Constrained Optimzation problem in Infinite dimensions

I'm trying to solve the following constrained optimzation problem:

$\displaystyle \max_{\bar{\sigma}}\left\{\left(\sum_{k=0}^{\infty }\bar{\gamma}^{k+1}\sigma_k\right)^2 \middle|\alpha-\sum_{k=0}^{\infty}\sigma^2_k=0\right\}$

where

$\displaystyle \bar{\sigma}=\{\sigma_k\}_{k=0}^{\infty}$,

$\displaystyle \bar{\gamma}}\in(0,1)$

$\displaystyle \sigma_k,\alpha>0$

My goal is to find an expression for the sequence supporting the optimum of the problem, but I keep getting stuck...