Radius of convergence

Definition

For a power series centered at zero

Consider a power series of the form:

$\sum _{k=0}^{\infty }a_{k}x^{k}$

We define the radius of convergence using three cases:

Case name	Definition	Comment about interval of convergence (points where the power series converges, absolutely or conditionally)
Finite radius of convergence	The radius of convergence is the largest positive real number $c$, if it exists, such that the power series is an absolutely convergent series for all $x$ satisfying $|x|<c$.	One of these four: $[-c,c]$, $(-c,c)$, $[-c,c)$, and $(-c,c]$.
Zero radius of convergence	If there is no nonzero real number $x$ for which the power series converges absolutely, then the radius of convergence is defined to be 0.	$\{0\}$
Infinite radius of convergence	If the power series converges absolutely for every choice of nonzero real number $x$, then the radius of convergence is defined to be $\infty$.	all of $\mathbb {R}$.

For a power series centered at a point other than zero

Consider a power series centered at $\mu$:

$\sum _{k=0}^{\infty }a_{k}(x-\mu )^{k}$

Case name	Definition	Comment about interval of convergence (points where the power series converges, absolutely or conditionally)
Finite radius of convergence	The radius of convergence is the largest positive real number $c$, if it exists, such that the power series is an absolutely convergent series for all $x$ satisfying $|x-\mu |<c$.	One of these four: $[\mu -c,\mu +c]$, $(\mu -c,\mu +c)$, $[\mu -c,\mu +c)$, and $(\mu -c,\mu +c]$.
Zero radius of convergence	If there is no nonzero real number $x$ for which the power series converges absolutely, then the radius of convergence is defined to be 0.	$\{\mu \}$
Infinite radius of convergence	If the power series converges absolutely for every choice of nonzero real number $x$, then the radius of convergence is defined to be $\infty$.	all of $\mathbb {R}$.

Facts

• The radius of convergence of a power series $\sum _{k=0}^{\infty }a_{k}(x-\mu )^{k}$ is given as ${\frac {1}{\lim \sup _{k\to \infty }|a_{k}|^{1/k}}}$. For more, see rules for determining interval of convergence.
• Radius of convergence of derivative of power series equals radius of convergence