1. The problem statement, all variables and given/known data Find the radius of convergence of the series: ∞ ∑ n^-1.z^n n=1 Use the following lemma: ∞ ∞ If |z_1 - w| < |z_2 - w| and if ∑a_n.(z_2 - w)^n converges, then ∑a_n.(z_1 - w)^n also n=1 n=1 converges. The contrapositive is also true. 2. Relevant equations 3. The attempt at a solution Hi, here's what I've done: Let r be the radius of convergence. The series converges when |z-w| < r and diverges for |z-w|> r. In this question, z = z, w = 0. Use the ratio test for convergence: lim |a_n+1 / a_n| = lim |zn / n+1 | n->∞ = |z| Thus the series converges when |z| < 1 diverges when |z| > 1 So let |z - 0| = |z_2 - w| Then, by the lemma, all |z_1 - 0| converges, where |z_1 - 0| < |z - 0| and the converse is true for a diverging series. Thus the radius of convergence = |z| --- I think I've covered everything, but I don't think I've made very satisfactory use of the lemma. Can anyone please point me in the right direction? Thanks for any help.