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(Solved): By computing the Fourier series of f(x)=x2, show that n=1n41=904. ...

$By computing the Fourier series of $ f(x)=x^{2} $, show that \[ \sum_{n=1}^{\infty} \frac{1}{n^{4}}=\frac{\pi^{4}}{90} . \]$

By computing the Fourier series of

f (x) = x^{2}

, show that

\sum_{n = 1}^{\infty} \frac{1}{n ^{4}} = \frac{π ^{4}}{90} .

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To compute the Fourier series of the function   , we need to find the Fourier coefficients. Here are the three steps to compute the series and obtain the desired result:

Step 1: Compute the Fourier coefficients
The Fourier coefficient    is given by the formula:

Substituting    into the formula, we have:

Using integration by parts twice, we obtain:

For n=0, the coefficient   ? is given by:


The Fourier coefficient an? measures the contribution of the cosine function with frequency n to the function f(x). For the function    we find that all the even coefficients    are zero. This means that the cosine functions with even frequencies do not affect the function

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(Solved): By computing the Fourier series of f(x)=x2, show that n=1n41=904. ...

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