Lista de Exercícios Sequências e Séries

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CA´LCULO DIFERENCIAL E INTEGRAL 3 - CD23NB - 2017/1
Professor: Geovani Raulino
Lista de Exerc´ıcios - 3ª Avaliac¸a˜o
Se´ries e Teste da divergeˆncia
1. Determine se a se´rie geome´trica e´ convergente ou divergente. Se for convergente, calcule a
soma.
(a)
∞∑
n=1
6(0, 9)n−1
(b)
∞∑
n=1
10n
(−9)n−1
(c)
∞∑
n=1
(−3)n−1
4n
(d)
∞∑
n=1
1
(
√
2)n
(e)
∞∑
n=1
pin
3n+1
(f)
∞∑
n=1
en
3n−1
2. Determine se a se´rie e´ convergente ou divergente. Se ela for convergente, calcule a sua soma.
(a)
∞∑
n=1
1
2n
(b)
∞∑
n=1
n+ 1
2n− 3
(c)
∞∑
n=1
n
n+ 5
(d)
∞∑
k=1
k(k + 2)
(k + 3)2
(e)
∞∑
n=1
3n + 2n
6n
(f)
∞∑
n=1
1 + 2n
3n
(g)
∞∑
n=1
n
√
2
(h)
∞∑
n=1
[2(0, 1)n + (0, 2)n]
(i)
∞∑
n=1
ln
(
n2 + 1
2n2 + 1
)
(j)
∞∑
n=1
(cos 1)n
(k)
∞∑
n=1
arctann
(l)
∞∑
n=1
(
3
5n
+
2
n
)
(m)
∞∑
n=1
(
1
en
+
1
n(n+ 1)
)
(n)
∞∑
n=1
en
n2
3. Determine se a se´rie e´ convergente ou divergente. Se for convergente, encontre a sua soma.
(a)
∞∑
n=2
2
n2 − 1
(b)
∞∑
n=1
2
n2 + 4n+ 3
(c)
∞∑
n=1
3
n(n+ 3)
(d)
∞∑
n=1
n
n+ 1
(e)
∞∑
n=1
(
e1/n − e1/(n+1))
(f)
∞∑
n=1
(
cos
1
n2
− cos 1
(n+ 1)2
)
4. Expresse o nu´mero como uma raza˜o de inteiros.
(a) 0, 2 = 0, 2222 . . .
(b) 0, 73 = 0, 73737373 . . .
(c) 3, 417 = 3, 417417 . . .
(d) 6, 254 = 6, 25454 . . .
(e) 0, 123456
(f) 7, 12345
1
O Teste da Integral
5. Use o Teste da Integral para determinar se a se´rie e convergente ou divergente.
(a)
∞∑
n=1
1
n4
(b)
∞∑
n=1
1
4
√
n
(c)
∞∑
n=1
1
(2n+ 1)3
(d)
∞∑
n=1
1√
n+ 4
(e)
∞∑
n=1
ne−n
(f)
∞∑
n=1
n+ 2
n+ 1
6. Determine se a se´rie e´ convergente ou divergente.
(a)
∞∑
n=1
2
n0,85
(b)
∞∑
n=1
n−1,4 + 3n−1,2
(c)
∞∑
n=1
3n + 2n
6n
(d)
∞∑
n=1
5− 2√n
n3
(e)
∞∑
n=1
n2
n3 + 1
(f)
∞∑
n=1
1
n2 + 4
(g)
∞∑
n=1
3n+ 2
n(n+ 1)
(h)
∞∑
n=1
lnn
n3
(i)
∞∑
n=1
1
n2 − 4n+ 5
(j)
∞∑
n=1
1
n lnn
(k)
∞∑
n=1
1
n(lnn)2
(l)
∞∑
n=1
e1/n
n2
(m)
∞∑
n=1
n2
en
(n)
∞∑
n=1
1
n3 + n
(o)
