# Using the Fourier Series To Find Some Interesting Sums - Comments

• Insights
• Svein
In summary, Svein's PF Insights post discusses using the Fourier Series to find interesting sums. The conversation following the post includes a suggestion for a follow-up article on finding a general formula for a specific summation and a discussion on different methods for evaluating summations, including using Parseval's formula. Svein then shares a trick for evaluating \sum \frac{1}{n^2} using representations of the Dirac delta function, ultimately arriving at the identity \sum_n \frac{1}{n^2} = \frac{\pi^2}{6}. The conversation concludes with praise for Svein's use of Parseval's formula for a simpler solution.
Svein
Science Advisor
Svein submitted a new PF Insights post

Using the Fourier Series To Find Some Interesting Sums

Continue reading the Original PF Insights Post.

ShayanJ and Greg Bernhardt
micromass said:
Nice article. I think a follow up could consist of finding a general formula for
$$\sum \frac{1}{n^{2p}}$$
You can do that with Fourier series too!
Yes - but I thought it best to keep it reasonably simple the first time through.

Very nice. I have done similar tricks for evaluating summations, but I didn't know the trick of using Parseval’s formula. My favorite trick is using representations of the Dirac delta function, and that's how I would evaluate $\sum \frac{1}{n^2}$. However, it's a lot more convoluted.

Start with the representation for the delta function:

$\delta(x) = \frac{1}{2\pi} + \frac{1}{\pi} \sum_n cos(nx)$

Now, integrate both sides from $-x$ to $+x$.

$sign(x) = \frac{x}{\pi} + \frac{2}{\pi} \sum_n \frac{1}{n} sin(nx)$

(where $sign(x) = \pm 1$ depending on whether $x>0$ or $x < 0$)

Integrate again, this time from $0$ to $x$:

$|x| = \frac{x^2}{2\pi} + \frac{2}{\pi} \sum_n \frac{1}{n^2} (1 - cos(nx))$

Using a trig identity, $1-cos(nx) = 2 sin^2(\frac{n}{2} x)$. So we have:

$|x| = \frac{x^2}{2\pi} + \frac{4}{\pi} \sum_n \frac{1}{n^2} sin^2(\frac{n}{2} x)$

Now, we set $x = \pi$ to get the identity:

$\pi = \frac{\pi}{2} + \frac{4}{\pi} \sum_n \frac{1}{n^2} sin^2(\frac{n}{2} \pi) = \frac{4}{\pi} \sum_{odd\ n} \frac{1}{n^2}$

So:

$\frac{\pi}{2} = \frac{4}{\pi} \sum_n \frac{1}{n^2} sin^2(\frac{n}{2} \pi) = \frac{4}{\pi} \sum_{odd\ n} \frac{1}{n^2}$

Drat! The sum on the right side is only over odd values of $n$, because $sin^2(\frac{n}{2} \pi) = 0$ when $n$ is even. But all is not lost. We can reason as follows:

$\sum_n \frac{1}{n^2} = \sum_{odd\ n} \frac{1}{n^2} + \sum_{even\ n} \frac{1}{n^2}$

but $\sum_{even\ n} \frac{1}{n^2} = \sum_{n} \frac{1}{(2n)^2} = \frac{1}{4} \sum_{n} \frac{1}{n^2}$. So we have:

$\sum_n \frac{1}{n^2} = \sum_{odd\ n} \frac{1}{n^2} + \frac{1}{4}\sum_{n} \frac{1}{n^2}$

So $\sum_{odd\ n} \frac{1}{n^2} = \frac{3}{4} \sum_n \frac{1}{n^2}$. Putting this back into our result, we have:

$\frac{\pi}{2} = \frac{4}{\pi} \sum_n \frac{1}{n^2} sin^2(\frac{n}{2} \pi) = \frac{4}{\pi} \frac{3}{4} \sum_{n} \frac{1}{n^2}$

So $\sum_n \frac{1}{n^2} = \frac{\pi^2}{6}$

Using Parseval is a lot simpler.

## 1. How does the Fourier Series work?

The Fourier Series is a mathematical tool used to represent a periodic function as a combination of sine and cosine waves. It works by decomposing a function into its constituent frequencies and amplitudes using a formula called the Fourier Transform.

## 2. What is the purpose of using the Fourier Series?

The Fourier Series is used to simplify complex mathematical problems by breaking them down into simpler, periodic functions. It is also used in signal processing, image analysis, and other areas of science and engineering to analyze and manipulate data.

## 3. Can the Fourier Series be used to find any type of sum?

No, the Fourier Series is specifically designed to calculate sums of periodic functions. It may not be effective for finding sums of non-periodic functions or those with discontinuities or singularities.

## 4. Are there any limitations to using the Fourier Series?

Yes, the Fourier Series has some limitations. It may not converge for all functions, and the convergence may be slow for some functions. It also has difficulties representing functions with sharp corners or discontinuities.

## 5. How can the Fourier Series be applied in real-world situations?

The Fourier Series has many practical applications, such as in signal processing, image compression, and data analysis. It can also be used in physics, engineering, and other fields to model and analyze periodic phenomena, such as sound waves, electromagnetic waves, and vibrations.

Replies
5
Views
2K
Replies
5
Views
3K
Replies
1
Views
2K
Replies
3
Views
2K
Replies
3
Views
833
Replies
7
Views
1K
Replies
3
Views
3K
Replies
3
Views
6K
Replies
12
Views
1K
Replies
5
Views
2K