1
$\begingroup$

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 ?

$\endgroup$
2
$\begingroup$

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

$\endgroup$
2
  • $\begingroup$ So, if lim A(n)/n exists and is > 0 then I can always conclude that the sum of the reciprocals is divergent? $\endgroup$ – Souvik Dey Oct 11 '12 at 6:04
  • 1
    $\begingroup$ @Souvik SeeTheorem on natural density. $\endgroup$ – Martin Sleziak Oct 11 '12 at 6:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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