# Determining convergence of (-1)^n/ln(n!) as n goes to infinity

1. Nov 26, 2011

### FuzzieLogic

1. The problem statement, all variables and given/known data

Determine whether (-1)^n/ln(n!) is divergent, conditionally convergent, or absolutely convergent.

2. Relevant equations
None, really? :S

3. The attempt at a solution

Okay, so I know the series converges by the Alternating Series test - terms are positive, decreasing, going to zero. I also know that it absolutely converges...because Wolfram Alpha told me so. =P I have no idea how to prove it though. The ratio test results in 1 so that's inconclusive, so I'm left with comparison or limit comparison test..but I don't know what to compare it to.

2. Nov 26, 2011

### Dick

I would compare it to 1/ln(n^n). That's less than 1/ln(n!), right? And 1/ln(n^n) diverges by an integral test. Do you have that?

3. Nov 26, 2011

### flyingpig

WolframAlpha tells you the answer, you have the solutions. I don't see what's wrong here

4. Nov 26, 2011

### Dick

Why are you posting here if you have nothing to contribute??

Last edited: Nov 27, 2011
5. Nov 26, 2011

### FuzzieLogic

Hi Dick and FlyingPig,

Dick - you are right, the integral does not converge.

FlyingPig, the problem is that WolframAlpha gave me an integer answer when I entered "the summation of 1/ln(n!) from 2 to infinity", so that was misleading.

Anyway thanks so much for helping me solve the problem.

6. Nov 26, 2011

### FuzzieLogic

This is what WolframAlpha gave me:

NIntegrate::ncvb: "NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in n near {n} = {2.176*10^6}. NIntegrate obtained 2.1848934935880644 and 0.0050671018999950595 for the integral and error estimates."

and then the integer 5.49193 as output. I guess that was an estimate for a sum that never actually converges.

7. Nov 26, 2011

### Dick

You are quite right. Wolfram Alpha can be misleading. So can flyingpig. Can you show the series diverges without trusting either one? What's the integral of 1/(n*ln(n))?

Last edited: Nov 26, 2011