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. 

Let $a_n=1+(-1)^n+\epsilon^{2n}$ for $0<\epsilon<1$.  Then $\sum_{n=1}^{+\infty}a_n=+\infty$ and  $\sum_{n=1}^{\infty}\sqrt{a_na_{n+1}}<\infty$.
