Infinite Series using Fourier Analysis

Vivek Palaniappan
Sep 9, 2018 · 3 min read

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.

Vivek Palaniappan

Written by

Looking into the broad intersection between engineering, finance and AI

Engineer Quant

Delve into engineering and quantitative analysis

Welcome to a place where words matter. On Medium, smart voices and original ideas take center stage - with no ads in sight. Watch
Follow all the topics you care about, and we’ll deliver the best stories for you to your homepage and inbox. Explore
Get unlimited access to the best stories on Medium — and support writers while you’re at it. Just $5/month. Upgrade