# Don't Exceed but Evaluate

Calculus Level 4

$\displaystyle \lim_{n \to \infty} n^{-n^2} \Bigg [ \bigg (n + 1 \bigg ) \left (n+ \frac 1 2 \right ) \left (n + \frac 1 4 \right ) \ldots \left ( n + \frac {1}{2^{n-1}} \right ) \Bigg ]^n$

If the above limit equals to $\alpha$, what is the value of $\lfloor 1000 \alpha \rfloor$?

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