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,
nanxn-1
f (x)dx = C + 
xn+1
