Questions on whether certain series diverge, and how to deal with divergent series using summation methods such as Ramanujan summation and others.

When a series of real or, more generally, complex numbers diverges, it is still possible, sometimes, to give a meaning to its sum. For instance, given a series $\displaystyle\sum_{n=0}^\infty a_n$, if the series $\displaystyle\sum_{n=0}^\infty a_nx^n$ converges for each $x\in[0,1)$ and if furthermore the limit$$\lim_{x\to1}\sum_{n=0}^\infty a_nx^n$$exists, it is natural to say that $\displaystyle\sum_{n=0}^\infty a_n$ is this limit. Besides, if the series $\displaystyle\sum_{n=0}^\infty a_n$ actually converges, then the two sums are the same.