The Basel problem asks to compute the summation of the reciprocal of the natural numbers squared, and it is known that

While there are numerous known proofs (our presentation resembles Proof 9 most closely), we give a self-contained one that motivates the usage of Fourier analysis for periodic functions.

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

and similarly

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.