# Infinite series

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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.

The simplest convergent infinite series is perhaps

1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2
It is possible to "visualize" its convergence on the real number line. This series is a geometric series and mathematicians usually write it as
```    ∞
∑  2-n  =  2
n=0
```

Formally, if an infinite series

```    ∞
∑  an
n=0
```

is given with real (or complex) numbers an, we say that the series converges towards S or that its value is S if the limit

```          N
lim   ∑  an
N→∞  n=0
```
exists and is equal to S. If this is not the case, we say the series diverges.

### Convergence criteria

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

2) If all the numbers an are positive and ∑ bn is a convergent series such that anbn for all n, then ∑ an converges as well. Conversely, if all the bn are positive, anbn for all n and ∑ bn diverges, then ∑ an diverges as well.

3) If the an are positive and there exists a constant C < 1 such that an+1/anC, then ∑ an converges.

4) If the an are positive and there exists a constant C < 1 such that (an)1/nC, then ∑ an converges.

5) If f(x) is a positive montone decreasing function defined on the interval [1, ∞) with f(n) = an for all n, then ∑ an converges if and only if the integral1 f(x) dx exists.

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

### Examples

The series

```    ∞   1
∑  ---
n=1  nr
```
converges if 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.
```    ∞
∑  zn
n=0
```
converges if and only if |z| < 1.

### Absolute convergence

The sum

```    ∞
∑  an
n=0
```
is said to converge absolutely if the series of absolute values
```    ∞
∑  |an|
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.

### 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]?.

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.

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