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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    


f (x)dx = C + x n+1    

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

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