radius of convergence

To the power series

\\sum_{k=0}^{\\infty}a_{k}(x-x_{0})^{k}

(1)

there exists a number $r\\in[0,\\infty]$ , its radius of convergence, such that the series converges absolutely for all (real or complex) numbers $x$ with $|x-x_{0}|<r$ and diverges whenever $|x-x_{0}|>r$ . This is known as Abel’s theorem on power series. (For $|x-x_{0}|=r$ no general statements can be made.)

The radius of convergence is given by:

r=\\liminf_{k\\to\\infty}\\frac{1}{\\sqrt[k]{|a_{k}|}}

(2)

and can also be computed as

r=\\lim_{k\\to\\infty}\\left|\\frac{a_{k}}{a_{k+1}}\\right|,

(3)

if this limit exists.

It follows from the Weierstrass $M$ -test (http://planetmath.org/WeierstrassMTest) that for any radius $r^{\\prime}$ smaller than the radius of convergence, the power series converges uniformly within the closed disk of radius $r^{\\prime}$ .