Does the infinite series $\sum_{n=1}^\infty \frac{\ln n}{n}$ converge?

Question

Does the series $\sum_{n=1}^\infty \frac{\ln n}{n}$ converge?

Is there a way to determine this besides a comparison test?

You can use the integral test if you don't like the comparison test — , Nov 12 '16 at 19:40

Olivier Oloa · Accepted Answer · 2016-11-12T20:04:22.020

3

Hint. The given series is divergent

by comparison with the harmonic series: $$ \frac1n<\frac{\log n}n , \qquad n \geq 3, $$
by the integral test: $$ \frac{\log^2 N}2=\int_1^N \frac{\log x}x\,dx\le\sum_{n=1}^N\frac{\log n}n $$ letting $N \to \infty$.

edited Nov 12 '16 at 20:04

answered Nov 12 '16 at 19:38

Olivier Oloa

3

Why is that a hint? – zhw. Nov 12 '16 at 19:40
This is more than a hint. This is nearly a complete answer. Nearly two complete answers for that matter. – Wojowu Nov 12 '16 at 20:10
1

Thank you. I was looking for a way besides the comparison test so this is perfect. – Sithe Nov 13 '16 at 02:19

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