∞∑
n=1
n
n4 + 1
Os Testes de Comparaca˜o
7. Determinar se a se´rie converge ou diverge.
(a)
∞∑
n=1
1
n2 + n+ 1
(b)
∞∑
n=1
n+ 1
n
√
n
(c)
∞∑
n=1
n− 1
n2
√
n
(d)
∞∑
n=1
9n
3 + 10n
(e)
∞∑
n=1
4 + 3n
2n
(f)
∞∑
n=1
cos2 n
n2 + 1
(g)
∞∑
n=1
n2 − 1
3n4 + 1
(h)
∞∑
n=1
n− 1
n4n
(i)
∞∑
n=1
sin2 n
n
√
n
(j)
∞∑
n=1
√
n
n− 1
(k)
∞∑
n=1
2 + (−1)n
n
√
n
(l)
∞∑
n=1
1√
n3 + 1
(m)
∞∑
n=1
1
2n+ 3
(n)
∞∑
n=1
1 + 4n
1 + 3n
(o)
∞∑
n=1
n+ 4n
n+ 6n
2
(p)
∞∑
n=1
√
n+ 2
2n2 + n+ 1
(q)
∞∑
n=1
n+ 2
(n+ 1)3
(r)
∞∑
n=1
n2 − 5n
n3 + n+ 1
(s)
∞∑
n=1
1 + n+ n2√
1 + n2 + n6
(t)
∞∑
n=1
e
1
n
n
(u)
∞∑
n=1
1
n!
Series Alternadas
8. Determinar se a se´rie converge ou diverge.
(a)
∞∑
n=1
(−1)nn
10n
(b)
∞∑
n=1
(−1)n−1
ln (n+ 4)
(c)
∞∑
n=1
(−1)n3n− 1
2n+ 1
(d)
∞∑
n=1
(−1)n n√
n3 + 2
(e)
∞∑
n=1
(−1)n√
n
(f)
∞∑
n=1
(−1)n
√
n
1 + 2
√
n
(g)
∞∑
n=1
(−1)n+1 n
2
n3 + 4
(h)
∞∑
n=1
(−1)n+1 n
2n
(i)
∞∑
n=2
(−1)n n
lnn
(j)
∞∑
n=1
(−1)n−1 lnn
n
(k)
∞∑
n=1
cosnpi
n3/4
(l)
∞∑
n=1
sin (npi
2
)
n!
Convergeˆncia Absoluta e os Testes da Raza˜o e da Raiz
9. Determinar se a se´rie e´ absolutamente convergente, condicionalmente convergente ou diver-
gente.
(a)
∞∑
n=1
n2
2n
(b)
∞∑
n=1
(−10)n
n!
(c)
∞∑
n=1
(−1)n
n4
(d)
∞∑
n=1
(−1)n+1
4
√
n
(e)
∞∑
n=1
(−1)n−12n
n4
(f)
∞∑
n=1
k
(
2
3
)k
(g)
∞∑
n=1
e−nn!
(h)
∞∑
n=1
(−1)n(1, 1)n
n4
(i)
∞∑
n=1
(−1)(n−1)n
n2 + 1
(j)
∞∑
n=1
(−1)ne 1n
n3
(k)
∞∑
n=1
sin 4n
4n
(l)
∞∑
n=2
10n
(n+ 1)42n+1
(m)
∞∑
n=1
(−1)(n+1)n22n
n!
(n)
∞∑
n=1
(−1)n arctann
n2
(o)
∞∑
n=1
3− cosn
n2/3 − 2
(p)
∞∑
n=1
(−1)n
lnn
(q)
∞∑
n=1
n!
nn
(r)
∞∑
n=1
cos (npi/3)
n!
3
(s)
∞∑
n=2
(−2)n
nn
(t)
∞∑
n=2
(
n2 + 1
2n2 + 1
)
(u)
∞∑
n=2
( −2n
n+ 1
)5n
(v)
∞∑
n=1
(
1 +
1
n
)n2
(w)
∞∑
n=2
n
(lnn)n
(x)
∞∑
n=1
(−1)n+1 1 · 3 · 5 . . . (2n− 1)
(2n− 1)!
(y)
∞∑
n=1
2 · 4 · 6 · 8 . . . 2n
n!
(z)
∞∑
n=1
(−1)n 2
nn!
5 · 8 · 11 · (3n+ 2)
10. Para quais das se´ries o Teste da Raza˜o na˜o e´ conclusivo.
(a)
∞∑
n=1
1
n3
(b)
∞∑
n=1
n
2n
(c)
∞∑
n=1
(−3)n−1√
n
(d)
∞∑
n=1
√
n
1 + n2
Se´ries de Poteˆncias
11. Encontre o raio e o intervalo de convergeˆncia das seguintes se´ries
(a)
∞∑
n=1
xn√
n
(b)
∞∑
n=0
(−1)nxn
(n+ 1)
(c)
∞∑
n=1
(−1)(n−1)xn
n3
(d)
∞∑
n=1
nnxn
(e)
∞∑
n=0
xn
n!
(f)
∞∑
n=1
xn
√
n
(g)
∞∑
n=1
(−1)nn4nxn
(h)
∞∑
n=1
xn
n3n
(i)
∞∑
n=1
(−2)nxn
4
√
n
(j)
∞∑
n=1
xn
5nn5
(k)
∞∑
n=2
(−1)nxn
4n lnn
(l)
∞∑
n=0
(−1)n(x− 3)n
2n+ 1
(m)
∞∑
n=0
(x− 2)n
n2 + 1
(n)
∞∑
n=0
(−1)nx2n
(2n)!
