# Thread: Convergence of a serie.

1. ## Convergence of a serie.

Hello. Need some help on solving this one:

Review the convergence of the following serie:

$1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...+\frac {1}{n!}+...$

2. that's a known series, try ratio test.

3. Ok, thanks.

I knew it was convergent in the first place. By ratio test i see that it is convergent.
I've tested for n=50 and I see that it converges to number $e$, it approximates up to 14 digits. But can I come to number $e$ somehow analitycally?

4. It's the Maclaurin series for $e^{x}$ evaluated at $x= 1$

5. $\displaystyle e^x = \sum\limits_{k = 0}^\infty {\frac{{x^k }}
{{k!}}}$