Hey guys I was wondering

let's take

$\displaystyle \lim_{x \to \infty} \sum_{n=0}^x \frac{x^n}{n!}$

and

$\displaystyle \lim_{x \to \infty}\sum_{n=0}^\infty \frac{x^n}{n!}$

I have goofed around with these to realise they're not the same value. the first one give me something with Euler gamma function in it, the second one is simply the taylor serie of $\displaystyle e^x$

So my first question is what's make them so differents?

If I had, let's say take

$\displaystyle \lim_{x \to \infty}\lim_{a \to \infty}\sum_{n=0}^x \frac{a^n}{n!}$ (Not even sure I can write something like this..)

This should give me the exponential function?

And Where could I read more some "dumb friendly" info about Euler gamma function?

edit: Actualy after tacking a shower I do understand the difference beetween the given limite.

Altough I can't seem to understand the Euler gamma function