(o)
∞∑
n=1
3n(x+ 4)n√
n
(p)
∞∑
n=1
n(x+ 1)n
4n
(q)
∞∑
n=1
(x− 2)n
nn
(r)
∞∑
n=1
(3x− 2)n
n3n
(s)
∞∑
n=1
n(x− a)n
bn
(t)
∞∑
n=1
n(x− 4)n
(n3 + 1)
(u)
∞∑
n=1
n!(2x− 1)n
(v)
∞∑
n=2
n2xn
2 · 4 · 6 . . . 2n
(w)
∞∑
n=1
(4x+ 1)n
n2
(x)
∞∑
n=2
x2n
n(lnn)2
(y)
∞∑
n=1
xn
1 · 3 · 5 . . . (2n− 1)
4
Representac¸a˜o de Func¸o˜es como Se´ries de Poteˆncias
12. Encontre uma representac¸a˜o em se´ries de poteˆncias para as seguintes func¸o˜es e determine o
intervalo de convergeˆncia.
(a) f(x) =
1
1 + x
(b) f(x) =
3
1− x4
(c) f(x) =
2
3− x
(d) f(x) =
1
x+ 10
(e) f(x) =
x
9 + x2
(f) f(x) =
x
1 + 2x2
(g) f(x) =
1 + x
1− x
(h) f(x) =
x2
a3 − x3
13. Expresse a func¸a˜o como a soma de uma se´rie de poteˆncias usando primeiro frac¸o˜es parciais.
Encontre o intervalo de convergeˆncia.
(a) f(x) =
3
x2 − x− 2 (b) f(x) =
x+ 2
2x2 − x− 1
14. (a) Use diferenciac¸a˜o para achar a representac¸a˜o em se´rie de poteˆncia para f(x) =
1
(1 + x)2
.
Qual e´ o raio de convergeˆncia?
(b) Use o item (a) para encontrar uma se´rie de poteˆncia para f(x) =
1
(1 + x)3
.
(c) Use o item (b) para encontrar uma se´rie de poteˆncia para f(x) =
x2
(1 + x)3
.
15. (a) Ache uma representac¸a˜o em se´rie de poteˆncia para f(x) = ln (1 + x). Qual e´ o raio de
convergeˆncia?
(b) Use o item (a) para encontrar uma se´rie de poteˆncia para f(x) = x ln (1 + x).
(c) Use o item (a) para encontrar uma se´rie de poteˆncia para f(x) = ln (x2 + 1)
16. Encontre uma representac¸a˜o em serie de poteˆncias para a func¸a˜o e determine o raio de con-
vergeˆncia.
(a) f(x) = ln (5− x)
(b) f(x) =
x2
(1− 2x)2
(c) f(x) =
x3
(x− 2)2
(d) f(x) = arctan (x/3)
17. Avalie a integral indefinida como uma se´rie de poteˆncias. Qual e o raio de convergeˆncia?
(a)
∫
t
1− t8dt
(b)
∫
ln (1− t)
t
dt
(c)
∫
x− tan−1 (x)
x3
dx
(d)
∫
tan−1 (x2)dx
5
GABARITO
1. (a) Conv. e S = 60
(b) Div.
(c) Conv. e S = 1
7
(d) Conv. e S =
√
2− 1
(e) Div.
(f) Conv. e S = 3e
3−e
2. (a) Div.
(b) Div.
(c) Div.
(d) Div.
(e) Conv. e S = 3
2
(f) Conv. e S = 5
2
(g) Div.
(h) Conv. e S = 17
36
(i) Div.
(j) Conv. e S = cos 1
1−cos 1
(k) Div.
(l) Div.
(m) Conv.e S = e
e−1
(n) Div.
3. (a) Conv. e S = 3
2
(b) Conv. e S = 5
6
(c) Conv. e S = 11
6
(d) Div.
(e) Conv. e S = e− 1
(f) Conv. e S = cos 1− 1
4. (a) 2
9
(b) 73
99
(c) 1138
333
(d) 344
55
(e) 41111
333000
(f) 237446
33333
5. (a) Conv.
(b) Div.
(c) Conv.
(d) Div.
(e) Conv.
(f) Div.
6. (a) Div.
(b) Conv.
(c) Conv.
(d) Div.
(e) Conv.
(f) Div.
7. (a) Conv.
(b) Div.
(c) Conv.
(d) Conv.
(e) Div.
(f) Conv.
(g) Conv.
(h) Conv.
