TITLE: Is the sequence $u_n=n\sum_{i=1}^\infty2^{-i}(1-2^{-i})^n$ convergent when $n\to \infty$?
QUESTION [6 upvotes]: Problem
Is the sequence $u_n=n\sum_{i=1}^\infty2^{-i}(1-2^{-i})^n$ convergent when $n\to \infty$?
Progress
If $i\leq \log n$ then $(1-2^{-i})^n<e^{-\frac{n}{2^i}}$. Unfortunately, I do not know what to do next

REPLY [2 votes]: Some numerics may help. According to a rough calculation of the sum (capping the summation at $i=10^4$), it seems that the sum does not diverge, but stabilizes close to
$$u_n \approx 1.44268$$

A closer look at the graph shows that it oscillates around this value