My question is just a concept that I don't understand. When differentiating a power series that starts at n=0 we bump that bound up to n=1. My question is do we always do that? or Do we only do that when the first term of the power series is a constant and thus when it is differentiated it becomes zero? My guess is the second case.
Huh? n = 0 is just the index. We can call out "starting point" a_{0} or a_{1} --- whichever we prefer. And yes, that term will disappear when you take the derivative of ∑a_{n}x^{n}. a_{0} + a_{1}x + a_{2}x^{2} + ... (a_{0} + a_{1}x + a_{2}x^{2} + ... )' = a_{1} + 2a_{2}x + ... It's as simple as that.