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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.

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    $\begingroup$ 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 $\endgroup$
    – Zim
    Jun 17 '20 at 16:55

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