How to work with the double sum?

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In summary, the given conversation discusses the equation \sum_{n=0}^{\infty} (-1)^n\sum_{k=0}^{n} n!/(n-k)!* ln^{n-k}(2) and how to work with the double sum. The possibility of combining the two sums into one is considered, but it is uncertain if it is possible. The attempt at expanding the double sum is shown, and it is concluded that the series diverges as the limit of the nth term is not zero.
  • #1
seanhbailey
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Homework Statement



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

Homework Equations





The Attempt at a Solution


How to work with the double sum?
 
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  • #2
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
seanhbailey said:

Homework Statement



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

Homework Equations





The Attempt at a Solution


How to work with the double sum?

seanhbailey said:
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.
 
  • #4
I am getting that the sum goes to infinity- is this right?
 
  • #5
Looks that way to me. The limit of the nth term of the series isn't zero, so the series diverges.
 

1. How do I determine the index ranges for a double sum?

The index ranges for a double sum are determined by the problem at hand. Typically, one index will depend on the other, but this is not always the case. It is important to carefully read the problem and understand the relationship between the two indices.

2. How do I simplify a double sum?

Simplifying a double sum can be done by carefully applying algebraic and arithmetic rules. Some common techniques include factoring, distributing, and using properties of sums. It is also helpful to look for patterns or common terms that can be combined.

3. How do I switch the order of a double sum?

Switching the order of a double sum is only possible if both indices are independent of each other. This can be done by rewriting the sum using the properties of sums and carefully considering the new index ranges. It is important to remember that the order of operations still applies in a double sum.

4. How do I handle limits in a double sum?

Limits in a double sum can be handled by considering the index ranges and applying the appropriate limits for each index. It may also be helpful to use the properties of sums to simplify the expression before evaluating the limits. In some cases, the limits may not be defined, requiring further analysis of the problem.

5. How do I know if I have the correct answer for a double sum?

To know if you have the correct answer for a double sum, you can use mathematical software or perform the calculation by hand to check your result. It is also important to carefully check the index ranges and ensure all terms have been correctly accounted for. In some cases, you may also be able to verify your answer using a different method or approach.

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