Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence

In the setting of Complex Analysis: Standing Notation, let $(a_{k})$ be a sequence in $\mathbb{C}$.

(i) Partial sums. For a natural number $n$, the $n$-th partial sum of $(a_{k})$ is the finite sum $s_{n}=\sum_{k=1}^{n}a_{k}$ in $\mathbb{C}$, and the sequence $(s_{n})$ is its sequence of partial sums.

(ii) Convergence and the sum. The series $\sum_{k=1}^{\infty}a_{k}$ converges if $(s_{n})$ is convergent, and diverges otherwise. If it converges, its sum, also written $\sum_{k=1}^{\infty}a_{k}$, is the limit of $(s_{n})$.

(iii) Absolute convergence. The series $\sum_{k=1}^{\infty}a_{k}$ converges absolutely if the series $\sum_{k=1}^{\infty}|a_{k}|$ of the moduli converges in $\mathbb{R}$.