A Rigorous Proof Using Fourier Series
Let over the interval x ∈ (–π,π). The Fourier series for this function (worked out in that article) is
Then, using Parseval's identity (with ) we have that
- ,
where
for n ≠ 0, and a0 = 0. Thus,
for n ≠ 0 and
Therefore,
as required.
Read more about this topic: Basel Problem
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