Real numbers/Series/Definition

Series

Let ${}{\left(a_{k}\right)}_{k\in \mathbb {N} }$ be a sequence of real numbers. The series ${}\sum _{k=0}^{\infty }a_{k}$ is the sequence ${}{\left(s_{n}\right)}_{n\in \mathbb {N} }$ of the partial sums

${}s_{n}:=\sum _{k=0}^{n}a_{k}\,.$

If the sequence ${}{\left(s_{n}\right)}_{n\in \mathbb {N} }$ converges, then we say that the series converges. In this case, we also write

$\sum _{k=0}^{\infty }a_{k}$

for its limit, and this limit is called the sum of the series.