I need to prove that n^1/n tends to 1 as n tends to infinity.

I have a hint that says write n^1/n=1+h and use Binomial Theorem

Im just not sure where to start.

Any ideas?

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- Nov 6th 2012, 02:57 AMalexandria21Proving n^1/n tends to 1
I need to prove that n^1/n tends to 1 as n tends to infinity.

I have a hint that says write n^1/n=1+h and use Binomial Theorem

Im just not sure where to start.

Any ideas? - Nov 6th 2012, 03:43 AMhediRe: Proving n^1/n tends to 1
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- Nov 6th 2012, 03:57 AMProve ItRe: Proving n^1/n tends to 1
$\displaystyle \displaystyle \begin{align*} \lim_{n \to \infty} n^{\frac{1}{n}} &= \lim_{n \to \infty} e^{\ln{\left( n^{\frac{1}{n}} \right)}} \\ &= \lim_{n \to \infty} e^{\frac{1}{n}\ln{n}} \\ &= \lim_{n \to \infty}e^{\frac{\ln{n}}{n}} \\ &= \lim_{n \to \infty}e^{\frac{\frac{1}{n}}{1}} \textrm{ by L'Hospital's Rule} \\ &= \lim_{n \to \infty}e^{\frac{1}{n}} \\ &= e^0 \\ &= 1 \end{align*}$