# Homework Help: Double Sum

1. Jan 30, 2010

### seanhbailey

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

What is $$\sum_{n=0}^{\infty} (-1)^n\sum_{k=0}^{n} n!/(n-k)!* ln^{n-k}(2)$$?

2. Relevant equations

3. The attempt at a solution
How to work with the double sum?

2. Jan 30, 2010

### seanhbailey

Could I combine the two sums into one? I am not sure how, but I have a feeling that is what I am supposed to do. Thanks.

3. Jan 30, 2010

### Staff: Mentor

I don't think you can, but I could be wrong.

I would start expanding it and see if that gets me anywhere. One thing that bothers me is that you will have numerous expressions with ln0(2). I don't know what that means, but maybe it's supposed to represent just plain 2.

If you start expanding the double sum, you get:
(+1)*( 2) ; n = 0
+ (-1)*(1!/1! * ln(2) + 1!/0! * 2) ; n = 1, k = 0, k = 1
+ (+1)*(2!/2! * ln2(2) + 2!/1! * ln(2) + 2!/0! * 2) ; n = 2, k = 0, 1, 2
and so on.

4. Jan 30, 2010

### seanhbailey

I am getting that the sum goes to infinity- is this right?

5. Jan 31, 2010

### Staff: Mentor

Looks that way to me. The limit of the nth term of the series isn't zero, so the series diverges.