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Radius of Convergence

For a power series defined as

$f\left(x\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{c}_n}{\left(x-a\right)}^n$

the radius of convergence $r$ is a nonnegative real number (or possibly, infinity), such that the series converges if

$\left|x-a\right|<r$

and diverges if

$\left|x-a\right|>r$ .