Proving the convergence of a recurrent sequence

Question

Define the sequence $\{U_n\}_{n \ge 0}$ by:

$$U_0 = U_1 = 1,\ and \ U_n = \sum_{i=1}^{n-1}\frac{U_i}{(i - 1)!},\ \forall \ n \ge 2$$

I calculated the first several values of the sequence, and it seems like it is converging to $3.(something)$. I tried to prove it increasing and bounded from above. It's easy to see that it's increasing. Now, how do I prove that it's bounded from above? Is there an easier way to prove convergence? I have a feeling that this is very easy and that there's something obvious I'm missing.

Thanks a lot.

Answer 1

Note that, $$U_{n+1}=\frac{U_n}{n!}+U_n=U_{n}\left(1+\frac{1}{n!}\right)\implies U_n=\prod_{i=1}^{n-1}\left(1+\frac{1}{i!}\right)$$ So $U_n\to \prod_{n=1}^{\infty}\left(1+\frac{1}{n!}\right)$ which I showed to be finite in this answer.