# Convergent series

1. Jan 23, 2008

### dobry_den

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

1. The problem statement, all variables and given/known data

Test the following series for convergence:

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

3. 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!

Last edited: Jan 23, 2008
2. Jan 23, 2008

Yes it is.