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I'm trying to determine whether the series $\sum_{n=2}^\infty \frac{1}{n \ln n}$ diverges, and I would like to prove this using only the comparison test (either direct or limit), and not using the integral test.

darpan sarkar
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  • What have you tried? – Yunxuan Zhang Jun 26 '25 at 11:32
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    If you know for example that $\sum_{n = 2}^{\infty} \frac{1}{n \ln n \ln \ln n}$ diverges, it's easy ;) But, for series like yours, the integral comparison test and the Cauchy condensation test are the most convenient ones. – Dermot Craddock Jun 26 '25 at 11:34
  • Most places use the Cauchy condensation or integral test for this series—tried $p$-series for comparison and some inequality tricks also, but couldn’t make it work. @YunxuanZhang – darpan sarkar Jun 26 '25 at 11:42
  • @DermotCraddock Yeah, it seems like I have to use such a series or those standard tests—doesn’t look like there’s another way around it. Anyway, thanks for the insight! – darpan sarkar Jun 26 '25 at 11:50
  • If you know the prime number theorem you can use the comparison test plus a proof the primes' reciprocals diverge. – J.G. Jun 26 '25 at 12:15
  • Yes the series in the title diverges. – Adam Rubinson Jun 26 '25 at 14:18

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