How will the limit $\lim \sin(A^{n})$ behave for $|A| > 1$? While answering to the following question

Problem. Find the range of $x$ for which $\displaystyle \sum_{n=0}^{\infty} (-1)^{n} \sin(x^{-n})$ converges.

Unlike some careless answerers who merely claimed that this converges only for $|x| > 1$, I faced a hard question:
Question. Is there any $|A| > 1$ such that $\sin (A^{n})$ converges? (Of course, I put $A = x^{-1}$ here.)
I literally have no idea on it since the behavior of the sequence depends on $n \mapsto A^{n} \text{ mod } 2\pi$ and I have no idea how it behaves for values of $A$.
Actually I suspect that there is a number $A > 1$ for which $\sin (A^{n})$ converges. But this is just a groundless guess, anyway.
Do you have any idea regarding this one?

It can be shown that 
$$
\mathcal{D}=\{|A|>1 : \sin(A^n) \textrm{ diverges.}\}
$$
is dense. (I conjecture that $\mathcal{C}=\{|A|>1: \sin(A^n) \textrm{ converges.}\} $ has measure zero, do not know how to deal with complex exponentials ---> now I do know.)
This follows from a result on equidistribution:

For real numbers $\theta>1$, is is shown in  Koksma's
  paper that
  $\theta^n$ is equidistributed modulo $1$ for almost all $\theta>1$. 
  Equidistribution modulo $2\pi$ is just a minor modification of Satz 2
  in the paper.

Then the density of $\mathcal{D}$ follows from considering $A=re^{i\pi\frac{k}{m}}$ for $k,m$ fixed integers. In fact 
$A^{2mn}=(r^{2m})^n$, and for almost all $r^{2m}>1$, $(r^{2m})^n$ is equidistributed modulo $2\pi$.
A stronger result of the above(in terms of error term) is shown in 
Leveque's paper. 
Edit: Proof of my conjecture
Let $A=re^{i \theta}$, so $A^n=r^ne^{in\theta}$. Suppose that $\theta\neq  k\pi$ for any integer $k$. Then there is a subsequence $n_j$ for $j=1,2,\cdots$ such that 
$$\sin n_j\theta >\frac 12 \ \ \textrm{ for all }j. $$
Now, $$\sin A^n = \frac{e^{-r^n\sin n\theta}e^{i r^n\cos n\theta}-e^{r^n\sin n\theta}e^{-i r^n\cos n\theta}}{2i}$$
Since $|v-w|\geq ||v|-|w||$ for any complex $v, w$, 
$$|\sin A^{n_j}|\geq \frac{1}{2} \left|e^{r^{n_j}\sin n_j\theta}-e^{-r^{n_j}\sin n_j\theta}\right| $$
Since $r>1$, the above $\rightarrow\infty$ as $j\rightarrow\infty$. 
Hence, if $A=re^{i\theta}$, with $\theta \neq\pi \textrm{ mod }2\pi$, then the sequence $\sin A^n$ diverges. This proves my conjecture. 
$$\mathcal{C}\textrm{ has measure zero.}$$
Further, we see that $\mathcal{C}\subset \mathbb{R}$, and has measure zero for the Lebesgue measure on $\mathbb{R}$. 
Edit2
I finally found references that partially settles your problem. 
Pisot and 
Kawapisz
We have (1) from Pisot, (2) from Kawapisz. 

(1) If a real number $\lambda$ is transcendental, then the series $$
  \sum_{n=0}^{\infty} \sin^2 (\pi\lambda x^n)$$ is divergent for all
  $x\geq 1$.
(2) If $x$ is algebraic, and there exists $\lambda\neq 0$ such that $$\sin (\pi\lambda x^n)$$ converges to $0$, then $\lambda \in \mathbb{Q}(x)$. 

With $\lambda=1/\pi$, there does not exist any real number $x\geq 1$ such that 
$$
\sum_{n=0}^{\infty} \sin^2 x^n$$
converges. 
Therefore, the a partial conclusion to your problem is:

The series $$ \sum_{n=0}^{\infty} (-1)^{n} \sin(x^{-n})$$ absolutely converges if and only
  if $|x|>1$. If $|x|<1$ and $x$ is algebraic, then the series does not converge.

Still, whether or not there exists a real number $x>1$ such that $\sin x^n$ converges, is open.  What we know so far is: if $x>1$ algebraic then $\sin x^n$ does not converge to $0$.
