# An application of the pigeonhole principle in analysis

The pigeonhole principle is stated here.

I found in a book the following application:

We can obtain the nature of the series $\sum \frac 1{n^2\sin^2(n)}$ from the pigeonhole principle.

The issue is that the book did not give any hint in how to derive this result from the pigeonhole principle. I don't even know if this series diverges or converges.

Do you know how one should proceed? Do you have any reference where this is done?

Maybe this is a duplicate, but I don't think so since I found nothing here.

## 1 Answer

The pigeonhole principle can be used to prove that there are infinitely many integers $n$ and $m$ such that

$$\left|\pi - \frac{n}{m} \right| < \frac{1}{m^2} \\ \implies |n - m \pi| < \frac{1}{m} \\ \implies\frac{n}{m} < \pi + \frac{1}{m^2} < \pi+1$$

A proof of this basic Diophantine approximation using the pigeonhole principle is given on p. 42 of http://www.maths.ed.ac.uk/~aar/papers/niven.pdf

Hence,

$$\left\lvert \sin n \right\rvert = \left\lvert\sin(n - m \pi)\right\rvert \leqslant |n - m \pi| < \frac{1}{m} = \frac{1}{n}\frac{n}{m} < \frac{1 + \pi}{n}.$$

This implies that for infinitely many $n$ we have

$$\frac{1}{n^2 \left\lvert\sin n\right\rvert^2} > \frac{1}{(1+ \pi)^2},$$

and the series must diverge.