I find to difficult to evaluate with $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right )$$ I tried to use the fact, that $$\frac{1}{1-n} \geqslant \ln(n)\geqslant 1+n$$ what gives $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right ) \geqslant \lim_{n\rightarrow\infty} n(1-\sqrt[n]{1+n}) =\lim_{n\rightarrow\infty}n *\lim_{n\rightarrow\infty}(1-\sqrt[n]{1+n})$$ $$(1-\sqrt[n]{1+n})\rightarrow -1\Rightarrow\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right )\rightarrow-\infty$$Is it correct? If not, what do I wrong?
Limit:$ \lim\limits_{n\rightarrow\infty}\left ( n\bigl(1-\sqrt[n]{\ln(n)} \bigr) \right )$
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1Your inequality $$\frac1{1-n}\ge\ln(n)\ge1+n$$ is false. – 2012-11-14
4 Answers
Use Taylor!
$$n(1-\sqrt[n]{\log n}) = n (1-e^{\frac{\log\log n}{n}}) \approx n\left(1-\left(1+\frac{\log\log n}{n}\right)\right) = - \log\log n$$
which clearly tends to $-\infty$.
Hint: We look at the behaviour of $$x\left(1-\sqrt[x]{\log x}\right)$$ for large $x$. Rewrite the expression as $$\frac{1-e^{\frac{\log\log x}{x}} }{\frac{1}{x}}.$$ Top and bottom both approach $0$ as $x\to\infty$, so the conditions for using L'Hospital's Rule hold. The rest is a calculation.
$\log(x) \leq x - 1$, so $\log(x) = n \log(x^{1/n}) \leq n (x^{1/n} - 1)$ for all integral $n \geq 1$. Take $x = \log(n)$ to get $\log(\log(n)) \leq n(\log(n)^{1/n}-1)$ or $n(1-\log(n)^{1/n}) \leq -\log(\log(n))$. This shows that your limit is $-\infty$.
Since it's so hard let's solve it in one line
$$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right )=-\lim_{n\rightarrow\infty}\left(\frac{\sqrt[n]{\ln(n)}-1}{\displaystyle\frac{1}{n}\ln(\ln (n)) }\cdot \ln(\ln (n))\right)=-(1\cdot \infty)=-\infty.$$
Chris.
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3This is a tad unclear. – 2012-11-16
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0It seems you're using $$\lim_{x \to 0}\frac{\log(1+x)}{x}=1$$, for example, and that $$(\log n)^{1/n}\to 1$$, which would be good to leave explicit, so that every one may understand. – 2012-11-16
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0@ Peter Tamaroff: since they are trivial limits I considered that everybody understand that. Anyway, I appreciate your message that makes my answer more clearer (if there is need for this). – 2012-11-16
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0Ok. However, they are not "trivial" limits. In fact, I'm not sure about the second one. – 2012-11-16
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0@PeterTamaroff: the second one? It's d'Alembert criterion that is evaluation just at a first sight. – 2012-11-16
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0(without some pen and paper) – 2012-11-16
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0@PeterTamaroff: or simply take n=e^u, where u tends to infinity and you're done elementarily. – 2012-11-16
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0$ u^{\displaystyle1/e^u}= u^{\displaystyle (1/u)\cdot(u/e^u)}=1^0=1 $ – 2012-11-16
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0You are mixing sequential and functional limits there. Also, apart from D Alambert, you need the relation between it and Cauchy's root test, and you will also need Stolz Cesaro, or consider the functional limit and use L Hopital. – 2012-11-16
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0@PeterTamaroff: first of all, it's Jean Le Rond d’Alembert, not D Alambert. Then in the above messages I pointed out that I considered these limits as trivial. You meet them in the first pages in any calculus books, and that's why I just used the results without any further proof. Of course, this doesn't mean that it's a bad thing to post their proofs one billion (or more) times. – 2012-11-16