Definition:Radius of Convergence/Real Domain
This page is about Radius of Convergence in the context of Power Series. For other uses, see Radius.
Definition
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.
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
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 15.2$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): radius of convergence