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

66 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES
00
2 4 6 8 10 n 2n 2n . 2 2 . -
3. -- +- - - +---+···= :E{-1) --.Now limb,. = lim-- = lun / = - =fO.Smce
5 6 7 8 9 n=l n + 4 n-oo n-oo n + 4 n-too 1 + 4 n 1
lim an =I 0 (in fact the limit does not exist), the series diverges by the Test for Divergence.
n-+oo
00 00 1 00 1
5. :E a,. = :E ( -1f- 1 2 + 1 = :E ( -1)"- 1 b,.. Now b,. = - 2 1
> 0, {bn} is decreasing, and lim b,. = 0, so the
n=l n=l n n=l n + n-oo
series converges by the Alternating Series Test.
-1)"b,.. Now lim b,. = lim ~- 1 3
jn = - =f 0. Since lim a,. =f 0
n=l n=l n + n=l n-+oo n-+oo + 1 n 2 n-+00
7. E a,.= E (
- 1)" 2
3 n -
1
1 = f (
(in fact the limit does not exist), the series diverges by the Test for Divergence.
00 00 00 1
9. :E a,.= :E ( - 1)"e-" = :E ( -1)"bn. Now b,. = n > 0, {bn} is decreasing, and lim b,. = 0, so the series converges
n = l n=l n =l e n-+oo
by the Alternating Series Test.
n2
11. b,. = n 3
+ 4
> 0 for n 2:: 1. {bn} is decreasing for n 2:: 2 since
lim b,. = lim 1
1 /4 n/
(x 3 + 4)(2x) - x 2 (3x 2 ) _ x(2x 3 + 8- 3x 3 ) _ x(8- x 3 )
(x3 + 4 )2 - (x3 + 4 )2 - (x3 + 4 )2 < 0 for x > 2. Also,
3
= 0. Thus, the series E (-1)"+1 ~converges by the Alternating Series Test.
n -+00 n-op + n n=l n 3 + 4
13. lim b~ = lim e 2 fn = e 0 = 1, so lim (-l)n- 1 e 2 1" does not exist. Thus, the series f: (-1)"- 1 e 2 1" diverges by the
n-+oo ra.-oo n-+oo n=l
Test for Divergence.
sin(n + ~)1r (-1)" 1
15. a,. =
1 + .,fti = 1 + .,fii," Now b,. = + Vn > 0 for n 2:: q, {b,.} is decreasing, and .. ~~~ b,. = 0, so the series
1
oo sin(n +.! )1r
:E fo converges by the Alternating Series Test.
n=O 1 + n
17. n~l ( - 1)" sin(~J bn =sin(~) > 0 for n 2:: 2 and sin(~) 2:: sin( n: 1
). and ,.~ sin(~) = sinO = 0, so the
series converges by the AJternating Series Test.
n" n·n···· ·n
19. - = > n =>
n! 1· 2 · ·· · · n -
by the Test for Divergence.
n"
lim- =oo =>
n.-+oo n!
( 1)" " · oo n
lim - n does not exist. So the series :E (-1}" ~, diverges
n ..... oo n. 1 n=l n.
21. The graph gives us an estimate for the sum of the series
E (-O.~) " of -0.55.
n=l n.
(0 8)"
bs = ~ :::::: 0.000 004, so
-1
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