Proving according to Cauchy criterion that an alternating harmonic series with factorial denominator converges.

Question

I need to prove according to Cauchy's criterion that the sequence:

$a_n = 1 - \frac{1}{2!} + \frac{1}{3!} - ......... + \frac{1}{(2n-1)!} - \frac{1}{(2n)!} $

converges. but I got stuck with the negative and positive terms of the sequence, I thought about triangle inequality but also didn't work. can someone give me some tips for solving this question?

Answer 1

\begin{align} |a_{n+m}-a_n|&\le \frac{1}{(2n+1)!} + \frac{1}{(2n+2)!}+\dots+\frac{1}{(2(n+m)-1)!}+\frac{1}{(2(n+m))!}\le \\ &\le \frac{1}{(2n)(2n+1)} + \frac{1}{(2n+1)(2n+2)} + \dots + \frac{1}{(2n+2m-2)(2n+2m-1)}+\\\ &+\frac{1}{(2n+2m-1)(2n+2m)}\le \\ &=\frac{1}{2n}-\frac{1}{2n+1} + \frac{1}{2n+1}-\frac{1}{2n+2}+\dots+ \frac{1}{2n+2m-2}-\frac{1}{2n+2m-1} +\\&+ \frac{1}{2n+2m-1}-\frac{1}{2n+2m}=\\ &=\frac{1}{2n}-\frac{1}{2n+2m} \le \frac{1}{2n} \end{align}