Asymptotic density and sum of the reciprocals

Question

Let A and B be two infinite proper-subsets of the set of positive integers. Let A(n) denote the number of those elements of the set A , which does not exceed n ; we use similar definition for B(n) . Also let lim A(n)/n > lim B(n)/n , as n→∞

If the sum of the reciprocals of the numbers in B is divergent then can we ever conclude that the sum of the reciprocals of the numbers in A is also divergent ?

score 2 · Accepted Answer · answered Oct 11 '12 at 06:00

2

Your inequality implies $\lim A(n)/n$ exists and is positive, which is enough to conclude divergence for $A$, regardless of what happens to $B$.

answered Oct 11 '12 at 06:00

Gerry Myerson

179,216

So, if lim A(n)/n exists and is > 0 then I can always conclude that the sum of the reciprocals is divergent? – Souvik Dey Oct 11 '12 at 06:04
1

@Souvik SeeTheorem on natural density. – Martin Sleziak Oct 11 '12 at 06:18

Asymptotic density and sum of the reciprocals

1 Answers1