1. Overview of Fourier Series
In mathematics, infinite series are very important. They are used extensively in calculators and computers for evaluating values of many functions.
The Fourier Series is really interesting, as it uses many of the mathematical techniques that you have learned before, like graphs, integration, differentiation, summation notation, trigonometry, etc.
If you get stuck, hopefully the Helpful Revision page will give you inspiration.
Infinite Series - Numbers
Useful Background
Check out the Series chapter, especially Infinite series. (In particular, note what it says about convergence of an infinite series.)
A geometric progression is a set of numbers with a common ratio.
Example: 1, 2, 4, 8, 16
A series is the sum of a sequence of numbers.
Example: 1 + 2 + 4 + 8 + 16
An infinite series that converges to a particular value has a common ratio less than 1.
Example: `1 + 1/3 + 1/9 + 1/27 + ... = 3/2`
When we expand functions in terms of some infinite series, the series will converge to the function as we take more and more terms.
Infinite Series Expansions of Functions
We learned before in the Infinite Series Expansions chapter how to re-express many functions (like sin x, log x, ex, etc) as a polynomial with an infinite number of terms.
We saw how our polynomial was a good approximation near some value x = a (in the case of Taylor Series) or x = 0 (in the case of Maclaurin Series). To get a better approximation, we needed to add more terms of the polynomial.
Fourier Series - A Trigonometric Infinite Series
In this chapter we are also going to re-express functions in terms of an infinite series. However, instead of using a polynomial for our infinite series, we are going to use the sum of sine and/or cosine functions.
Fourier Series is used in the analysis of signals in electronics. For example, later we will see the Fast Fourier Transform, which talks about pulse code modulation which is used when recording digital music.
Example
We will see functions like the following, which approximates a saw-tooth signal:
`f(t)=1+2\ sin t-` `sin 2t+2/3\ sin 3t` `-1/2sin 4t` `+2/5sin 5t+...`
Fourier series approximation of a saw-tooth signal
How does it work?
As we add more terms to the series, we find that it converges to a particular shape.
Taking one extra term in the series each time and drawing separate graphs, we have:
`f(t) = 1` (first term of the series)
We next add the next term, which is `2 sin t`:
`f(t) = 1 + 2\ sin t` (first 2 terms of the series)
We continue and the next term, which is `- sin 2t`:
`f(t) = 1 + 2 sin t - sin 2t` (first 3 terms of the series)
`f(t)=1+2\ sin t` `-sin 2t+2/3sin 3t` (first 4 terms)
`f(t)=1+2\ sin t` `-sin 2t+2/3sin 3t` `-1/2sin 4t` (first 5 terms)
`f(t)=1+2\ sin t` `-sin 2t+2/3sin 3t` `-1/2sin 4t+2/5sin 5t` (first 6 terms)
We say that the infinite Fourier series converges to the saw tooth curve.
That is, if we take more and more terms, the graph will look more and more like a saw tooth. If we could take an infinite number of terms, the graph would look like a set of saw teeth.
You can explore this example here: Fourier graph interactive.
Where did this series come from?
I'm glad you asked. We'll see how this works, and where the terms in the series come from, in the next sections.