Theorem 1: Derivatives And Integrals Of Power Series

9.8 DERIVATIVES AND INTEGRALS OF POWER SERIES

In the last section we concentrated on the problem of finding the interval of convergence of a power series. We shall now find the sums of some important power series. Our general plan will be as follows.

First, find the sums of two basic power series:

Then, starting with these basic power series, find the sums of other power series by differentiation and integration. (Based on Theorem 1.)

An especially useful property of power series is that they can be differentiated and integrated like polynomials. If we have a power series for a function f(x), we can use Theorem 1 to immediately write down the power series for the derivative f'(x) and integral dt.

Discussion

This theorem says that a power series can be differentiated and integrated term by term. Also, the radius of convergence remains the same. To differentiate or integrate each term of a power series we simply use the Power Rule.

We postpone the proof of Theorem 1 until later.

Example 1: Derivative and Integral