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Using the Fourier Series To Find Some Interesting Sums

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- Thread starter Svein
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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.

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Using the Fourier Series To Find Some Interesting Sums

Continue reading the Original PF Insights Post.

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Svein said:

Using the Fourier Series To Find Some Interesting Sums

Continue reading the Original PF Insights Post.

Nice article. I think a follow up could consist of finding a general formula for

[tex]\sum \frac{1}{n^{2p}}[/tex]

You can do that with Fourier series too!

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Yes - but I thought it best to keep it reasonably simple the first time through.micromass said:Nice article. I think a follow up could consist of finding a general formula for

[tex]\sum \frac{1}{n^{2p}}[/tex]

You can do that with Fourier series too!

- #4

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Start with the representation for the delta function:

[itex]\delta(x) = \frac{1}{2\pi} + \frac{1}{\pi} \sum_n cos(nx)[/itex]

Now, integrate both sides from [itex]-x[/itex] to [itex]+x[/itex].

[itex]sign(x) = \frac{x}{\pi} + \frac{2}{\pi} \sum_n \frac{1}{n} sin(nx)[/itex]

(where [itex]sign(x) = \pm 1[/itex] depending on whether [itex]x>0[/itex] or [itex]x < 0[/itex])

Integrate again, this time from [itex]0[/itex] to [itex]x[/itex]:

[itex]|x| = \frac{x^2}{2\pi} + \frac{2}{\pi} \sum_n \frac{1}{n^2} (1 - cos(nx))[/itex]

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

[itex]|x| = \frac{x^2}{2\pi} + \frac{4}{\pi} \sum_n \frac{1}{n^2} sin^2(\frac{n}{2} x)[/itex]

Now, we set [itex]x = \pi[/itex] to get the identity:

[itex]\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}[/itex]

So:

[itex]\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}[/itex]

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

[itex]\sum_n \frac{1}{n^2} = \sum_{odd\ n} \frac{1}{n^2} + \sum_{even\ n} \frac{1}{n^2}[/itex]

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

[itex]\sum_n \frac{1}{n^2} = \sum_{odd\ n} \frac{1}{n^2} + \frac{1}{4}\sum_{n} \frac{1}{n^2}[/itex]

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

[itex]\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}[/itex]

So [itex]\sum_n \frac{1}{n^2} = \frac{\pi^2}{6}[/itex]

Using Parseval is a lot simpler.

- #5

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Nice Insight @Svein!

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.

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.

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.

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.

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.

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