trying to find if a particular series converges or diverges

**Oiler** · October 11th 2011, 09:28 PM

Hey all,

I have the following series $\sum_{n=1}^{\infty}(\frac{ln(n)}{n})^2$ . I used the integral test, but since the function has a maximum at e not sure if its the right test. I got 2/e which is < 1 hence the series converges.

Thanks, for all thelp..

**FernandoRevilla** · October 11th 2011, 10:53 PM

Denote $u_n=\left(\dfrac{\ln n}{n}\right)^2$ and $v_n=n^{3/2}$ then, $\frac{u_n}{v_n}=\frac{\ln ^2 n}{n^{1/2}}\to 0$ as $n\to +\infty$ . Being $\sum_{n=1}^{+\infty}v_n$ convergent, also $\sum_{n=1}^{+\infty}u_n$ is convergent.

Remark: This is a particular case of the Bertrand series.

**Danny** · October 12th 2011, 08:25 AM

Originally Posted by Oiler

Hey all,

I have the following series $\sum_{n=1}^{\infty}(\frac{ln(n)}{n})^2$ . I used the integral test, but since the function has a maximum at e not sure if its the right test. I got 2/e which is < 1 hence the series converges.

Thanks, for all thelp..

You can apply the integral test if you start your series at n = 3 so $\sum_{n=3}^{\infty}(\frac{ln(n)}{n})^2$ .
Adding a few terms (or even subtracting a few) won't change the converence of the series.

**process91** · October 12th 2011, 08:43 AM

Echoing Danny's comments, the more general requirement for the intergral test is that there must exist some N such that, for all $n \ge N$ , $a_n >0$ . That is, it only matters that the "tail" of the sequence is positive.

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