Answer
$$ a_n= n \pi $$
(other examples are possible.)
Work Step by Step
Consider
$$ a_n= n \pi $$
then the series $\{a_n \}$ diverges.
\begin{align*}
\lim_{n\to \infty} a_n &= \infty \ne 0
\end{align*}
However, for $\{\sin a_n \}$
\begin{align*}
\lim_{n\to \infty} \sin a_n&= \lim_{n\to \infty} \sin n \pi \\
&=0
\end{align*}
Hence, $\{\sin a_n \}$ converges.
