(a) Show that power series representations are unique. In other words, suppose that for all x € (-R, R) we have ∞ n=0 ²₂x² = [b₁x². n=0
(a) Show that power series representations are unique. In other words, suppose that for all x € (-R, R) we have ∞ n=0 ²₂x² = [b₁x². n=0
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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![(a) **Show that power series representations are unique.** In other words, suppose that for all \( x \in (-R, R) \) we have
\[
\sum_{n=0}^{\infty} a_n x^n = \sum_{n=0}^{\infty} b_n x^n.
\]
Deduce that for all \( n = 0, 1, 2, 3, \ldots \) we have \( a_n = b_n \).
(b) Let \( f(x) = \sum_{n=0}^{\infty} a_n x^n \) and suppose that this power series converges on \( (-R, R) \). Suppose that for all \( x \in (-R, R) \) we have \( f'(x) = f(x) \), and that \( f(0) = 1 \). What are the values of \( a_n \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2Ffe0a37a0-7345-4b1f-8462-bfdc385d7bf4%2Fd13otmc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) **Show that power series representations are unique.** In other words, suppose that for all \( x \in (-R, R) \) we have
\[
\sum_{n=0}^{\infty} a_n x^n = \sum_{n=0}^{\infty} b_n x^n.
\]
Deduce that for all \( n = 0, 1, 2, 3, \ldots \) we have \( a_n = b_n \).
(b) Let \( f(x) = \sum_{n=0}^{\infty} a_n x^n \) and suppose that this power series converges on \( (-R, R) \). Suppose that for all \( x \in (-R, R) \) we have \( f'(x) = f(x) \), and that \( f(0) = 1 \). What are the values of \( a_n \)?
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