Series of Real Numbers

In the setting of The Real Numbers: Standing Notation and Background, let $(a_{k})_{k\in\mathbb{N}}$ be a sequence of real numbers.

1. (Partial sums)(i) 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 $(s_{n})_{n\in\mathbb{N}}$ is a sequence of real numbers, called the sequence of partial sums of $(a_{k})$.

2. (Convergence and sum)(ii) The series $\sum_{k=1}^{\infty}a_{k}$ converges if its sequence of partial sums converges, and diverges otherwise. When it converges, the limit of the sequence of partial sums is unique by uniqueness of limits; that limit is the sum of the series, and is also written $\sum_{k=1}^{\infty}a_{k}$, so that

$$\sum_{k=1}^{\infty}a_{k}=\lim_{n\to\infty}s_{n}.$$

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