Infinite Series using Fourier Analysis

For our first application of Fourier Analysis, we shall look at the following series:

In other words, the sum of the squares of odd numbers:

To find the sum, let us consider the following function:

At x=π, the value of the function is the average of the values from both definitions:

Now let us compute the Fourier Series of the function. From my previous post Fourier Analysis I, we know that

Where the coefficients are given by:

Plugging in the given function and computing the integrals, we obtain:

When x=0,

For x=π,

Now, subtracting the second sum from the first, we obtain:

If you have a better or faster solution, do respond below.

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