Sequence $(a_n)$ s.t $\sum\sqrt{a_na_{n+1}}<\infty$ but $\sum a_n=\infty$ I am looking for a positive sequence  $(a_n)_{n=1}^{\infty}$ such that  $\sum_{n=1}^{\infty}\sqrt{a_na_{n+1}}<\infty$ but  $\sum_{n=1}^{\infty} a_n>\infty$.
Thank you very much. 

The simplest example I can think of is $\{1,0,1,0,...\}$. If you want your elements to be strictly positive, use some fast-converging sequence such as $n^{-4}$ in place of the zeroes.
