44. For which positive integers k is the following series convergent? laga Σ (n!)? sw Mong pe n=1 (kn)! 8

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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of positive numbers that conver
to . Determine whether the given series is absolutely converg
Tkm)!
00
b cos nT
41. E
42. E
(-1)"n!
n-1
n-1 n"bib2b3... b,
43. For which of the following series is the Ratio Test inconc
sive (that is, it fails to give a definite answer)?
00
(a) E
(b) E
2"
n=1 n
enitzsT 10l v
n=1
(-3)"-
VnW
Vn
(d) E
1 + n?
(c) E
n=1
n=1
44. For which positive integers k is the following series
convergent?
(n!)?
Σ
n=1 (kn)!
low
45. (a) Show that E-o x"/n! converges for all x.
(b) Deduce that lim, x"/n! = 0 for all x.
n=0
46. Let E a, be a series with positive terms and let r, = an-
Suppose that lim, rn = L< 1, so E a, converges by
Ratio Test. As usual, we let R, be the remainder after n
that is,
orl
R,
an+1 + an+2 + an+3 + · . .
(a) If {rn} is a decreasing sequence and r+1 < 1, show
0 summing a geometric series, that
Transcribed Image Text:of positive numbers that conver to . Determine whether the given series is absolutely converg Tkm)! 00 b cos nT 41. E 42. E (-1)"n! n-1 n-1 n"bib2b3... b, 43. For which of the following series is the Ratio Test inconc sive (that is, it fails to give a definite answer)? 00 (a) E (b) E 2" n=1 n enitzsT 10l v n=1 (-3)"- VnW Vn (d) E 1 + n? (c) E n=1 n=1 44. For which positive integers k is the following series convergent? (n!)? Σ n=1 (kn)! low 45. (a) Show that E-o x"/n! converges for all x. (b) Deduce that lim, x"/n! = 0 for all x. n=0 46. Let E a, be a series with positive terms and let r, = an- Suppose that lim, rn = L< 1, so E a, converges by Ratio Test. As usual, we let R, be the remainder after n that is, orl R, an+1 + an+2 + an+3 + · . . (a) If {rn} is a decreasing sequence and r+1 < 1, show 0 summing a geometric series, that
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