me function f(x) = 2n+1 (-1)" x²+1 1 n 2 = X- 2x has derivatives of all orders, and the Taylor series for f about x = 0 is x + 2 X³ X5 X² --+ 2 4 8 + a. Using the Ratio Test, determine the interval of convergence of the Taylor series about x = 0 for f. b. Let g(x) be the function such that g(0) = 0 and whose derivative is f(x) = 2x x + 2 Write the first four nonzero terms of the Taylor series for g(x) about x = 0. Use the first two nonzero terms of this series to estimate g (¹) c. The series found in part (b) is an alternating series whose terms decrease in absolute value to 0. Show that the 1 approximation found in part (b) differs from g 1500 (1) by less than
me function f(x) = 2n+1 (-1)" x²+1 1 n 2 = X- 2x has derivatives of all orders, and the Taylor series for f about x = 0 is x + 2 X³ X5 X² --+ 2 4 8 + a. Using the Ratio Test, determine the interval of convergence of the Taylor series about x = 0 for f. b. Let g(x) be the function such that g(0) = 0 and whose derivative is f(x) = 2x x + 2 Write the first four nonzero terms of the Taylor series for g(x) about x = 0. Use the first two nonzero terms of this series to estimate g (¹) c. The series found in part (b) is an alternating series whose terms decrease in absolute value to 0. Show that the 1 approximation found in part (b) differs from g 1500 (1) by less than
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![The function f(x)
n=0
(-1)" x²n+1
(−1)"
22
= X
=
2x
has derivatives of all orders, and the Taylor series for f about x = 0 is
2
x² + 2
x³ x5
X
+
2
7
x²
X
4 8
a. Using the Ratio Test, determine the interval of convergence of the Taylor series about x = 0 for f.
2x
6.
b. Let g(x) be the function such that g(0) = 0 and whose derivative is f(x)
=
2
x + 2
Write the first four nonzero
terms of the Taylor series for g(x) about x = 0. Use the first two nonzero terms of this series to estimate g
8 (²-).
c. The series found in part (b) is an alternating series whose terms decrease in absolute value to 0. Show that the
1
approximation found in part (b) differs from g
1500
(=) by less than](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c4923bb-9172-4c62-ab1a-cda8928d94fc%2F7927db13-22a1-4ec6-a0cd-d64df83c40e6%2F2ytyuv7_processed.png&w=3840&q=75)
Transcribed Image Text:The function f(x)
n=0
(-1)" x²n+1
(−1)"
22
= X
=
2x
has derivatives of all orders, and the Taylor series for f about x = 0 is
2
x² + 2
x³ x5
X
+
2
7
x²
X
4 8
a. Using the Ratio Test, determine the interval of convergence of the Taylor series about x = 0 for f.
2x
6.
b. Let g(x) be the function such that g(0) = 0 and whose derivative is f(x)
=
2
x + 2
Write the first four nonzero
terms of the Taylor series for g(x) about x = 0. Use the first two nonzero terms of this series to estimate g
8 (²-).
c. The series found in part (b) is an alternating series whose terms decrease in absolute value to 0. Show that the
1
approximation found in part (b) differs from g
1500
(=) by less than
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