Question

Show that ∞n=0 (x^n)/(n!)converges for all x ∈ R.

Answer 1

Using the ration test, the radius of convergence of the series $\displaystyle \sum_{n=0}^{+\infty}\frac{x^n}{n!}$ is

$$r=\lim_{n\rightarrow\infty}\left|\frac{c_n}{c_{n+1}}\right|=\lim_{n\rightarrow\infty}\left|\frac{\frac{1}{n!}}{\frac{1}{(n+1)!}}\right|=\lim_{n\rightarrow\infty}n+1=+\infty.$$

That is, for all $x\in\mathbb{R}$, the series is convergent.