Infinite series sum of (-1)^n/lnx

[JoeTrumpet]

Homework Statement

Find if the series is absolutely convergent, conditionally convergent, or divergent.

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

Homework Equations

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.

 

[n_bourbaki]

Try thinking some more about comparision tests.

 

[HallsofIvy]

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

 

[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)!

 

[HallsofIvy]

Yes, the "alternating series" test says simply that if terms of $\sum a_n$ are alternating in sign and |a_n| 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.

 

[JoeTrumpet]

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

 

[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.