Summation with sine & cosine sequences

Note for integer $n$ , $\cos n\pi=\sin\left(\frac{\pi}{2}-n\pi\right)=(-1)^n$ , so $a_n=2(-1)^n$ . Similarly, $\cos\left(2n\pi-\frac{\pi}{3}\right)=\cos\left(\frac{\pi}{3}\right)=\frac{1}{2}$ , so $b_n=5$ .

Thus, $\displaystyle\sum_{n=1}^{\infty}\left(\frac{a_n}{b_n}\right)^n=\sum_{n=1}^{\infty}(-1)^{n^2}\left(\frac{2}{5}\right)^n=\sum_{n=1}^\infty(-1)^n\left(\frac{2}{5}\right)^n$ is a geoemtric series with ratio $-\frac{2}{5}$ , so

$\displaystyle\sum_{n=1}^\infty\left(\frac{a_n}{b_n}\right)^n=\frac{-\frac{2}{5}}{1+\frac{2}{5}}=\boxed{-\frac{2}{7}}$ .