Polynomials are easy to differentiate and integrate, applying the respective sum rules a finite number of times to reduce to the case of a monomial. We would like to be able to do the same thing for power series (including Taylor series in particular). It is a theorem that this always works within the radius of convergence of the power series. We state the result below.
Suppose f (x) = a _{n} x ^{n} is a power series with radius of convergence r . Then for all x with | x| < r ,
f'(x) = na _{n} x ^{n-1} |
and
f (x)dx = C + x ^{n+1} |
where C is an arbitrary constant, reflecting the non-uniqueness of the antiderivative.