An **infinite series** is a sum of infinitely many terms. Such a sum can have a finite value, and if it has, it is said to *converge*. The fact that infinite series can converge resolves several of Zeno?'s paradoxes.
*S*. If this is not the case, we say the series *diverges*.
### Convergence criteria

### Examples

*r* > 1 and diverges for *r* < 1, which can be shown with the integral criterium 5) from above.
This series gives rise to the Riemann zeta function.
*z*| < 1.
### Absolute convergence

*absolutely* if the series of absolute values
### Power series

### Generalizations

The simplest convergent infinite series is perhaps

- 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2

∞ ∑ 2^{-n}= 2n=0

Formally, if an infinite series

∞ ∑a_{n}n=0

is given with real (or complex) numbers *a*_{n}, we say that the series **converges towards S ** or that its

exists and is equal toNlim ∑a_{n}N→∞n=0

1) If the series ∑ *a*_{n} converges, then the sequence (*a*_{n}) converges to 0 for *n*→∞; the converse is in general not true.

2) If all the numbers *a*_{n} are positive and ∑ *b*_{n} is a convergent series such that *a*_{n} ≤ *b*_{n} for all *n*, then ∑ *a*_{n} converges as well. Conversely, if all the *b*_{n} are positive, *a*_{n} ≥ *b*_{n} for all *n* and ∑ *b*_{n} diverges, then ∑ *a*_{n} diverges as well.

3) If the *a*_{n} are positive and there exists a constant *C* < 1 such that *a*_{n+1}/*a*_{n} ≤ *C*, then ∑ *a*_{n} converges.

4) If the *a*_{n} are positive and there exists a constant *C* < 1 such that (*a*_{n})^{1/n} ≤ *C*, then ∑ *a*_{n} converges.

5) If *f*(*x*) is a positive montone decreasing function defined on the interval [1, ∞) with *f*(*n*) = *a*_{n} for all *n*, then ∑ *a*_{n} converges if and only if the integral
∫_{1}^{∞} *f*(*x*) d*x* exists.

6) The series ∑ *a*_{n} of real numbers is called *alternating* if the signs of the *a*_{n} alternate. Such a series converges if the sequences |*a*_{n}| is monotone decreasing and converges towards 0. The converse is in general not true.

The series

∞ 1 ∑ ---converges ifn=1n^{r}

The geometric series

∞ ∑converges if and only if |z^{n}n=0

The sum

∞ ∑is said to convergea_{n}n=0

∞ ∑ |converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum. If a series is not absolutely convergent, then there is always some reordering of the terms so that the reordered series diverges.a_{n}|n=0

Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called [power series]?.

Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required. However, the formal operation with non-convergent series has been retained in rings of [formal power series]? which are studied in abstract algebra. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of [generating functions]?.

The notion of series can be defined in every abelian [topological group]?; the most commonly encountered case is that of series in a Banach space.