Understanding the Fourier Series

In this article, we will look closer at a specific series — specifically the Fourier Series. We will look at how to find them, and then also how to check whether it is convergent or not.

Definition of a Fourier Series

First of all, we will need to know what is meant by a Fourier Series. So what exactly is it? A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. This may seem a bit abstract, so let us explain it more clearly.

We remember that a periodic function is a function that repeats its values at regular intervals. A Fourier Series is then an expansion of an infinite sum of sines and cosines, which we can use to approximate our periodic function.

Finding the Fourier Series

The question is then, how do we find the Fourier Series? Let us first look at how the Fourier series of the function f(x) looks:

It is then clear that we need to find the coefficients: a_0, a_n, and b_n. The formulas are given by:

However, sometimes we can make it easier for ourselves to find the function. We can look at whether the function is even or uneven because we have the following rules:

• Even Function: The Fourier series expansion of an even function with the period of 2π does not involve any terms with sines. It has the form:
• Uneven Function: The Fourier series expansion of an…