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?