# Necessary and sufficient conditions for the convergence of a series of complex terms

Given a sequence $$( a_n )_{n \in \mathbb{N}}$$ with $$a_n \in \mathbb{C}$$ for all $$n \in \mathbb{N}$$ and its associated series $$\sum_{n=1}^{\infty} a_n$$ how many necessary and sufficient critieria, apart from the standard Cauchy convergence criterion, are known? I am aware only of Tauber's second theorem which states that the series of complex terms $$\sum_{n=0}^\infty a_n$$ is summable and the value of its sum is $$s \in \mathbb{C}$$ if and only if

1. $$\lim_{z\to 1^-}\sum_{n=0}^\infty a_nz^n=s$$ and
2. $$a_1+2a_2+\cdots+na_n=o(n)$$.

and I wonder if there are other similar conditions.

• I have actually never seen this analogue for complex numbers. However, there is a really well-written section on Tauberian theorems (and their generalizations by Hardy, Littlewood, Karamata, and more) in Chapter 7.2 of "Invitation to Classical Analysis" by Peter Duren
– Zim
Jun 17 '20 at 16:55