(d) Let h(x) = f(x³ +1). Find the closed form expression for (not an infinite series expression) h'(x).

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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Please JUST solve D! Thank you!
(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
2
3
4
(−1)″ −¹(x − 1)″
n
f(x)=(x-1)-
=
∞
Σ
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), .
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 2 3 4 (−1)″ −¹(x − 1)″ n f(x)=(x-1)- = ∞ Σ 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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