One of the questions in my problem sets involved finding the value of the sum $$\displaystyle \sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{2n-1} $$ by considering the Fourier series of $ f(x) = | x | $ for $ x \in \left[ - \pi , \pi \right] $ and other various questions of this style (e.g. sum of reciprocal of squares by considering Fourier series of $ f(x) = x $ on the same interval).
Verifying this result given a trial function is simple enough but it isn't obvious to me as to how these functions were obtained in the first place. My question concerns how we would go about doing this process in reverse, namely given a summation, how would we go about finding the function whose Fourier series we can use to evaluate it (of course, assuming the series can be evaluated in such a manner. Perhaps this could even be a follow on question, how can we spot when this technique can be applied to summing a series.)
This is pure speculation:
I would guess that one started with taking the Fourier series of particular functions, getting the Fourier series, then playing around to see what series one can get. I remember doing that when working through these sorts of problems in Stein and Shakarchi's Fourier Analysis.
Then when one gets good at this, notices patterns or at least gets an idea about what kinds of functions give Fourier series one can get to things like finding closed forms for $\zeta(2n)$ and stuff like that.