Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges

Solution 1:

The trick is to use the classic theorem of Hurwitz on rational approximation of irrationals, that there are infinitely many $m,n$ with $|\frac\pi2-\frac{m}{n}|\lt\frac1{n^2}$ (more specifically, you need infinitely many such pairs with $n\equiv 1\bmod 4$, but this result is known; see below). Any such pair will have $|m-\frac{n\pi}{2}|\lt\frac1n\lt\frac2m$; but then $\sin(\frac{n\pi}{2}) = 1$ (this is where the restriction on $n$ comes from), and by quick application of either the addition identity for sin or the mean value theorem, we get $\sin(m)\gt1-\frac2m$ for any such pair. This means that infinitely often we have $\frac{1+\sin(m)}{2}\gt1-\frac1m$, and $\left(\frac{1+\sin(m)}{2}\right)^m\gt\left(1-\frac1m\right)^m$, and you should be able to bound the RHS of this away from zero. This means that you've found infinitely many $a_m$ bounded away from zero, and that's enough to show divergence.

IMPORTANT CAVEAT: as @robjohn points out in the comments, it's not as easy as I thought at first to get the Hurwitz result in an arbitrary residue class! The paper "On The Approximation Of Irrational Numbers With Rationals Restricted By Congruence Relations" gives the following result:

For any irrational number $\xi$, any $s\geq 1$, and any integers $a$ and $b$, there are infinitely many integers $u,v$ satisfying $\left|\xi-\frac uv\right|\lt \frac{2s^2}{v^2}$ with $u\equiv a\pmod s$ and $v\equiv b\pmod s$.

We can take $\xi=\frac\pi2$, $s=4$, $b=1$ here and get infinitely many $u,v$ with $\left|\frac\pi2-\frac uv\right|\lt\frac{8}{v^2}$ and $v\equiv 1\pmod 4$; this is a factor of 8 away from the result above, but it's still good enough to give infinitely many $m$ with $\left(\frac{1+\sin(m)}{2}\right)^m\gt\left(1-\frac8m\right)^m$ and that's good enough to give a bound away from zero and prove divergence.

Solution 2:

There is a "power saving" of $\sqrt{n}$, in our favor, compared to the usual Diophantine approximation problems, because of the quadratic flatness of $\sin(x)$ near its maxima. [Updated: but this might not be a large enough saving to get around the use of sophisticated arguments. Analysis is still in progress.]

If $k \mod 2\pi = \frac{\pi}{2} + u$ for small $u$, then $1 - \sin(k)$ is of order $u^2$. In fact $\frac{1+\sin k}{2} \sim (1 - \frac{u^2}{4})$, but the constant $1/4$ is irrelevant. For a divergence proof what we would like to know is if $u = u_k$ is infinitely often small enough so that $(1 - u_k^2)^k$ is bounded below, i.e., is $ku_k^2$ bounded. The equivalent Diophantine question is

does $\frac{\pi}{2}$ have infinitely many rational approximations $\frac{k}{4n+1}$ accurate (up to a constant factor) to within $\frac{1}{n\sqrt{n}}$?

This is weaker than the $O(n^{-2})$ approximation from continued fractions and pigeonhole arguments and suggests the possibility of a more direct construction.

Originally I wrote $1/\sqrt{n}$ as the accuracy requirement, which is much too easy. Anything $1/n$ or larger has a simple construction. Whether $n^{-u}$ with $u \in (1,2)$ can be achieved without exponentially thin subsequences (such as continued fractions) seems like a very interesting question.

Solution 3:

You can show that it diverges using the first condition, the limit of $a_k$ as $k\to\infty$. You don't need to compute the limit. You need to show that either it does not exist or if exists it is not 0. In this case the limit does not exist(? see the comments below)