Radius of Convergence of Power Series over Factorial/Real Case

Theorem

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

Let $\ds \map f x = \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!}$.

Then $\map f x$ converges for all $x \in \R$.

That is, the interval of convergence of the power series $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!}$ is $\R$.

This is a power series in the form $\ds \sum_{n \mathop= 0}^\infty a_n \paren {x - \xi}^n$ where $\sequence {a_n} = \sequence {\dfrac 1 {n!} }$.

$\ds \lim_{n \mathop \to \infty} \size {\dfrac {a_{n + 1} } {a_n} }$

$=$

$\ds \lim_{n \mathop \to \infty} \size {\dfrac {\dfrac 1 {\paren {n + 1}!} } {\dfrac 1 {n!} } }$

$\ds $

$=$

$\ds \lim_{n \mathop \to \infty} \size {\dfrac {n!} {\paren {n + 1}!} }$

$\ds $

$=$

$\ds \lim_{n \mathop \to \infty} \size {\dfrac 1 {n + 1} }$

$\ds $

$=$

$\ds 0$

Hence the result.

$\blacksquare$