Limit of recursive sequence involving factorial in sequence definition

Question

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 difficulty in calculating limit. Can anyone explain me how I should approach this problem?

Answer 1

So you want $3+\frac{3}{2}(\frac{1}{2!}+ \frac1{3!}+\frac{1}{4!}+\dots)=3+\frac{3}{2}(e-2)=3+\frac{3e}{2}-3=\frac{3e}{2}$