# Proving $\lim_{n \to \infty} n[n^{1/n}-1] = +\infty$

I understand that this may require the use of the more popular limit:

$$\lim_{n \to \infty} (1 + \frac{a}{n})^n = e^a$$

but can't seem to find the correct way to use it.

I've tried setting $b_n$ := $(n^{1/n} - 1)$ so that $(1 + b_n)^n = n \to \infty$, but that didn't seem to help.

Any hints on this method or solutions using other methods would be greatly appreciated. Thanks.

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Have you tried to take logarithims of both sides, put it in $0/0$ or $\infty/\infty$ form then apply Lopital's rule? –  Baby Dragon Nov 10 '12 at 17:10
I suspect that "Lopital" should be written "l'Hôpital"! :) –  Giovanni De Gaetano Nov 10 '12 at 17:16
@GiovanniDeGaetano: That's how we write it, but it is my understanding that being obsessed with how one spells names is a modern phenomenon. l'Hôpital himself may possibly have used a variety of spellings. But “Lopital” may not have been among them. –  Harald Hanche-Olsen Nov 10 '12 at 17:26
@HaraldHanche-Olsen You mention a fact I wasn't aware of, but which is certainly true in the spirit! For example Wikipedia mention the second possible spelling: "L'Hospital". Thank you very much for that! :) Regarding the specific spelling "Lopital" I doubt it was used, essentially for etymological reasons, but it doesn't look like a bad error anymore! –  Giovanni De Gaetano Nov 10 '12 at 17:33

You want $n(n^{\frac{1}{n}}-1)>B$, or $n^{\frac{1}{n}}-1>B/n$, or $n>(1+B/n)^n$, which is true for large $n$ as the left hand side goes to infinity and right hand side goes to $e^B$.

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Thanks very much Max. –  Conan Wong Nov 11 '12 at 6:31

Using power series we get $$\frac1x\left(e^x-1\right)=1+\frac12x+\frac16x^3+\dots$$ thus, setting $x=\frac{\log(n)}{n}$ yields $$\frac{n}{\log(n)}\left(e^{\frac1n\log(n)}-1\right) =1+\frac12\left(\frac1n\log(n)\right)+\frac16\left(\frac1n\log(n)\right)^3+\dots$$ Therefore, $$\lim_{n\to\infty}\frac{n}{\log(n)}\left(n^{1/n}-1\right)=1$$ and $$n\left(n^{1/n}-1\right)\sim\log(n)$$ so that $$\lim_{n\to\infty}n\left(n^{1/n}-1\right)=\infty$$

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Why does $n\left(n^{1/n}-1\right)\sim\log(n)$? –  Joe Nov 10 '12 at 20:31
@Joe: that is simply another way of saying $$\lim_{n\to\infty}\frac{n}{\log(n)}\left(n^{1/n}-1\right)=1$$ –  robjohn Nov 10 '12 at 22:24
Oh, wow. Simply glossed over that when I was tired earlier. I thought you had wrote $\left(n^{1/n}-1\right)\sim\log(n)$. –  Joe Nov 10 '12 at 23:55
Thanks very much robjohn. –  Conan Wong Nov 11 '12 at 6:32

Here is a way: Define $f_n(x)=n^x$ and use the mean value theorem; $$n^{1/n}-1=f_n(1/n)-f_n(0)=\frac1nf_n'(c_n)$$ for some $c_n$ between $0$ and $1/n$. I hope you can compute $f_n'(x)$ and then take it the rest of the way yourself?

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Thanks very much Harald. –  Conan Wong Nov 11 '12 at 6:33