# Homework Help: Infinite series sum of (-1)^n/lnx

1. Jul 21, 2008

### JoeTrumpet

1. The problem statement, all variables and given/known data
Find if the series is absolutely convergent, conditionally convergent, or divergent.

The sum from two to infinity of (-1)^n/lnx.

2. Relevant equations

3. The attempt at a solution
I don't know how to integrate 1/lnx, so that failed. The ratio and root test don't seem to simplify matters any further. I wasn't sure of anyway I could use the comparison test or limit ratio test. I used Alternate Series Test so I at least know that it's convergent, but I don't know if it's conditionally convergent or absolutely convergent.

2. Jul 21, 2008

### n_bourbaki

Try thinking some more about comparision tests.

3. Jul 21, 2008

### HallsofIvy

That is an alternating series (that's crucial) and ln(x) is an increasing function. What does that tell you?

4. Jul 21, 2008

### JoeTrumpet

Hm, would it be reasonable to say that because n > ln(n) for all n from 2 to infinity, 1/ln(n) > 1/x, which is divergent, thus making 1/ln(n) divergent and the sum from 2 to infinity of ((-1)^n)/ln(n) conditionally convergent? It didn't really occur to me to compare a transcendental function to an algebraic function (if it's even legal, anyway). Thanks for the help (assuming my solution was okay)!

Last edited: Jul 21, 2008
5. Jul 21, 2008

### HallsofIvy

Yes, the "alternating series" test says simply that if terms of $\sum a_n$ are alternating in sign and |an| is decreasing, then the series converges. In fact, it is easy to see that, given any N, the entire sum lies between the partial sum up to N and the partial sum up to N+1.

6. Jul 21, 2008

### JoeTrumpet

But would that test be useful if you're distinguishing between whether something is absolutely convergent or conditionally convergent?

7. Jul 21, 2008

### Dick

It will possibly tell you if it's conditionally convergent. You'll need another test to answer whether is absolutely convergent or not. Something like the integral test or a comparison test.