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

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

    Quote Originally Posted by kanli View Post
    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 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<a_n<\sqrt{\frac{2}{n-1}}.

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