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## 'Discrete Trends' printed from http://nrich.maths.org/

As $n$ is an integer try finding $n^{1\over n}$ for some small
values of $n$. What do you find? If you think you might have found
the maximum value then you'll need to use the first part of the
question to prove it really is the maximum. As the problem is about
discrete (whole number) values you can find a solution without
calculus. To show that

$$n^{1/n} < 1 + \sqrt {{2\over {n-1}}}$$

write $n^{1/n} = 1 + \delta$ and use the Binomial Theorem. If
$n> 1$ then $\delta> 0$. Throw away all but one term of the
Binomial expansion to get the inequality.