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 -periodic 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 .
We say that is the Fourier series of . 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 asserts that can be approximated in -norm by its Fourier series. In fact, by applying the Weierstrass M-test, we can make the following statement with a stronger convergence guarantee:
Furthermore, the Fourier transform itself encodes many of the properties of the original function . In particular, we note that the symmetry of is preserved in its Fourier transform.
If is odd, then for every , .
Now we have the machinery to compute .
For , we have
Using integration by parts, we have for every ,
From Euler’s formula, the Fourier series of can be written as
Since is even, by Proposition, . From our calculation, the Fourier series of is explicitly
Note that is convergent from the Cauchy condensation test. Thus, by the Convergence Theorem, we conclude that
where the convergence is uniform (and in particular, pointwise). Setting , we see that
Lastly, observe that
finishing the proof.