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An outline
In mathematics '''convergence''' of an [[infinite series]] occurs when,
In mathematics '''convergence''' of an [[infinite series]] occurs when,
<math>\lim_{N\rightarrow\infty}\sum^{N}_{n=1}a_{n}=L</math>
:<math>\lim_{N\rightarrow\infty}\sum^{N}_{n=1}a_{n}=L</math>
for <math>L</math> a finite number, <math>|L|<\infty</math>.
for <math>L</math> a finite number, <math>|L|<\infty</math>. It will occur only if (<math>\Leftarrow</math>),
:<math>\lim_{n\rightarrow\infty}a_{n}=0</math>
However there is not one unique convergence condition (if or <math>\Rightarrow</math>)
==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,
:<math>\sum^{\infty}_{n=1}a_{n}=\sum^{\infty}_{n=1}b_{n}(x-a)^{n}</math>,
This will converge if <math>\lim_{n\rightarrow\infty}|\frac{a_{n+1}}{a_{n}}|<1</math>.
===Alternating Series convergence===
An alternating series is of the form,
:<math>\sum^{\infty}_{n=1}(-1)^{n}a_{n}</math>
This will converge if <math>0\leq a_{n+1}<a_{n}</math>.
===Root test===
==Complex series convergence==
==Intervals of convergence==
[[Category:Calculus]]
[[Category:Calculus]]