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
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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 \(x\).
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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