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:



We now have the solution to the summation. This is quite an interesting approach to find sums of infinite series, but is in fact a common way to formally evaluate infinite sums. It was also used to find a rigorous proof for the Basel problem, which is the sum of squares of reciprocals. For an excellent video on the Basel problem, do check this out.
If you have a better or faster solution, do respond below.

