Definition:Radius of Convergence/Real Domain

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This page is about Radius of Convergence in the context of Power Series. For other uses, see Radius.


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.