Tag Archives: series

A divergent sequence

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 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.

What’s not so clear to me is: how does

\sum_{n=1}^{N} \frac{1}{n^{2 + \cos n}}

grow with N?

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