TITLE: Determine whether the series converges or diverges.
QUESTION [8 upvotes]: Determine whether the series converges or diverges. 
$$
 \sum _{n=1}^{\infty }\:\left(\frac{19}{n!}\right)
$$
I know that this question a lot easier if I use ratio test but I  have not learned ratio test yet. The only option I have is divergence, comparison, limit comparison, and integral test. How can I prove that this series converges by using the limited tests. 
Thanks in advance.

REPLY [3 votes]: Use inequality
$$ n! > 2^n, \ n\geqslant 4.$$
Then for $n\geqslant 4$
$$\frac{1}{n!} < \frac{1}{2^n},$$ 
so
$$\sum_{n=1}^{\infty }\frac{19}{n!} = 19\sum_{n=1}^{\infty }\frac{1}{n!} < 19\left(1 + \frac{1}{2} + \frac{1}{6} + \sum_{n=4}^{\infty }\frac{1}{2^n}  \right)=19\left(1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{8}\right).$$