Solução_Calculo_Stewart_6e

F.
502 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
11 Review
1. (a) See Definition 11.1.1.
(b) See Definition 11.2.2.
(c) The terms of the sequence {a n} approach 3 as n becomes large.
(d) By adding sufficiently many terms of the series, we can make the partial sums as close to 3 as we like.
2. (a) See Definition 11.1.11.
(b) A sequence is monotonic if it is either increasing or decreasing.
(c) By Theorem 11.1.12, every bounded, monotonic sequence is convergent.
3. (a) See (4) in Section 11.2.
(b) The p-series ∞
n=1
1
is convergent if p>1.
np 4. If a n =3,then lim
n→∞ a n =0and lim
n→∞ s n =3.
5. (a) Test for Divergence: If lim
n→∞ a n does not exist or if lim
n→∞ a n 6= 0,thentheseries ∞
n=1 a n is divergent.
(b) Integral Test: Suppose f is a continuous, positive, decreasing function on [1, ∞) and let a n = f(n). Then the series
∞
n=1 a n is convergent if and only if the improper integral ∞
f(x) dx is convergent. In other words:
1
(i) If ∞
f(x) dx is convergent, then ∞
1 n=1 a n is convergent.
(ii) If ∞
f(x) dx is divergent, then ∞
1 n=1
an is divergent.
(c) Comparison Test: Suppose that a n and b n are series with positive terms.
(i) If b n is convergent and a n ≤ b n for all n,then a n is also convergent.
(ii) If b n is divergent and a n ≥ b n for all n,then a n is also divergent.
(d) Limit Comparison Test: Suppose that a n and b n are series with positive terms. If lim
n→∞ (a n/b n )=c,wherec is a
finite number and c>0, then either both series converge or both diverge.
(e) Alternating Series Test: If the alternating series ∞
n=1 (−1)n−1 b n = b 1 − b 2 + b 3 − b 4 + b 5 − b 6 + ··· [b n > 0]
satisfies (i) b n+1 ≤ b n for all n and (ii) lim b n =0, then the series is convergent.
n→∞
(f ) Ratio Test:
(i) If lim
n→∞ an+1
= L1 or lim
a n
n→∞ a
n+1
= ∞, then the series ∞ a n is divergent.
a n
n=1
(iii) If lim =1, the Ratio Test is inconclusive; that is, no conclusion can be drawn about the convergence or
n→∞ an+1
a n
divergence of a n.