Check the solution to "I want to show thatlimn→∞n(nn-1)=0"

Cash Duncan

Answered question

2022-04-07

I want to show that
$\lim_{n \to \infty} \sqrt{n} (\sqrt[n]{n} - 1) = 0$

Kendall Wilkinson

Beginner2022-04-08Added 17 answers

The OP's attempt can be pushed to get a complete proof.

n = {(1 + x_{n})}^{n} \geq 1 + n x_{n} + \frac{n (n - 1)}{2} x_{n}^{2} + \frac{n (n - 1) (n - 2)}{6} x_{n}^{3} > \frac{n (n - 1) (n - 2) x_{n}^{3}}{6} > \frac{n^{3} x_{n}^{3}}{8}

provided n is "large enough" 1. Therefore, (again, for large enough n,)

x_{n} < 2 n^{- \frac{2}{3}}

, and hence

\sqrt{n} x_{n} < 2 n^{- \frac{1}{6}}

. Thus

\sqrt{n x_{n}}

approaches 0 by the sandwich (squeeze) theorem.
In fact, you should be able to show that for all

n \geq 12

, we have

\frac{n (n - 1) (n - 2)}{6} > \frac{n^{3}}{8} \Leftrightarrow (1 - \frac{1}{n}) (1 - \frac{2}{n}) \geq \frac{3}{4}

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