The Basel problem via Fourier series

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, If is continuous and -periodic, then     where , and is the -th Fourier coefficient equal to . We say that is the Fourier series of . When we define the […]

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