Radius of Convergence of Power Series over Factorial

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Theorem

Real Case

Let $\xi \in \R$ be a real number.

Let $\displaystyle f \left({x}\right) = \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!}$.


Then $f \left({x}\right)$ converges for all $x \in \R$.


That is, the interval of convergence of the power series $\displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!}$ is $\R$.


Complex Case

Let $\xi \in \C$ be a complex number.

Let $\displaystyle \map f z = \sum_{n \mathop = 0}^\infty \dfrac {\paren {z - \xi}^n} {n!}$.


Then $\map f z$ converges absolutely for all $z \in \C$.

That is, the radius of convergence of the power series $\displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {z - \xi}^n} {n!}$ is infinite.