Convergence of the series $\sum_{k=1}^{\infty} \left( \frac{1+\cos k}{2+\cos k}\right)^{2k}$. Check my work.
Find out whether $\sum_{k=1}^{\infty}\left(\frac{1+\cos k}{2+\cos k}\right)^{2k}$ converges absolutely, conditionally or diverges.