The Basel problem asks to compute the summation of the reciprocal of the natural numbers squared, and it is known that
Recall that a function is if for every , . From Fourier analysis, every such function can be expressed as a series of cosines and sines. More precisely,
where , and is the -th Fourier coefficient equal to .
When we define the inner product of two functions to be , then it is easily seen that the “characters” form an orthonormal system, and the -th Fourier coefficient can be viewed as the projection of along the -th character. The Fourier theorem then asserts that can be approximated in -norm by any sufficiently large orthonormal system. In fact, by applying the Weierstrass M-test, we can make the following statement with a stronger convergence guarantee:
Now we have the machinery to compute .
For , we have
Note that . Using integration by parts, we have
Thus, Note that is convergent from the Cauchy condensed test. Thus, we can conclude that
where the convergence is uniform (and in particular, pointwise). Setting finishes the proof.