# Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted

The *n*th partial sum *S*_{n} is the sum of the first *n* terms of the sequence; that is,

A series is **convergent** (or **converges**) if the sequence of its partial sums tends to a limit; that means that, when adding one 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 there exists a number such that for every arbitrarily small positive number , there is a (sufficiently large) integer such that for all ,

If the series is convergent, the (necessarily unique) number is called the *sum of the series*.

The same notation

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.