(i) Conv.
(j) Div.
(k) Conv.
(l) Conv.
(m) Div.
(n) Div.
(o) Conv.
(p) Conv.
(q) Conv.
(r) Div.
(s) Div.
(t) Div.
(u) Conv.
8. (a) Conv.
(b) Conv.
(c) Div.
(d) Conv.
(e) Conv.
(f) Div.
(g) Conv.
(h) Conv.
(i) Div.
(j) Conv.
(k) Conv.
(l) Conv.
6
9. (a) AC
(b) AC
(c) AC
(d) CC
(e) D
(f) AC
(g) D
(h) D
(i) CC
(j) AC
(k) AC
(l) AC
(m) AC
(n) AC
(o) D
(p) CC
(q) AC
(r) AC
(s) AC
(t) AC
(u) D
(v) D
(w) AC
(x) AC
(y) D
(z) AC
10. (a), (d)
11. (a) R = 1, I = [−1, 1)
(b) R = 1, I = (−1, 1]
(c) R = 1, I = [−1, 1]
(d) R = 0, I = {x = 0}
(e) R =∞, I = (−∞,∞)
(f) R = 1, I = (−1, 1)
(g) R =
1
4
, I = (
−1
4
,
1
4
)
(h) R = 3, I = [−3, 3)
(i) R =
1
2
, I = (−1/2, 1/2]
(j) R = 5, I = [−5, 5]
(k) R = 4, I = (−4, 4]
(l) R = 1, I = (2, 4]
(m) R = 1, I = [1, 3]
(n) R =∞, I = (−∞,∞)
(o) R =
1
3
, I = [
−13
3
,
−11
3
)
(p) R = 4, I = (−5, 3)
(q) R =∞, I = (−∞,∞)
(r) R = 1, I = [−1/3, 5/3)
(s) R = b, I = (a− b, a+ b)
(t) R = 1, I = [3, 5]
(u) R = 0, I = {x = 1/2}
(v) R = 1, I = [−1, 1]
(w) R =
1
4
, I = [−1/2, 0]
(x) R = 1, I = [−1, 1]
(y) R =∞, I = (−∞,∞)
12. (a)
∞∑
n=0
(−1)n xn I = (−1, 1)
(b)
∞∑
n=0
3 x4n I = (−1, 1)
(c)
∞∑
n=0
2 xn
3n+1
I = (−3, 3)
(d)
∞∑
n=0
(−1)n xn
10n+1
I = (−10, 10)
(e)
∞∑
n=0
(−1)n x2n+1
9n+1
I = (−3, 3)
(f)
∞∑
n=0
(−2)n x2n+1 (−1√
2
, 1√
2
)
(g)
∞∑
n=0
(1 + x) xn I = [−1, 1)
(h)
∞∑
n=0
x3n+2
a3n+3
I = (−a, a)
13. (a)
∞∑
n=0
xn
(
(−2)n+1 2− 1
2n+1
)
, I = (−1, 1)
(b)
∞∑
n=0
(−1)n+1(2x)n -
∞∑
n=0
xn, I = (−1
2
, 1
2
)
14. (a)
∞∑
n=0
(−1)n(n+ 1)xn I = (−1, 1)
(b)
∞∑
n=0
(−1)n(1/2)(n+ 2)(n+ 1)xn
7
(c)
∞∑
n=2
(−1)nn(n− 1)xn
15. (a)
∞∑
n=1
(−1)n−1 xn
n
I = (−1, 1)
(b)
∞∑
n=2
(−1)n xn
(n− 1)
(c)
∞∑
n=1
(−1)n−1 x2n
n
16. (a) ln 5−
∞∑
n=1
xn
n 5n
R = 5,
(b)
∞∑
n=0
n 2n−1 xn+1 R = 1/2
(c)
∞∑
n=3
(n− 2)xn
2n−1
R = 2
(d)
∞∑
n=0
1
(2n+ 1)
(x
3
)2n+1
R = 3
17. (a)
∞∑
n=0
t8n+2
8n+ 2
+ C R = 1
(b)
∞∑
n=0
(−1) tn+1
(n+ 1)2
+ C R = 1
(c)
∞∑
n=1
(−1)n+1 x2n−1
4n2 − 1 + C R = 1
(d)
∞∑
n=0
(−1)n x4n+2
2n+ 1
+ C R = 1
Observac¸a˜o:
Exerc´ıcios que devera˜o ser entregues no dia da avaliac¸a˜o 28/06:
1.a, 2.b, 3.c, 4.c, 5.e, 7.g, 8.d, 9.a, 9.i, 11.a, 11.n, 12.g, 14, 16.a, 17.a
8