1,2,3,4 ???

Is $ \alpha\in [1,2]$ :

http://www.mathlinks.ro/viewtopic.php?t=150706

Anyone with a solution for this?

I have proved that $\alpha \in \left(\frac{1}{1+2\sqrt { \frac{2}{9} } },2 \right)$ Any $\alpha$ in this interval will satisfy the above relation. Does anybody know the answer, is this correct. ?