calculus - Absolute convergence of the series $sin(a_n)$ implies...?

Can someone please give me a counterexample for the following statement?

If $\sum _1 ^\infty sin(a_n) $ absolutely converges , then $\sum_1^\infty a_n$ also absolutely converges.

I tried working with $a_n = \frac{ (-1)^n } {n} $ and $a_n = \frac{(-1)^n lnn}{n}$ , but without any success...

Got any idea?

Thanks a lot !