Someone asked on Twitter:
Indeed this series diverges, just as the tweeter says: there’s a positive-density subset of n such that the summand exceeds .
More subtle: what about
This should still diverge. Argument: the probability that a real number x chosen uniformly from a large interval has is on order , not ; so there will be a subset of integers with density on order where the summand exceeds , and summing over those integers along should give a sum on order , which can be made as large as you like by bringing close to 0.
What’s not so clear to me is: how does
grow with N?