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Discrete Trends

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Show that if $n$ is a positive integer then

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

Show that $n^{1/n}\rightarrow 1$ as $n\rightarrow \infty$.

Find the maximum value of $n^{1/n}$ and prove that it is indeed the maximum.