# Quick question on a series

1. Dec 12, 2011

### miglo

1. The problem statement, all variables and given/known data
$$\sum_{n=1}^{\infty}\frac{(-1)^n}{\ln(n+1)}$$

2. Relevant equations
absolute convergence test
nth term test/divergence test

3. The attempt at a solution
so the absolute convergence test says that if the absolute value of the series converges then the original series converges absolutely
so with the series i have, in absolute value is $$\sum_{n=1}^{\infty}\frac{1}{\ln(n+1)}$$
then using the nth term test/divergence test the sequence $a_n=\frac{1}{\ln(n+1)}$ goes to zero as n goes to infinity therefore the series converges, so i have absolute convergence
but my book says that it only converges conditionally, what am i doing wrong? or is the book wrong?

2. Dec 12, 2011

### Dick

The nth term test can only show you that a series diverges. It can't prove it converges. I'd suggest you try a comparison test or an integral test to show 1/ln(n+1) diverges.

3. Dec 12, 2011

### miglo

oohhh right, its been awhile for me since ive done problems on series
cant believe i forgot how the nth term test works, thanks!

4. Dec 12, 2011

### McAfee

Also, try the alternating series test.