A very famous mathematical problem known as the “Basel Problem” is solved by Euler in 1734. Basically, it asks for the exact value of .

Three hundred years ago, this was considered a very hard problem and even famous mathematicians of the time like Leibniz, De Moivre, and the Bernoullis could not solve it.

Euler showed (using another method different from ours) that bringing him great fame among the mathematical community. It is a beautiful equation; it is surprising that the constant , usually related to circles, appears here.

## Squaring the Fourier sine series

Assume that

Then squaring this series formally,

To see why the above hold, see the following concrete example:

## Integrate term by term

We assume that term by term integration is valid.

Recall that

So

Similarly

So (Parseval’s Identity)

## Apply Parseval’s Identity to

By Parseval’s identity,

Simplifying, we get

This is the original Euler’s proof of the Basel Problem:

https://tomcircle.wordpress.com/2014/04/10/the-basel-problem/

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How did u get the right side (bn) of the last Parseval’s Identity when f (x) = x ?

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Use known Fourier series of f(x)=x

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