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

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(a_{k})$ be a sequence in $\mathbb{R}$.

¶ For $n\in\mathbb{N}$, the $n$-th partial sum of $(a_{k})$ is the finite sum $s_{n}=\sum_{k=1}^{n}a_{k}$, and the sequence $(s_{n})$ is its sequence of partial sums.

¶ 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 $(s_{n})$.

¶ The series $\sum_{k=1}^{\infty}a_{k}$ converges absolutely if the series $\sum_{k=1}^{\infty}|a_{k}|$ converges.