I don't quite understand your confusion. By series index, do you mean the beginning value of the sum. Like for you mean the n=0 part? Please explain your confusion a little more.

Taylor series do always start with n=0 if my above assumption is true. Note that by the formula for generating every term in a Taylor series that when you apply the fact that 0!=1 and that (the 0th derivative of f(x) is f(x)) then you are simply left with f(a), which can be 0 for a lot of things.