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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?

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In what region do you want the bounds to be accurate? e.g. for small $s$, $1$ works, but presumably that's unsatisfying. For $s$ near $1$, you can expand the series out to a finite number of terms, but that's not particularly exciting either. For $s$ very large, you might want to approximate the product as the exponential of an integral. – Will Sawin Apr 15 '12 at 19:54
Is $\log_2 s$ good enough for you? – fedja Apr 16 '12 at 1:49

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