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

70 D CHAPTER 11 INFINITE SEQUENCES AND SERIES
00
2 · 4 · 6 · · · · · (2n)
00
(2 · 1) · (2 · 2) · (2 · 3) · · · ·. (2. n) oo .2nn! oo . .
29· n~l n! = n~l nl = .~
1
-;;;[ = n~l 2n, wh1ch diverges by the Test for
Divergence since lim 2n = oo.
n -oo
31 B h . d fi . . lim I an+l I li 15n + 11 5 .
. y t e recurs1ve e mt1on, -- = m --- = - > 1, so the series diverges by the Ratio Test.
n-oo a,. n--400 4n + 3 4
00
b" cosn1r
00
bn 1
•
33. The series I: " = I: (-1)" 2':., where bn > 0 for n ~ 1 and lim bn = - .
n=l n n=l n n-oo 2
I a n+l l 1 . I ( - l )n+lb~+l n I 1
li
. b · n 1 ( ) 1 . ~ b~ cos n 1r .
rn -- = 1m + · (
. 1
< 1, so the senes LJ iS
n-+oo an n-oo n - n ~ n -~o oo n + n = l n
absolutely convergent by the Ratio Test.
1 ) b = 1m n -- 1 = - 2 1 = - 2
35. (a) lim 1 1 / (n + 1 3
3
1
) 1 = lim n = lim = 1. Inconclusive
3
n _,oo 1/n3 n -+oo (n + 1) n ->oo (1 + 1/ n)
3
(b) lim I (n 2
++ 1
1 ) ·
2 " I = lim n 2
+ 1 = lim ( - 2
1 + -2
1 ) = -2
1 . Conclusive (convergent)
n - oo n n n -too n n --+oo n
(c) lim I ~ · ( V:
1
1 = 3 lim ·; n = 3 lim ~/ = 3. Conclusive (divergent)
n .... oo v n + 1 - 3 n - n->oo n + 1 n -+oo v
(d) li
I
I+Vn
.Jn + 1 1 + n21 lim. [R1 1/n2 + 1 ] 1 r l .
m ·--- = + - · = . nconc us1ve
n .... oo 1 + (n + 1) 2 Vn n->oo n 1jn2 + (1 + 1/n) 2
1 - = lxl · 0 = 0 < 1, so by the Ratio Test the
n ->oo a,. n ->oo n + . X" n->oo n + 1 n ->oo n + 1
37. (a) lim \ an+l I = lim I ( xn+~)' · n! I = lim 1-x -1 = lx l lim -
00 x n
series I; I converges for all x .
n = O n.
(b) Since the series of part (a) always convergt