If $ \sum\limits_{n=1}^{\infty} a_n $ with $a_n >0$ is convergent, then is $ \sum\limits_{n=1}^{\infty} \sqrt {a_n a_{n+1}} $ always convergent?
Convergent or divergent series: $\sum_{n=1}^{\infty} \sqrt{a_n \cdot a_{n+1}}$?