Convergence/Divergence of $\sum_{n=1}^\infty \left(\frac {1+\cos(n)}3 \right)^n$ I need to see if this series $\sum_{n=1}^\infty\limits \left(\frac {1+\cos(n)}3 \right)^n$ either converges or diverges. I was thinking that because the inside terms are going to fluctuate between $(0,\frac 23)$, the inside is never negative, so it's going to diverge because a sum of positive numbers raised to a power are strictly increasing? Is my logic correct here and/or if there is a theorem that strengthens my argument, it would be appreciated.

**Hint:** $$ \sum_{n=1}^\infty \left|\frac{1+\cos(n)}3 \right|^n \leq\sum_{n=1}^\infty \left(\frac23\right)^n<\infty, $$ hence the series converges absolutely, hence converges.