If the power series az and bz converge for z SolutionAssume

If the power series a(z) and b(z) converge for |z|

Solution

Assume without loss of generality that c = 0, and suppose |x| < R. Choose such that |x| < < R, and let r = |x| , 0 < r < 1. To estimate the terms in the differentiated power series by the terms in the original series, we rewrite their absolute values as follows: nanx n1 = n ( |x| )n1 |an n | = nrn1 |an n |. The ratio test shows that the series nrn1 converges, since limn [ (n + 1)r n nrn1 ] = limn [(1 + 1 n ) r ] = r < 1, so the sequence (nrn1 ) is bounded, by M say. It follows that nanx n1 M |an n | for all n N. 6.4. Differentiation of power series 81 The series |an n | converges, since < R, so the comparison test implies that nanx n1 converges absolutely. Conversely, suppose |x| > R. Then |anx n| diverges (since anx n diverges) and nanx n1 1 |x| |anx n | for n 1, so the comparison test implies t