1. The problem statement, all variables and given/known data Suppose sum(a_n*x^n) represents a power series with radius of convergence (-R, R). Is it true that the series sum(n*a_n*x^n) is convergent? Prove or give a counter example. 2. Relevant equations 3. The attempt at a solution Let b_n = n*a_n*x^n Using ratio test: lim[b_(n+1)/b_n] = lim [(n+1)/n] * [lim x*a_(n+1)/a_n] < 1 because lim x*a_(n+1)/a_n <1 from hypothesis. Any gaps in logic?