Let an be a strictly monotonic increasing sequence of positive integers. Let bn be the least common multiple of a1, a2, ... , an. Prove that ∑ 1/bn converges.
Solution
We need a crude upper bound on the number of divisors for N. N is too crude. But we can do better by noticing that N has at most √N divisors ≤ √N and hence at most 2√N divisors in total (every divisor d > √N has a matching divisor N/d < √N). Now we know that bn has at least n distinct divisors (namely a1, ... , an). Hence bn ≥ n2/4. But we know that ∑ 1/n2 converges.
© John Scholes
jscholes@kalva.demon.co.uk
5 Feb 2002