Solução_Calculo_Stewart_6e

F.
474 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
15.
17.
19.
∞
n=1
cos nπ
n 3/4
= ∞
n=1
Alternating Series Test.
∞
n=1
π
(−1) n sin . b n =sin
n
(−1) n
n . b 3/4 n = 1 is decreasing and positive and lim
n3/4 π
n
converges by the Alternating Series Test.
π
> 0 for n ≥ 2 and sin ≥ sin
n
n→∞
π
n +1
1
=0, so the series converges by the
n3/4
π
,and lim sin =sin0=0,sotheseries
n→∞ n
n n
n! = n · n ·····n
1 · 2 ·····n ≥ n ⇒ lim n n
n→∞ n! = ∞ ⇒ lim (−1) n n n
does not exist. So the series diverges by the Test for
n→∞ n!
Divergence.
21.
n a n s n
1 1 1
2 −0.35355 0.64645
3 0.19245 0.83890
4 −0.125 0.71390
5 0.08944 0.80334
6 −0.06804 0.73530
7 0.05399 0.78929
8 −0.04419 0.74510
9 0.03704 0.78214
10 −0.03162 0.75051
By the Alternating Series Estimation Theorem, the error in the approximation
∞
n=1
(−1) n−1
n 3/2
decimal places, rounded up).
≈ 0.75051 is |s − s 10 | ≤ b 11 =1/(11) 3/2 ≈ 0.0275 (to four
23. The series ∞
n=1
(−1) n+1
satisfies (i) of the Alternating Series Test because
n 6
1
(n +1) < 1 and (ii) lim
6 n6 n→∞
1
n 6 =0,sothe
series is convergent. Now b 5 = 1 5 =0.000064 > 0.00005 and b 6 6 = 1 ≈ 0.00002 < 0.00005,sobytheAlternatingSeries
66 Estimation Theorem, n =5. (That is, since the 6th term is less than the desired error, we need to add the first 5 terms to get the
sum to the desired accuracy.)
25. The series ∞
n=0
(−1) n
10 n n! satisfies (i) of the Alternating Series Test because 1
10 n+1 (n +1)! < 1
1
and (ii) lim
10 n n! n→∞ 10 n n! =0,
sotheseriesisconvergent.Nowb 3 = 1
10 3 3! ≈ 0.000 167 > 0.000 005 and b 4 = 1 =0.000 004 < 0.000 005,soby
10 4 4!
the Alternating Series Estimation Theorem, n =4(since the series starts with n =0,notn =1). (That is, since the 5th term
is less than the desired error, we need to add the first 4 terms to get the sum to the desired accuracy.)
27. b 7 = 1 7 5 = 1
16,807
∞
n=1
(−1) n+1
n 5 ≈ s 6 = 6
≈ 0.000 059 5, so
n=1
(−1) n+1
n 5 =1− 1 32 + 1
243 − 1
1024 + 1
3125 − 1
7776 ≈ 0.972 080. Addingb 7 to s 6 does not change
the fourth decimal place of s 6 , so the sum of the series, correct to four decimal places, is 0.9721.