Absolute Convergence of Series with (-1)^n and 1/ln(n+1) Terms

[miglo]

Homework Statement

$$\sum_{n=1}^{\infty}\frac{(-1)^n}{\ln(n+1)}$$

Homework Equations

absolute convergence test
nth term test/divergence test

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?

 

[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.

 

[miglo]

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

 

[McAfee]

Also, try the alternating series test.