# Thread: trying to find if a particular series converges or diverges

1. ## trying to find if a particular series converges or diverges

Hey all,

I have the following series $\displaystyle \sum_{n=1}^{\infty}(\frac{ln(n)}{n})^2$. 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..

2. ## Re: trying to find if a particular series converges or diverges

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

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

3. ## Re: trying to find if a particular series converges or diverges

Originally Posted by Oiler
Hey all,

I have the following series $\displaystyle \sum_{n=1}^{\infty}(\frac{ln(n)}{n})^2$. 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 $\displaystyle \sum_{n=3}^{\infty}(\frac{ln(n)}{n})^2$.
Adding a few terms (or even subtracting a few) won't change the converence of the series.

4. ## Re: trying to find if a particular series converges or diverges

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