Question

is 1 + x + (x-1)^2 + (x-2)^3 + (x-3)^4+... a power series?

Answer 1

It is an finite series with clearly defined terms, but it does not fit the definition of a power series here:

http://en.wikipedia.org/wiki/Power_series

because the terms that get raised to progressively higher powers are not the same.