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Differentiation and Integration of Power Series

Differentiation and Integration of Power Series

Differentiation and Integration of Power Series

Differentiation and Integration of Power Series

Differentiation and Integration of Power 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.