The nth root of n tends to 1

Builds on Limit of a sequence Squeeze theorem

Example

For each integer $n \ge 1$ , let $a_n = n^{1/n}$ be the positive $n$ -th root of $n$ . Then $a_n \to 1$ as $n \to \infty$ . For the proof, write $a_n = 1 + h_n$ with $h_n \ge 0$ . For $n \ge 2$ , the binomial theorem gives $n = (1 + h_n)^n \ge \binom{n}{2} h_n^2 = \frac{n(n-1)}{2} h_n^2$ . Thus $0 \le h_n \le \sqrt{2/(n-1)}$ , and the squeeze theorem gives $h_n \to 0$ .

Remarks

The first terms are $1, \sqrt{2} \approx 1.414, 3^{1/3} \approx 1.442, 4^{1/4} \approx 1.414$ . The largest term is at $n = 3$ . In the same way, $c^{1/n} \to 1$ for each real $c > 0$ . Mathlib proves the version for a real variable, $x^{1/x} \to 1$ as $x \to \infty$ , in the lemma tendsto_rpow_div.

Reading path

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Sequence definition
Real sequence definition
Limit of a sequence definition
Squeeze theorem statement
The nth root of n tends to 1 example

Tag	`0AVEYZ2T0Q`
URI	`m2://0AVEYZ2T0Q`
Kind	example
Writing	W0 stub
MSC	40A05
Instance of	Convergent sequence
Uses	Limit of a sequence, Squeeze theorem

The nth root of n tends to 1

Example

Remarks

Reading path

Further reading

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