All the complex numbers zn are non-zero and |zm - zn| > 1 (for any m ≠ n). Show that ∑ 1/zn3 converges.
We show that it is absolutely convergent. Take a disk radius 1/2 centred on each zn. The disks must be disjoint since |zm - zn| > 1 for distinct m, n. Now consider how many zn can lie in the annulus N ≤ |z| < N+1. If zn lies in the annulus, then at least half the corresponding disk must also, so if M is the number of zn, then M π/8 < π ( (N+1)2 - N2) = (2N+1) π, and the sum of 1/|zn|3 for these zn is at most M/N3 = 8(2N + 1)/N3. But ∑ 8(2N + 1)/N3 converges, since ∑ 1/N2 converges.
9th Putnam 1949
© John Scholes
11 Mar 2002