Convergence of $e^x$

Question

I am working with the Maclaurin series for $f(x)= e^x$. I am in the point of proving that the series converges to $f(x)$ for all $x$, using Taylor's theorem with remainder I have to show the following: $$ \mathop {\lim }\limits_{n \to \infty } \frac{{\left| x \right|^{n + 1} }}{{(n + 1)!}} = 0. $$ How do you work out the divergence test?

Answer 1

Are you familiar with the ratio test for infinite series? From this you can prove pretty easily that $\sum_{n = 1}^{\infty} \dfrac{x^n}{n!}$ converges absolutely for all $x \in \mathbb{R}$. Then by the divergence test, its general term $\dfrac{x^n}{n!}$ must tend to $0$ as $n \to \infty$.

Answer 2

It suffices to show $\frac{|x|^n}{n!} \to 0$. Use the squeeze theorem with $$0 \le \frac{|x|^n}{n!} \le \frac{|x|^n}{n(n-1) \cdots \lceil n/2 \rceil} \le \frac{|x|^n}{(n/2)^{n/2}}$$