG!\"\"/F1;\"\"!,&F'F-F-F2F--F%6#
F5F-" }{TEXT -1 133 ". A condi\347\343o inicial m(0) n\343o entra na h
ist\363ria, desde que n\343o nula, e podemos tomar m(0)=1. Tamb\351m e
stamos assumindo claramente que " }{XPPEDIT 18 0 "a(i)<>0" "6#0-%\"aG6
#%\"iG\"\"!" }{TEXT -1 13 " para todo i." }}{PARA 0 "" 0 "" {TEXT -1 
15 "Bem, agora como" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "x(n) = x(0)+sum(
beta(j),j = 0 .. n-1);" "6#/-%\"xG6#%\"nG,&-F%6#\"\"!\"\"\"-%$sumG6$-%
%betaG6#%\"jG/F3;F+,&F'F,F,!\"\"F," }{TEXT -1 46 ", podemos substituir
 o que j\341 sabemos para ter" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "m(n)*y
(n) = m(0)*y(0)+sum(m(j+1)*b(j),j = 0 .. n-1);" "6#/*&-%\"mG6#%\"nG\"
\"\"-%\"yG6#F(F),&*&-F&6#\"\"!F)-F+6#F1F)F)-%$sumG6$*&-F&6#,&%\"jGF)F)
F)F)-%\"bG6#F;F)/F;;F1,&F(F)F)!\"\"F)" }{TEXT -1 10 ", ou seja," }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "y(n)=
product(a(i),i=0..n-1)*(y(0)+sum(product(1/a(i),i=0..j)*b(j),j=0..n-1)
)" "6#/-%\"yG6#%\"nG*&-%(productG6$-%\"aG6#%\"iG/F/;\"\"!,&F'\"\"\"F4!
\"\"F4,&-F%6#F2F4-%$sumG6$*&-F*6$*&F4F4-F-6#F/F5/F/;F2%\"jGF4-%\"bG6#F
DF4/FD;F2,&F'F4F4F5F4F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 59 "O Maple nos diz isso de uma tacada, \+
usando o comando rsolve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "
ed:=y(n+1)=a(n)*y(n)+b(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#edG/-
%\"yG6#,&%\"nG\"\"\"F+F+,&*&-%\"aG6#F*F+-F'F0F+F+-%\"bGF0F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsolve(ed,y(n));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*&-%(productG6$-%\"aG6#%#n0G/F*;\"\"!,&%\"nG
\"\"\"F0!\"\"F0,&-%$sumG6$*&-%\"bG6#%#n1GF0-F%6$F'/F*;F-F:F1/F:F,F0-%
\"yG6#F-F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 51 "Resolvendo ED's n\343o lineares de um modo ou de out
ro" }}{PARA 0 "" 0 "" {TEXT -1 175 "J\341 vimos que h\341 uma f\363rmu
la para ED's de um tipo simples, que podemos usar. Tamb\351m vimos que
 o comando rsolve \351 o an\341logo do dsolve e que tenta resolver ED'
s em forma fechada." }}{PARA 0 "" 0 "" {TEXT -1 123 "Entretanto, \351 \+
f\341cil arrumar problemas sem solu\347\343o em forma fechada. O an
\341logo discreto da equa\347\343o log\355stica d\341 por exemplo:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "edlog:=y(n+1)=y(n)*(1-y(n));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&edlogG/-%\"yG6#,&%\"nG\"\"\"F+F
+*&-F'6#F*F+,&F+F+F-!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "rsolve(edlog,y(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'rsolveG6
$/-%\"yG6#,&%\"nG\"\"\"F,F,*&-F(6#F+F,,&F,F,F.!\"\"F,F." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 75 "Digamos que sabemos que y(0)=0.5 e querem
os y(2). Uma forma simples \351 fazer" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "y[0]:=.5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#
\"\"!$\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "y[1]:=y
[0]*(1-y[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"\"$\"#D!
\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "y[2]:=y[1]*(1-y[1]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"#$\"%v=!\"%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "E se quisermos y(127) em vez? Bem,
 a\355 podemos subir um degrau na escada da sofistica\347\343o e fazer
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "upsilon[0]:=.5;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(upsilonG6#\"\"!$\"\"&!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "for i from 0 to 126 do\n  up
silon[i+1]:=upsilon[i]*(1-upsilon[i]):\nend do:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 95 "#confere se os primeiros upsilons s\343o os y'
s\nupsilon[0]-y[0];\nupsilon[1]-y[1];\nupsilon[2]-y[2];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "# agora o novo\nupsilon[127];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+Bip\"[(!#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 247 "Podemos at\351 plotar os pontos discretos. Infelizmente n\343o
 h\341 um comando t\343o vers\341til como DEplot para recurs\365es (o \+
comando REplot \351 bem mais limitado, s\363 funcionando para ED's lin
eares). Se algum comando for novo para voc\352, utilize o help do Mapl
e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "pontos:=[seq([j,upsil
on[j]],j=0..127)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "point
plot(pontos,color=red);" }}{PARA 13 "" 1 "" {GLPLOT2D 340 228 228 
{PLOTDATA 2 "6$-%'POINTSG6\\s7$$\"\"!F($\"\"&!\"\"7$$\"\"\"F($\"#D!\"#
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F*F($\"+&\\#fC6FC7$$\"\"'F($\"+qm@\")**!#67$$\"\"(F($\"+3)p\\)*)FM7$$
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\"#XF($\"+m[-t>FM7$$\"#YF($\"+&f'4M>FM7$$\"#ZF($\"+*H*o'*=FM7$$\"#[F($
\"+'*\\rg=FM7$$\"#\\F($\"+$R#4E=FM7$$\"#]F($\"+fiu#z\"FM7$$\"#^F($\"+n
ogg<FM7$$\"#_F($\"+-&4'H<FM7$$\"#`F($\"+7Sp*p\"FM7$$\"#aF($\"+:W!3n\"F
M7$$\"#bF($\"+T&))Gk\"FM7$$\"#cF($\"+8x*eh\"FM7$$\"#dF($\"+fky*e\"FM7$
$\"#eF($\"+\\A^k:FM7$$\"#fF($\"+j_.S:FM7$$\"#gF($\"+x\"=j^\"FM7$$\"#hF
($\"+pfK$\\\"FM7$$\"#iF($\"+Wd-r9FM7$$\"#jF($\"+xlQ\\9FM7$$\"#kF($\"+i
$z$G9FM7$$\"#lF($\"+'owzS\"FM7$$\"#mF($\"+-F:)Q\"FM7$$\"#nF($\"+BI))o8
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