If $a_n$ monotonic, $a_n>0$, and $\sum\sqrt {a_na_{n+1}}$ converges, prove $\sum a_n$ converges.
Given $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $\sum \sqrt{a_n a_{n+1}} < \infty \implies \sum a_n < \infty$