# 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:

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

I've tested for n=50 and I see that it converges to number $\displaystyle e$, it approximates up to 14 digits. But can I come to number $\displaystyle e$ somehow analitycally?
4. It's the Maclaurin series for $\displaystyle e^{x}$ evaluated at $\displaystyle x= 1$
5. $\displaystyle \displaystyle e^x = \sum\limits_{k = 0}^\infty {\frac{{x^k }} {{k!}}}$