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How can I show that $\sum \limits_{n=2}^\infty\frac{1}{n\ln n}$ is divergent without using the integral test?

I tried using the comparison test but I could not come up with an inequality that helps me show the divergence of a series. I also tried using the limit comparison test but I was not successful.

Please do not give me solutions; just a hint so that I can figure it out myself.

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  • $\begingroup$ This is just equal to \left(\frac{1}{n}\right)$ \left(\frac{1}{\text{ln} \, n}\right)$. Do note that $\text{ln} \, n \lt n$ therefore $\frac{1}{\text{ln} \, n} \gt \frac{1}{n}$ thus it would be diverging faster than the harmonic series. Not that rigorous though, so you would have to do something else to prove it. $\endgroup$ – Aldon Sep 3 '15 at 15:18
  • $\begingroup$ @Aldon: oh; I see this now. I was so stupid to not think of this. Thanks $\endgroup$ – user265696 Sep 3 '15 at 15:21
  • $\begingroup$ Good thing you understood it even if I made a typo. P.S. Comparison tests are awesome. $\endgroup$ – Aldon Sep 3 '15 at 15:30
  • $\begingroup$ See also: math.stackexchange.com/questions/574503/… $\endgroup$ – Martin Sleziak Oct 31 '15 at 20:30
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My usual way is to use Cauchy's condensation test and recall that the harmonic series is divergent (by the same reason, if you like it).

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    $\begingroup$ In fact, this is such a spectacular method that you can use it to resolve $\sum_n 1/(n (\log n ) (\log \log n) \ldots (\log \log \ldots \log n)^p)$ for any $p$ and any number of iterations of the logarithm. $\endgroup$ – user2566092 Sep 3 '15 at 15:15
  • $\begingroup$ @user2566092 Spectacular is a bit overreaching, IMO, since the integral test works equally well for your example :). $\endgroup$ – Erick Wong Sep 3 '15 at 15:24

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