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
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.
So (Parseval’s Identity)
Apply Parseval’s Identity to
By Parseval’s identity,
Simplifying, we get