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) = anxn is a power series with radius of convergence r. Then for all x with | x| < r,

f'(x) = nanxn-1    


f (x)dx = C + xn+1    

where C is an arbitrary constant, reflecting the non-uniqueness of the antiderivative.