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
