Definition: Convergent Complex Series

A complex series \(\sum_{k=0}^\infty x_k\) is called convergent, if the complex sequence \((s_n)_{n\in\mathbb N}\) of partial sums \[s_n:=\sum_{k=0}^n x_k,\quad\quad n\in\mathbb N\] is a convergent complex sequence. For convergent complex series, the notation

\[\sum_{k=0}^\infty x_k\]

can, depending on the context, denote two things:

  1. the convergent real series itself or
  2. the convergent real series itself or
  1. Proposition: A General Criterion for the Convergence of Infinite Complex Series

Definitions: 1
Proofs: 2
Propositions: 3

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983