Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence $(a_{0},a_{1},a_{2},\ldots )$ defines a series S that is denoted

$S=a_{0}+a_{1}+a_{2}+\cdots =\sum _{k=0}^{\infty }a_{k}.$ The nth partial sum Sn is the sum of the first n terms of the sequence; that is,

$S_{n}=\sum _{k=1}^{n}a_{k}.$ A series is convergent (or converges) if the sequence $(S_{1},S_{2},S_{3},\dots )$ of its partial sums tends to a limit; that means that, when adding one $a_{k}$ 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 there exists a number $\ell$ such that for every arbitrarily small positive number $\varepsilon$ , there is a (sufficiently large) integer $N$ such that for all $n\geq N$ ,

$\left|S_{n}-\ell \right|<\varepsilon .$ If the series is convergent, the (necessarily unique) number $\ell$ is called the sum of the series.

The same notation

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