Test Series Convergence: 1! + 2! + ... + n! / (2n)!

[dobry_den]

Hi, could you please check if my solution is correct?

Homework Statement

Test the following series for convergence:

$$\sum_{n=1}^{\infty}\frac{1!+2!+...+n!}{(\left 2n \right)!}$$

The Attempt at a Solution

I can use a slightly altered series

$$\sum_{n=1}^{\infty}\frac{nn!}{(\left 2n \right)!}$$

whose every term is >= than the corresponding term in the original series.. and thus if this altered series converges, then the original one should so as well...

Then, if I use the limit ratio test for the second series:

$$\lim_{n \rightarrow \infty}\frac{(\left n+1\right)(\left n+1 \right)!}{(\left 2n+2 \right)!}\frac{(\left 2n\right)!}{n(\left n \right)!} = 0$$

This means that the altered series is convergent, and thus the original series is also convergent.

Is this reasoning correct? Thanks in advance!

 

[e(ho0n3]

Yes it is.