Proving n^1/n tends to 1

**alexandria21** · November 6th 2012, 02:57 AM

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?

**hedi** · November 6th 2012, 03:43 AM

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**Prove It** · November 6th 2012, 03:57 AM

Originally Posted by alexandria21

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?

$\displaystyle \begin{align*} \lim_{n \to \infty} n^{\frac{1}{n}} &= \lim_{n \to \infty} e^{\ln{\left( n^{\frac{1}{n}} \right)}} \\ &= \lim_{n \to \infty} e^{\frac{1}{n}\ln{n}} \\ &= \lim_{n \to \infty}e^{\frac{\ln{n}}{n}} \\ &= \lim_{n \to \infty}e^{\frac{\frac{1}{n}}{1}} \textrm{ by L'Hospital's Rule} \\ &= \lim_{n \to \infty}e^{\frac{1}{n}} \\ &= e^0 \\ &= 1 \end{align*}$

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