Converging and Diverging Infinite Series help

**Cursed** · March 15th 2009, 04:29 AM

Determine whether the series converges or diverges:

1. $\sum_{n=2} \frac{ln(n)}{n}$

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I tried the $n$ -th term test:

$\lim_{n \to \infty} \frac{ln(n)}{n}$
$\lim_{n \to \infty} (\frac{1}{n})(ln(n))$

As the $n$ in $\frac{1}{n}$ approaches $\infty$ , the denominator gets larger and larger, therefore making the value of $\frac{1}{n}$ smaller and smaller. So then you're essentially multiplying by zero.

So I need to try a new test, but what test do I try?

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2. $\sum_{n=1} \frac{1}{(ln2)^n}$

Can someone give me a hint to help me get started on this one?

**Ruun** · March 15th 2009, 04:42 AM

Hi!

Check Convergence tests - Wikipedia, the free encyclopedia for other convergence tests you can try.

The second one, remember that $ln(2)$ it's a number. Then

$\sum_{n=1} \frac{1}{(ln2)^n} = \sum_{n=2} \frac{1^n}{ln(2)^n}$

Maybe the second term will make think you in a very special case of series.

**CaptainBlack** · March 15th 2009, 05:06 AM

Originally Posted by Cursed

Determine whether the series converges or diverges:

1. $\sum_{n=2} \frac{ln(n)}{n}$

as $\ln(n)$ is positive and strictly increasing we have $\frac{\ln(n)}{n}>\frac{\ln(2)}{n}$ and as the harmonic series is divergent your series is divergent.

CB

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