Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ?

share|cite|improve this question
up vote 2 down vote accepted

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

share|cite|improve this answer
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 6:04
@Souvik SeeTheorem on natural density. – Martin Sleziak Oct 11 '12 at 6:18

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.