MacClaurin series discrepancy

This can happen with a bunch of different series, but let's just look at a really easy one as an example.

$\displaystyle f(x)=e^{x+1}$

There are two ways to calculate the Maclaurin polynomial:

1.) $\displaystyle e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$ so $\displaystyle e^{x+1} = \sum_{n=0}^{\infty}\frac{(x+1)^n}{n!}$

2.) $\displaystyle f^{(n)}(x) = e$ for all $\displaystyle n$ so the polynomial is $\displaystyle \sum_{n=0}^{\infty}\frac{ex^n}{n!}=e\sum_{n=0}^{\i nfty}\frac{x^n}{n!}$

Clearly, these are the same if you're summing from $\displaystyle 0$ to $\displaystyle \infty$, but what if you are just trying to approximate the function? Which is a better estimate? (I would assume it's the second one.)

Or, the true reason behind my asking this: My friend's professor asked on an exam what the coefficient on the $\displaystyle x^2$ term was, yet depending on how you calculate this, you could get different answers. In the first method you get $\displaystyle \frac{1}{2}$; in the second, you get $\displaystyle \frac{e}{2}$.

I'm assuming that if $\displaystyle a_n^2$ is the coefficient on $\displaystyle x^2$ (in the first method) when taking the sum from $\displaystyle 0$ to $\displaystyle n$, these coefficients will form a sequence such that $\displaystyle \lim_{n\to\infty}a_n^2 = \frac{e}{2}$

Does anyone have anything to add to this or is that reasoning all pretty much fine?