I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. One of the students, Denis Zelent, invented a very interesting one: $$ \sum_{n = 1}^\infty \frac{1}{n^2 \sin n} \, . \tag{1} $$

Question(short version): Has convergence of this series been studied in literature?

My immediate answer was that this must have something to do with the irrationality measure $\mu$ of $\pi$. Obviously, $\mu \geqslant 2$, and the best currently known upper bound for $\mu$ is $\mu \leqslant 7.6063\!\ldots\,$, due to Salikhov; see [V. Kh. Salikhov, *On the Irrationality Measure of $\pi$*. Russ. Math. Surv 63(3):570–572, 2008]. It is widely believed that $\mu = 2$.

In fact, the inequality $\mu < 3$ is equivalent to convergence of $1 / (n^2 \sin n)$ to zero. Thus if we knew that $\mu \geqslant 3$, the series (1) would diverge. On the other hand, if we had $\mu < 2$ (which is of course absurd), then one could easily show that $1 / (n^2 \sin n) = O(n^{-1 - \varepsilon})$ for some $\varepsilon > 0$, which would imply absolute convergence of the series (1).

My student searched the web and realized that his question is related to the well-known open problem, asking whether the *Flint Hills series*
$$
\sum_{n = 1}^\infty \frac{1}{n^3 \sin^2 n} \tag{2}
$$
converges. An extension of this problem asks for convergence of a more general series
$$
\sum_{n = 1}^\infty \frac{1}{n^p |\sin n|^q} , \tag{3}
$$
which is equivalent to absolute convergence of the series (1) when $p = 2$ and $q = 1$. For more details, see [Max. A. Alexeyev, *On convergence of the Flint Hills series*, arXiv:1104.5100, 2011].

To summarise, this is what we have found so far:

lack of convergence of $1 / (n^2 \sin n)$ to zero would imply that $\mu \geqslant 3$, which is very unlikely;

convergence (in particular: absolute convergence) of the series (1) would imply $\mu \leqslant 3$, which means it is certainly an open problem;

lack of absolute convergence of the series (1) would not have any consequences for the estimates of $\mu$.

Question(long version)

- Does absolute convergence of the series (1) imply any tighter bounds on the estimates of the irrationality measure $\mu$ of $\pi$?
- Vice versa: Assuming that $\mu$ is known, can one tell whether the series (1) converges absolutely?
- Same questions with
absolute convergencechanged intoconvergence. In other words: are cancellations of any help here?- Does the series (1) has a fancy name, similar to Flint Hills series and Cookson Hills series? (And if not: can Denis choose an appropriate mountain range?)

**Edited**: I just noticed David Simmons's answer to an MO question on the Flint Hills series, which reduces the question of its convergence to a similar question for a series involving convergents of $\pi$. The same argument should work for the absolute convergence of the series (1), but I do not see right away if it leads to an answer to question 1.