Fix integer $s.$ I have encountered the following infinite sum.

$$\sum_{k=0}^\infty \Pi_{l=1}^k\left[1-(1-2^{-l})^s\right]$$

Is there a useful lower bound on this expression? For instance, if $s=1,$ this gives the series

$$\sum_{k=0}^\infty 2^{-k(k+1)/2}.$$

Are there good closed form expressions that describe lower bounds on this quantity?