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

• October 14th 2012, 07:20 PM
kanli
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.

Suppose that $n\ge 2$. Define $a_n = \sqrt[n]{n} - 1$ so it is clear that $a_n>0$.
Thus $n=(1+a_n)^n\ge\frac{n(n-1)}{2}a_n^2$ so that $0.
Thus $(a_n)\to 0$ giving $\sqrt[n]{n}\to 1.$