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.

There are two ways to calculate the Maclaurin polynomial:

1.) so

2.) for all so the polynomial is

Clearly, these are the same if you're summing from to , 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 term was, yet depending on how you calculate this, you could get different answers. In the first method you get ; in the second, you get .

I'm assuming that if is the coefficient on (in the first method) when taking the sum from to , these coefficients will form a sequence such that

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