(A calculator is not allowed for this problem) The function f is defined by the power series (x-1)2(x-1)3 (x-1)4 f(x)=(x-1)- + +... 2 3 4 Σ (-1)^-1(x-1)" n n = 1 for all real numbers x for which the series converges. (a) Determine the interval of convergence for f(x). Justify your answer. (b) Given g(x) = f'(x), find the first three terms and the general term of the power series for g(x). (c) Find a rational function that is identical to g(x) over its interval of convergence
(A calculator is not allowed for this problem) The function f is defined by the power series (x-1)2(x-1)3 (x-1)4 f(x)=(x-1)- + +... 2 3 4 Σ (-1)^-1(x-1)" n n = 1 for all real numbers x for which the series converges. (a) Determine the interval of convergence for f(x). Justify your answer. (b) Given g(x) = f'(x), find the first three terms and the general term of the power series for g(x). (c) Find a rational function that is identical to g(x) over its interval of convergence
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please use these questions to find non-zero terms!
Transcribed Image Text:(A calculator is not allowed for this problem)
The function f is defined by the power series
(x-1)2(x-1)³
(x-1)4
f(x)=(x-1)-
+
+...
2
3
4
=
Σ
(-1)^-1(x-1)"
n
n = 1
for all real numbers x for which the series
converges.
(a) Determine the interval of convergence for f(x).
Justify your answer.
(b) Given g(x) = f'(x), find the first three terms
and the general term of the power series for g(x).
(c) Find a rational function that is identical to g(x)
over its interval of convergence
(d) Let h(x) = f(x³ +1). Find the closed form
expression for (not an infinite series expression)
h'(x).
(HINT: use your result from part (c))
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