Is the series $\sum_{n=1}^{\infty} \sin(n^4)\sin(4^n)$ convergent or divergent? Is the series $$ \sum_{n=1}^{\infty} \sin(n^4)\sin(4^n) $$ convergent or divergent?
I tried expanding the sine functions and got no clue, and any test that I know of isn't helping me with this series. I even posted it on MathStackExchange and someone there advised me to seek help here.
 A: The terms of the series don't go to zero.
There is a uniquely ergodic dynamical system (with Lebesgue measure as the unique invariant measure) on the 4 dimensional torus and a function $f$ on the torus such that $f(T^n(0))=\sin(n^4)$. Since this dynamical system, there exists an $L$ such that for any $x$, one of $f(x),\ldots,f(T^{L-1}x)$ satisfies $f(x)\ge \frac 12$ (for example).
In particular for any $n$, at least one of $\sin(n^4),\ldots,\sin((n+L-1)^4)$ is at least $\frac 12$.
Now for any $n$, consider $a_n=\min(|\sin(4^n)|,\ldots,|\sin(4^{n+L-1})|)$.
If this is at least $1/4^{L+2}$, then one of the terms in the series between $n$ and $n+L$ is at least $1/(2\cdot 4^{L+2})$.
If it is less than $1/4^{L+2}$, find the first term $m$ after $n$ such that $|\sin(4^m)|>1/4^{L+2}$. In particular, since $|\sin(4x)|\le 4|\sin x|$, we see $|\sin(4^m)|\le 1/4^{L+1}$. Then $|\sin(4^{m+i})|\ge 1/4^{L+2}$ for $i=0,1,\ldots,L-1$. Amongst these terms, one of the $\sin(j^4)$ must exceed $\frac 12$, so that there is a term of size at least $1/(2\cdot 4^{L+2})$.
