Convergence

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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)

Contents

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}<a_{n}.

Root test

Complex series convergence

Intervals of convergence

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