EXAMPLE 5 Express 1/(1 − x)2 as a power series by differentiating the equation below. What is the radius of convergence? 1 1 − x = 1 + x + x2 + x3 + = [infinity] xn n = 0 |x| < 1 SOLUTION Differentiating each side of the equation, we get 1 (1 − x)2 = 1 + Correct: Your answer is correct. + 3x2 + = [infinity] n = 1 . If we wish, we can replace n by n + 1 and write the answer as 1 (1 − x)2 = [infinity] n = 0 . According to this theorem, the radius of convergence of the differentiated series is the same as the radius of convergence of the original series, namely, R = .