I have some difficulty with showing that the sequence $$ a_n = \frac{\ln(n)}{n^{1/n}} $$ is divergent. Can anyone help me out with this? Thanks!
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Have you checked the behavior of the terms $a_n$ as $n\to\infty$? What kind of tests/ checks/ heuristics do you have for whether series converge or diverge? – Eugene Shvarts Apr 21 '15 at 02:26
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First we can observe that $n^{1/n}\to1$. In fact $n^{1/n}=e^{\frac{\ln(n)}{n}}$ and $\frac{\ln(n)}{n}\to0$.
Therefore the denominator tends to $1$ while the numerator tends to $+\infty$.
Alamos
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@OinkOink You can learn about the limit of $n^{1/n}$ for example here: http://math.stackexchange.com/questions/115822/how-to-show-that-lim-n-to-infty-n-frac1n-1 – Martin Sleziak Apr 21 '15 at 09:50