# What is the radius of convergence of the series sum_(n=0)^oo(x^n)/(n!)?

Sep 22, 2014

The radius of convergence is $\infty$.

Let us look at some details.

Let a_n=x^n/{n!}. Rightarrow a_{n+1}=x^{n+1}/{(n+1)!}

By Ratio Test,

lim_{n to infty}|{a_{n+1}}/{a_n}| =lim_{n to infty}|{x^{n+1}/{(n+1)!}}/{x^n/{n!}}| =lim_{n to infty}|x/{n+1}|=0<1

Since ${\lim}_{n \to \infty} | \frac{{a}_{n + 1}}{{a}_{n}} | < 1$ regardless of the value of $x$, thr radius of convergence is $\infty$.