Definition:Radius of Convergence
This page is about Radius of Convergence in the context of Power Series. For other uses, see Radius.
Definition
Real Domain
Let $\xi \in \R$ be a real number.
Let $\ds \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$.
Let $I$ be the interval of convergence of $\map S x$.
Let the endpoints of $I$ be $\xi - R$ and $\xi + R$.
(This follows from the fact that $\xi$ is the midpoint of $I$.)
Then $R$ is called the radius of convergence of $\map S x$.
If $\map S x$ is convergent over the whole of $\R$, then $I = \R$ and thus the radius of convergence is infinite.
Complex Domain
Let $\xi \in \C$ be a complex number.
For $z \in \C$, let:
- $\ds \map f z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$
be a power series about $\xi$.
The radius of convergence is the extended real number $R \in \overline \R$ defined by:
- $R = \ds \inf \set {\cmod {z - \xi}: z \in \C, \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n \text{ is divergent} }$
where a divergent series is a series that is not convergent.
As usual, $\inf \O = +\infty$.
Linguistic Note
The plural of radius is radii, pronounced ray-dee-eye.
This irregular plural form stems from the Latin origin of the word radius, meaning ray.
The ugly incorrect form radiuses can apparently be found, but rarely in a mathematical context.
Sources
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.3.2$: Power series: Theorem $1.11$