The Taylor Series
Differentiation and Integration of Power Series
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.
