Lista 4 - Séries de potências

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Universidade Federal do Rio de Janeiro
INSTITUTO DE MATEMA´TICA
Departamento de Me´todos Matema´ticos
4a Lista de Exerc´ıcios de Ca´lculo II - Matema´tica
Profa.: Moˆnica
1. Determine o intervalo de convergeˆncia das seguintes se´ries de poteˆncias:
(a)
∞∑
n=0
nxn
2n
(b)
∞∑
n=1
(x− 1)n
n
(c)
∞∑
n=1
(x+ 1)n
n2
(d)
∞∑
n=1
nxn
(e)
∞∑
n=0
xn
n!
(f)
∞∑
n=0
2nxn
n!
(g)
∞∑
n=0
n!xn
3n
(h)
∞∑
n=1
3n
2
xn
2
(i)
∞∑
n=0
xn
2n
(j)
∞∑
n=1
(2x− 6)n
n3n
(k) 1 +
102x
1 · 3 +
104x2
1 · 3 · 5 +
106x3
1 · 3 · 5 · 7 + ...
(l)
∞∑
n=1
xn
n+
√
n
(m)
∞∑
n=1
nk
n!
xn , k ∈ IN
2. Determine o raio de convergeˆncia das se´ries:
(a)
∞∑
n=1
n!
nn
xn (b)
∞∑
n=1
(n!)2
(2n)!
xn (c)
∞∑
n=0
1 · 3 · 5 · · · (2n+ 1)
1 · 4 · 7 · · · (3n+ 1) (x+ 2)
n
3. Usando se´rie geome´trica e derivac¸a˜o ou integrac¸a˜o termo a termo, calcule a soma
das se´ries, determinando o intervalo de convergeˆncia.
(a)
∞∑
n=0
xn
(b)
∞∑
n=1
nxn
(c)
∞∑
n=1
n2xn
(d)
∞∑
n=0
(−1)nxn
(e)
∞∑
n=1
xn
n
4. Desenvolva em se´rie de Taylor de centro 0, dando sempre o intervalo de con-
vergeˆncia. (Sugesta˜o: na maioria dos itens na˜o e´ necessa´rio calcular as derivadas
da func¸a˜o)
(a) ex
(b) e−x
(c) ex
2
(d) sen x
(e) cos x
(f) 1
1+x2
(g) arctg x
(h) 1
2+3x
(i) 5x
4−x
(j) cosh x
(k) senhx
(l) ln (1 + x)
(m) ln (1− x)
(n) ln
(
1+x
1−x
)
(o)
∫ x
0
arctg x
x
dx
5. Desenvolva em se´rie de Taylor de centro 1, dando o intervalo de convergeˆncia.
(a) 5x4 + x3 − x2 + 2x+ 3
(b) 1
x
(c) 1
x2
(d) ln x
(e) 1
x+3
6. Encontre uma se´rie de poteˆncias para x ex e mostre que:
∞∑
n=1
1
(n+ 2)n!
=
1
2
.
7. Encontre a se´rie de poteˆncias para x arctg x e mostre que:
∞∑
n=1
(−1)n+1 x
2n+1
(2n− 1)(2n+ 1) =
1
2
[
(x2 + 1) arctg x− x] .
Respostas
1. (a) (-2,2) ; (b) [0,2) ; (c) [-2,0] ; (d) (-1,1) ; (e) IR ; (f) IR ; (g) {0} ; (h) {0} ;
(i) (-2,2) ; (j) [3/2,9/2) ;(k) IR ; (l) [-1,1) ; (m) IR.
2. (a) e ; (b) 4 ; (c) 3/2 .
3. (a) 1
1−x , para |x| < 1 ; (b) x(1−x)2 , para |x| < 1 ; (c) x(x+1)(1−x)3 , para |x| < 1 ;
(d) 1
1+x
, para |x| < 1 ; (e) − ln(1− x), para −1 ≤ x < 1.
4.
(a)
∞∑
n=0
xn
n!
, x ∈ IR
(b)
∞∑
n=0
(−1)nxn
n!
, x ∈ IR
(c)
∞∑
n=0
x2n
n!
, x ∈ IR
(d)
∞∑
n=0
(−1)nx2n+1
(2n+ 1)!
, x ∈ IR
(e)
∞∑
n=0
(−1)nx2n
(2n)!
, x ∈ IR
(f)
∞∑
n=0
(−1)nx2n, |x| < 1
(g)
∞∑
n=0
(−1)nx2n+1
2n+ 1
, |x| ≤ 1
(h)
∞∑
n=0
(−1)n3nxn
2n+1
, |x| < 2
3
(i)
∞∑
n=0
5xn+1
4n+1
, |x| < 4
(j)
∞∑
n=0
x2n
(2n)!
, x ∈ IR
(k)
∞∑
n=0
x2n+1
(2n+ 1)!
, x ∈ IR
(l)
∞∑
n=1
(−1)n−1xn
n
, |x| ≤ 1
(m)
∞∑
n=1
−xn
n
, −1 ≤ x < 1
(n)
∞∑
n=0
2x2n+1
2n+ 1
, −1 ≤ x < 1
(o)
∞∑
n=0
(−1)nx2n+1
(2n+ 1)2
, |x| ≤ 1
5. (a) 10 + 23(x− 1) + 32(x− 1)2 + 21(x− 1)3 + 5(x− 1)4, x ∈ IR
(b)
∞∑
n=0
(−1)n(x− 1)n, 0 < x < 2
(c)
∞∑
n=0
(−1)n(n+ 1)(x− 1)n, 0 < x < 2
(d)
∞∑
n=0
(−1)n(x− 1)n+1
n+ 1
, 0 < x ≤ 2
(e)
∞∑
n=0
(−1)n(x− 1)n
4n+1
, 0 ≤ x ≤ 2.
6. Integre x ex.
7. Integre x arctg x.