# Limit of recursive sequence involving factorial in sequence definition

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?

• Do you know that $e = 1+1+1/2+1/6+...+1/n!+...$? Aug 13, 2015 at 1:18
• Yeah, I knew that. I expanded the left side like 10 times, but it didn't occur to me that it's resulting in series expansion of $e$. Aug 13, 2015 at 1:23
• That's what experience is good for. Aug 13, 2015 at 1:23

So you want $3+\frac{3}{2}(\frac{1}{2!}+ \frac1{3!}+\frac{1}{4!}+\dots)=3+\frac{3}{2}(e-2)=3+\frac{3e}{2}-3=\frac{3e}{2}$
• I was expanding the sequence and then I totally failed to observe that I was getting expansion of $e$. Thanks @dREaM. Aug 13, 2015 at 1:18