а.) Determine the Fourier series expansion for the periodic extension of (x, B) = cosh(ßx) for -n < x < T, where B is a fixed positive constant. b.) Then use this to compute exact values for the sums 00 (-1)" n² + B? and n² + B2 n=1 2 n=1 as functions of ß for 0 < B. Hint: Try evaulating f(x,ß) at different values of x.
а.) Determine the Fourier series expansion for the periodic extension of (x, B) = cosh(ßx) for -n < x < T, where B is a fixed positive constant. b.) Then use this to compute exact values for the sums 00 (-1)" n² + B? and n² + B2 n=1 2 n=1 as functions of ß for 0 < B. Hint: Try evaulating f(x,ß) at different values of x.
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 Fourier Series
a.) Determine the Fourier series expansion for the periodic extension of
\[ f(x, \beta) = \cosh(\beta x) \]
for \(-\pi < x < \pi\), where \(\beta\) is a fixed positive constant.
b.) Then use this to compute exact values for the sums
\[
\sum_{n=1}^{\infty} \frac{1}{n^2 + \beta^2} \quad \text{and} \quad \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2 + \beta^2}
\]
as functions of \(\beta\) for \(0 < \beta\). **Hint:** Try evaluating \(f(x, \beta)\) at different values of \(x\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5d528f6-f704-4253-8470-44f2fbc0de86%2Ff9d18bc7-b43b-428b-9c97-c8195e7169b6%2Fr3u2zk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### A Fourier Series
a.) Determine the Fourier series expansion for the periodic extension of
\[ f(x, \beta) = \cosh(\beta x) \]
for \(-\pi < x < \pi\), where \(\beta\) is a fixed positive constant.
b.) Then use this to compute exact values for the sums
\[
\sum_{n=1}^{\infty} \frac{1}{n^2 + \beta^2} \quad \text{and} \quad \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2 + \beta^2}
\]
as functions of \(\beta\) for \(0 < \beta\). **Hint:** Try evaluating \(f(x, \beta)\) at different values of \(x\).
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