Convergence

In mathematics convergence of an infinite series occurs when,

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

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,

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

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,

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

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