I'm trying to calculate limit to this function and but I've not been able to figure out how to approach this.

The definition of sequence $S$ is

$S(1) = 3 $

And $ \forall \geq 2, S(n) = S(n-1) + \frac{3}{2(n!)} $

I need to calculate the following limit.

$\lim\limits_{n \to \infty} S(n)$

I have been able to prove that the sequence is monotonously increasing and is bounded also. But I'm having in calculating limit. Can anyone explain me how I should approach this problem?