In mathematics, a **series** is a sum of a sequence of terms.

Examples of simple series include arithmetic series which is a sum of a arithmetic progression which can be written as:

**infinite series**is a sum of infinitely many terms. Such a sum can have a finite value; if it has, it is said to

*converge*; if it does not, it is said to

*diverge*. The fact that infinite series can converge resolves several of Zeno's paradoxes.

The simplest convergent infinite series is perhaps

*equal*to 2, but it does prove that it is

*at most*2 -- in other words, the series has an upper bound.

This series is a geometric series and mathematicians usually write it as:

*a*

_{n}, we say that the series

**converges towards**or that its

*S***value is**if the limit

*S**S*. If there is no such number, then the series is said to

*diverge*.

Table of contents |

2 Convergence criteria 3 Examples 4 Absolute convergence 5 Power series 6 Generalizations |

### Some types of infinite series

- A
*geometric series*is one where each successive term is produced by multiplying the previous term by a constant number. Example: 1 + 1/2 + 1/4 + 1/8 + 1/16... - The
*harmonic series*is the series 1 + 1/2 + 1/3 + 1/4 + 1/5... - An
*alternating series*is a series where terms alternate signs. Example: 1 - 1/2 + 1/3 + 1/4 - 1/5...

### Convergence criteria

- If the series ∑
*a*_{n}converges, then the sequence (*a*_{n}) converges to 0 for*n*→∞; the converse is in general not true. - 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. If all the*b*_{n}are positive,*a*_{n}≥*b*_{n}for all*n*and ∑*b*_{n}diverges, then ∑*a*_{n}diverges as well. - 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. - If the
*a*_{n}are positive and there exists a constant*C*< 1 such that (*a*_{n})^{1/n}≤*C*, then ∑*a*_{n}converges. - If
*f*(*x*) is a positive monotone 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*is finite. - A series of the form ∑ (-1)
^{n}*a*_{n}(with*a*_{n}≥ 0) is called*alternating*. Such a series converges if the sequence*a*_{n}is monotone decreasing and converges towards 0. The converse is in general not true. - See also ratio test.

### Examples

The series

*r*> 1 and diverges for

*r*≤ 1, which can be shown with the integral criterion 5) from above. As a function of

*r*, the sum of this series is Riemann's zeta function.

The geometric series

*z*| < 1.

The telescoping series

*b*

_{n}converges to a limit

*L*as

*n*goes to infinity. The value of the series is then

*b*

_{1}-

*L*.

### Absolute convergence

is said to**converge absolutely**if the series of absolute values

If a series converges, but not absolutely, then one can always find a reordering of the terms so that the reordered series diverges. Even more: if the *a*_{n} are real and *S* is any real number, one can find a reordering so that the reordered series converges with limit *S* (Riemann).

### Power series

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. See also radius of convergence.

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.

### Generalizations

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.