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.

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