TITLE: Suppose $\sum x_n$ converges (not necessarily absolutely), does $\sum \sin x_n$ necessarily converge?
QUESTION [14 upvotes]: The question is:

If $\sum x_n$ converges, does $\sum \sin x_n$ converge?

I know that if $\sum x_n$ converges absolutely, then $\sum \sin x_n$ converges. My intuition is that we cannot completely abandon absolute convergence, but I am struggling to come up with a counterexample.

REPLY [5 votes]: Let's define a sequence $a_n$ three terms at a time. The $n$th grouping will look like $$\frac{1}{n^{1/3}}, \frac{-1}{2 n^{1/3}}, \frac{-1}{2n^{1/3}}.$$
Then $\sum a_n$ converges (in fact the sum is $0$). Now recall $\sin x = x - x^3/6+O(x^5).$ The $n$th grouping for $\sin a_n$ then looks like 
$$\frac{1}{n^{1/3}}-\frac{1}{6n} + O(1/n^{5/3}),\frac{-1}{2n^{1/3}} + \frac{1}{48n} + O(1/n^{5/3}), \frac{-1}{2n^{1/3}} + \frac{1}{48n} + O(1/n^{5/3}).$$
The sum of those terms is $-1/(8n) + O(1/n^{5/3}).$ Therefore $\sum \sin a_n $ diverges to $-\infty.$