How do you prove $\lim_{n\rightarrow\infty} n(2^{1/n} - 1) = \log 2$ ?
Background: in computer science, if you allocate CPU time to $n$ processes by rate-monotonic scheduling, all the processes get sufficient amount of time when the quantity $U$ called CPU utilization is less than or equal to $n(2^{1/n} - 1)$. It is monotonously decreasing and tends to $\log 2$ when $n\rightarrow\infty$, so if $U \le \log 2$, you can be sure all the processes will be given sufficient CPU time.