divergent | Quomodocumque

Someone asked on Twitter:

A conjectured divergence. #math #calculus pic.twitter.com/kAr1zTY0EA
— Sam Walters ☕️ (@SamuelGWalters) February 12, 2020

Indeed this series diverges, just as the tweeter says: there’s a positive-density subset of n such that the summand exceeds $n^{-1/2}$ .

More subtle: what about

$\sum_{n=1}^{\infty} \frac{1}{n^{2 + \cos n}}?$

This should still diverge. Argument: the probability that a real number x chosen uniformly from a large interval has $\cos x < -1+\epsilon$ is on order $\epsilon^{1/2}$ , not $\epsilon$ ; so there will be a subset of integers with density on order $\epsilon^{1/2}$ where the summand exceeds $n^{-1-\epsilon}$ , and summing over those integers along should give a sum on order $\epsilon^{-1/2}$ , which can be made as large as you like by bringing $\epsilon$ close to 0.