Are the following Fourier series convergent (on an interval, e.g. $-\pi\leq x\leq\pi$ ), if yes, what are their limit functions?
(a) $\sum_{n=1}^\infty\frac{(-1)^{n}}{n^2}\cot(\frac{n}{2})\sin(nx)$
(b) $\sum_{n=1}^\infty\frac{1}{n}\cot(\frac{n}{2})\sin^2(\frac{nx}{2})$
(we know that they are convergent for some values of $x$, e.g. $x=1$)
Subdivide the infinite interval of summation N or Z into several consecutive intervals of the form $10^n\ldots10^{n+1}$, and numerically evaluate the sum on each such sub-interval, using a computer software of your own choosing. If the value of each such sub-series decreases at an exponential rate, conclude empirically that the series converges. This is definitely the case with the first series. As for the second, the opposite seems to be the case, with the possible exception of $x\in\mathbb Z$, though even that is somewhat doubtful. As far as closed form solutions are concerned, this is unlikely to happen in general, since the plot of the first series resembles that of an everywhere continuous but nowhere differentiable Weierstrass function. Particular values, however, are an entirely different matter altogether; thus, for instance, we have $S_1\bigg(\dfrac12\bigg)=\dfrac1{16}-\dfrac{\pi^2}{12}$ .