help proof lim n to infinity (n^2) ^ (1/n)

I know the limit goes to 1. I can do it by l'hopital rule. I am wondering if there are other ways to prove this.

I am thinking about geometric mean arithmetic mean inequality. but that only gives me a upper bound which approaches 1. I think I need a lower bound but I don't know what is a good lower bound.

Please advice. thanks

Re: help proof lim n to infinity (n^2) ^ (1/n)

Quote:

Originally Posted by

**kanli** I know the limit goes to 1. I can do it by l'hopital rule. I am wondering if there are other ways to prove this.

Suppose that $\displaystyle n\ge 2$. Define $\displaystyle a_n = \sqrt[n]{n} - 1$ so it is clear that $\displaystyle a_n>0$.

Thus $\displaystyle n=(1+a_n)^n\ge\frac{n(n-1)}{2}a_n^2$ so that $\displaystyle 0<a_n<\sqrt{\frac{2}{n-1}}$.

Thus $\displaystyle (a_n)\to 0$ giving $\displaystyle \sqrt[n]{n}\to 1.$