I'm trying to find:
$ \lim_{n\rightarrow\infty} (\sqrt[n]{1}+\sqrt[n]{2}+\cdots+\sqrt[n]{2007}-2006)^n $
(Problem from CMJ)
We have:
$ k^{1/n}=1+\frac{\ln k}{n}+O(1/n^2) $
$ \left( \sum_{k=1}^{2007}k^{1/n}-2006 \right)^n= \left(1+\frac{1}{n}\sum_{k=1}^{2007}\ln k+O(1/n^2) \right)^n \sim_{n\rightarrow\infty} \exp \left( \sum_{k=1}^{2007} \ln k \right)(=2007!)$
I'm quite sure of my result but I could not check it numerically.
Do you agree with this limit?