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Find all possible values of the positive constant k such that the series

$$\sum_{n=1}^{\infty}\ln\left(1+\frac{1}{n^k}\right)$$

is convergent.


Definitely not root test. Tried ratio test, $L = 1$ which is not conclusive.

Tried integral test but the integral looks too hideous to evaluate.

Any other suggestions?


How do I come up with the series to compare with? Keep practising?

For this question, in my mind, I would compare it with another logarithm and not even think about riemann series.

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    Compare the general term $\ln(1+n^{-k})$ to a power of $n$ and use the known results about [Riemann series](http://fr.wikipedia.org/wiki/S%C3%A9rie_de_Riemann) (link in French, apparently the page does not exist in English).2011-11-13
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    Use $\backslash$ln.2011-11-13
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    @Didier: That doesn't work for $k\le1$, since we'd need the comparison to go the other way to demonstrate divergence.2011-11-13
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    @joriki, sorry? $\ln(1+x)\geqslant \frac12x$ for every $x$ in $(0,1)$.2011-11-13
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    @Didier: I see, sorry, I overlooked that we only need it in $(0,1)$.2011-11-13

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