Prove convergence of alternating sequence with factorial in denominator $\frac{\left(-5\right)^n}{n!}$

Question

How do I prove convergence of below alternating sequence with a factorial in the denominator?

$\{\frac{\left(-5\right)^n}{n!}\}$

How can I apply the principles described in this post to this question? What sequences can I use to apply Squeeze Theorem?

Answer 1

Note that

$$\frac{\left(-5\right)^n}{n!}=(-1)^n\frac{5^n}{n!}\to0$$

indeed

$$-\frac{5^n}{n!}\le (-1)^n\frac{5^n}{n!}\le \frac{5^n}{n!}$$

and by ratio test

$$\frac{5^{n+1}}{(n+1)!}\frac{n!}{5^n}=\frac{5}{n+1}\to 0$$