n = (1 + δ)^{n} > \frac{n (n - 1)}{2} δ^{2}

δ^{2} < \frac{2}{n - 1}

and

n^{1 / n} = 1 + δ < 1 + \sqrt{\frac{2}{n - 1}} .

Hence

δ \to 0

n \to \infty

and

n^{\frac{1}{n}} \to 1

n \to \infty .

For $n = 1, 2,$ etc. we have $n = 1, 2^{1 / 2}, 3^{1 / 3}, 4^{1 / 4} = 2^{1 / 2} . . .$ and we see that the values increase to a maximum of $3^{1 / 3}$ and then start to decrease. We need to prove that there is no large value of $n$ where the value is larger than this and clearly it is impossible to check all values of $n$ . However, using the earlier result, if $n \geq 19$ then $n^{1 / n} < 1 + \sqrt{\frac{2}{18}} = 1 + 1 / 3 < \sqrt{2}$ so the maximum occurs within the set where $n = 1$ to 18. It can be checked that the maximum is $3^{1 / 3}$ .