diverge or converge and sum of the series

[Laurenmath]

I believe you use the ratio test
I need to find out if the summation series converges or diverges

Summation from N=1 to infinity of n!/ (2n! + 1)

please help

 

[pka]

Note that \(\displaystyle \L
\frac{{n!}}{{\left( {2n} \right)! + 1}} < \frac{{n!}}{{\left( {2n} \right)!}}\).

Then \(\displaystyle \L
a_n = \frac{{n!}}{{\left( {2n} \right)!}}\quad \Rightarrow \quad \frac{{a_{n + 1} }}{{a_n }} = \frac{{n + 1}}{{\left( {2n + 2} \right)\left( {2n + 1} \right)}}\).

Now use basic the comparison tests.