Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence ${\displaystyle (a_{1},a_{2},a_{3},\ldots )}$ defines a series S that is denoted

${\displaystyle S=a_{1}+a_{2}+a_{3}+\cdots =\sum _{k=1}^{\infty }a_{k}.}$

The nth partial sum S_n is the sum of the first n terms of the sequence; that is,

${\displaystyle S_{n}=a_{1}+a_{2}+\cdots +a_{n}=\sum _{k=1}^{n}a_{k}.}$

A series is convergent (or converges) if and only if the sequence ${\displaystyle (S_{1},S_{2},S_{3},\dots )}$ of its partial sums tends to a limit; that means that, when adding one ${\displaystyle a_{k}}$ after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if and only if there exists a number ${\displaystyle \ell }$ such that for every arbitrarily small positive number ${\displaystyle \varepsilon }$, there is a (sufficiently large) integer ${\displaystyle N}$ such that for all ${\displaystyle n\geq N}$,

${\displaystyle \left|S_{n}-\ell \right|<\varepsilon .}$

If the series is convergent, the (necessarily unique) number ${\displaystyle \ell }$ is called the sum of the series.

The same notation

${\displaystyle \sum _{k=1}^{\infty }a_{k}}$

is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b.

Any series that is not convergent is said to be divergent or to diverge.