Recall that . Perhaps the shortest proof of applies (the sophisticated) Parseval’s identity of the Fourier series to . By contrast, the simple (but long) proof in this paper, which was recently popularized by the video below, uses basic ideas from Euclidean geometry.
The following argument interpolates between these two, resulting in a proof that is both simple and short.
Take even, put , and define by for and otherwise. Then the geometric sum formula gives
Next, we apply Parseval’s identity of the discrete Fourier transform, whose proof also follows from the geometric sum formula:
(An alternative geometric proof of this identity appears in Wastlund’s paper for the case where is a power of 2.) For , we have , suggesting we select an increasing function so that
(In the last sum, we take odd modulo .) Provided this holds, then we see from the terms of that , from which it follows that .
To prove , we bound the truncation error for any :
where the last step applies the fact that and assumes . As such, we may take (say) to obtain the desired convergence.