How to work with the double sum?

[seanhbailey]

Homework Statement

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

Homework Equations

The Attempt at a Solution

How to work with the double sum?

 

[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.

 

[Mark44]

Homework Statement

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

Homework Equations

The Attempt at a Solution

How to work with the double sum?

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.

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.

 

[seanhbailey]

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

 

[Mark44]

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