Definition: Complex Infinite Series

Let \((x_n)_{n\in\mathbb N}\) be a complex sequence. 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 called the (infinite) complex series \[\sum_{k=0}^\infty x_k\quad\quad( * ).\]

Note: If the sequence of partial sums is convergent, the expression \( ( * ) \) also denotes the limit to which the sequence converges.

Definitions: 1 2
Proofs: 3 4 5
Propositions: 6 7 8

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