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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Question
For number 16, I am having trouble figuring out where the (k-1)! came from. (Circled in pink in answer key)
Transcribed Image Text:16. f(x) = In (1+4 x), a = 0
%3D
![11.3.16
a. Note that f (0) = 0, f'(0)
16x?
= 4, f"(0) =
128x
-16, f"(0) = 128, and f"(0) = -1526. Thus, the series is
%3D
%3D
1536x4
given by 4x
6.
24
(-1)*+1 (k – 1)!(4x)k
k!
Σ
(-1)k+1 (4x)*
k=1
k=1
c. The absolute value of the ratio of consecutive terms is
4 |a| k
which has limit 4 x as k → o, so the
k +1'
interval of convergence is (-1/4,1/4]. Note that for x = 1/4 we have the alternating harmonic series,
while for I = -1/4 we have negative 1 times the harmonic series, which diverges.
11.3.17](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61b75115-d70f-4fe2-af93-2076876ad69a%2Fc8bb5c74-4c6f-4154-86ce-b8d8a2b8bd07%2Fxrr4k4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:11.3.16
a. Note that f (0) = 0, f'(0)
16x?
= 4, f"(0) =
128x
-16, f"(0) = 128, and f"(0) = -1526. Thus, the series is
%3D
%3D
1536x4
given by 4x
6.
24
(-1)*+1 (k – 1)!(4x)k
k!
Σ
(-1)k+1 (4x)*
k=1
k=1
c. The absolute value of the ratio of consecutive terms is
4 |a| k
which has limit 4 x as k → o, so the
k +1'
interval of convergence is (-1/4,1/4]. Note that for x = 1/4 we have the alternating harmonic series,
while for I = -1/4 we have negative 1 times the harmonic series, which diverges.
11.3.17
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