# Quick series question

1. Dec 4, 2007

### frasifrasi

why does the series 1/(nln(n)) diverge? I thought it converged since the limit goes to 0.

2. Dec 4, 2007

### rs1n

No, the SEQUENCE

$$\left\{ \frac{1}{n\ln n} \right\}$$

converges because the limit of the terms go to 0.

However, the SERIES

$$\sum_{n=2}^\infty \frac{1}{n\ln n}$$

diverges using the integral test.

3. Dec 4, 2007

### rs1n

For the series

$$\sum_{n=1}^\infty a_n$$

the condition that

$$\lim_{n\to\infty} a_n = 0$$

is necessary for convergence, however it is not sufficient. That is, satisfying the limit condition is not enough to conclude that the series converges.

4. Dec 4, 2007

5. Dec 4, 2007

### frasifrasi

I see, so

1/n will diverge since p <= 1 and 1/nln(n) is smaller than that, so it will converge as well--is that a correct comparison test?

6. Dec 4, 2007

### HallsofIvy

No, it's not. if an converges and bn< an then bn converges. If an diverges and bn> an then bn diverges. If an diverges and bn< an, you don't have any information as to whether bn converges or not.
As rs1n said, use the integral test.