Real numbers/Series/Definition

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

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

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

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

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