Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

CHAPTER 11 REVIEW 0 99
oo f (n) (O)
xn [a= 0 in part (b)]
(c) L -- 1
-
n=O n .
(d) See Theorem 11.10.8.
(e) See Tay.lor's Inequality ( 11.10.9).
11. (a)-( f) See Table I on page 786 [ ET 762].
12. See the binomial series ( 11 .10.17) for the expansion. The radius of convergence for the binomial series is 1.
TRUE-FALSE QUIZ
1. False. See Note 2 after Theorem I 1.2.6.
3. True. If lim an = L, then as n -7+ 00, 2n + 1 -+ oo, so a2n + l -+ L.
n - oo
5. False. For example, take en = ( - 1)" j (n6n).
7. False, smce . hm . I _+_ an 1 I = lim . I 1 · n3 ~ 1 = lun .· I n3 . · -1/n31 - = hm . 1 = 1.
n-oo a,. n - oo (n + 1) 3 1 n--+oo (n + 1) 3 1j n 3 n -oo (1 + 1/ n) 3
9. False. See the note after Example 2 in Section 11.4.
11. True. See (9) in Section II. I .
13. True.
111
/ (0) 1
By Theorem 11.10.5 the coefficient of x 3 is ~ = 3
=> !'"(0) = 2.
Or: Use Theorem 11 .9.2 to differentiate f three times.
15. False. For example, let a.. = bn = ( -1 r. Then {an} and { bn} are divergent, but a,.b,. = 1, so {a,. b .. } is convergent.
17. True by Theorem 11.6.3. [2:: ( -1)" a,. is absolutely convergent and hence convergent.]
19. True. 0.99999 . . . = 0.9 + 0.9(0.1) 1 + 0.9(0.1? + 0.9(0.1) 3 + · · · = f: (0.9)(0.1)n-l = __2.:Q_ = 1 by the formula
n=l 1 - 0.1
for the sum of a geometric series [S = a 1 / (1 -;- r)] with ratio r satisfying lrl < 1.
21 . True. A finite number of terms doesn't affect convergence or divergence of a series.
EXERCISES
1.
{ 2 + n 3 } . . 2 + n 3 · . 2/ n 3 + 1 1
1 2 3 converges smce lim
+ n = 1 2 3 lim 11 3 2 = - 2 .
n --+00 + n n--+00 n +
3
3. lim a,. = lim ~ = lim / ~ = oo, so the sequence diverges.
n--+OC n-+oo 1 + n n-+oo 1 n + 1 .
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