Example 1: Derivative and Integral

Differentiate and integrate the power series , and find the radii of convergence.

By the Ratio Test this power series has radius of convergence r = 1, for

Derivative:

Integral:

For convenience we rewrote the derivative as a power series in x^m where m = n - 1, and the integral as a power series in x^m where m = n + 1. Both the derivative and integral also have radius of convergence r = 1.