# Convergence

(Redirected from Convergent)

In mathematics convergence of an infinite series occurs when,

$\lim_{N\rightarrow\infty}\sum^{N}_{n=1}a_{n}=L$

for L a finite number, $|L|<\infty$. The terms in the limit are known as partial sums. If the partial sums approach $\infty$ or $-\infty$, the series is said to diverge. If the partial sums do not approach a single value, finite or infinite, then the series is said to have no limit.

Convergence of the series will occur only if ($\Leftarrow$),

$\lim_{n\rightarrow\infty}a_{n}=0$

However there is not one unique convergence condition (if or $\Rightarrow$)

## Real series convergence

There are two common convergence, power series and alternating series. However there are many others.

### Power Series convergence

A power series is of the form,

$\sum^{\infty}_{n=1}a_{n}=\sum^{\infty}_{n=1}b_{n}(x-a)^{n}$,

This will converge if $\lim_{n\rightarrow\infty}|\frac{a_{n+1}}{a_{n}}|<1$.

### Alternating Series convergence

An alternating series is of the form,

$\sum^{\infty}_{n=1}(-1)^{n}a_{n}$

This will converge if $0\leq a_{n+1}.