Is $\sum\limits_{n=1}^\infty \frac {\log n}{n}$ convergent or not?

Question

Problem: Is it convergent or not ?$$\sum_{n=1}^\infty \frac {\log n}{n}$$

Solution:$$ \lim_{n\to\infty} \frac {\log n}{n}=0$$

So it can be convergent or divergent

Other tests are not looking good

please help

A more general result say that series of the form $\sum \frac{1}{n^{\alpha} (\log n)^\beta }$ converges if and only if ($\alpha >1$) or ($\alpha = 1$ and $\beta >1$). — user37238, Sep 18 '13 at 08:06

Did · Accepted Answer · 2016-07-31T21:30:21.103

6

Or not.

Use the fact that $\frac{\log n}n\geqslant\frac1n$ for every $n\geqslant3$ and the fact that the harmonic series $\sum\limits_{n\geqslant1}\frac1n$ diverges.

edited Jul 31 '16 at 21:30

answered Sep 18 '13 at 08:01

Did

279,727

score 5 · Answer 2 · answered Sep 18 '13 at 08:03

5

Integral test $$ \int_1^\infty \frac {\ln x}x dx = \left . \frac {\ln^2 x}2 \right |_1^\infty \to \infty $$ therefore divergent.

answered Sep 18 '13 at 08:03

Kaster

9,722

score 2 · Answer 3 · answered Sep 18 '13 at 08:00

2

Hint: $$\frac{\log{n}}{n} > \frac{1}{n}$$ for any $n > 2$. What can you say about $\sum_{n = 0}^{\infty} \frac{1}{n}$?

answered Sep 18 '13 at 08:00

$\sum_{n = 0}^{\infty} \frac{1}{n}$ is divergent – rst Sep 18 '13 at 08:01

score 1 · Answer 4 · answered Sep 18 '13 at 08:01

1

Hint: For all $n\geq 3$, we have that $\log(n)\geq 1$. Use the comparison test.

answered Sep 18 '13 at 08:01

Zev Chonoles

129,973

score 1 · Answer 5 · answered Sep 18 '13 at 08:01

1

Hint: use Cauchy condensation test

answered Sep 18 '13 at 08:01

Myshkin

35,974
27
154
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Is $\sum\limits_{n=1}^\infty \frac {\log n}{n}$ convergent or not?

5 Answers5

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