Triple Integral

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Main Question or Discussion Point

I need to prove the following:

[tex]\int_{0}^1\int_{0}^1\int_{0}^1\frac{1}{1-xyz}dxdydz=\sum_{n=1}^{\infty}\frac{1}{n^3}[/tex]

Or, as a generalization:

[tex]\int_{0}^1\cdots\int_{0}^1\frac{1}{1-\prod_{k=1}^mx_k}\prod_{k=1}^mdx_k=\sum_{n=1}^{\infty}\frac{1}{n^m}[/tex]

...if there is such a generalization.

I don't know where to begin, any suggestions?

Thanks a lot for your help.
 
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Answers and Replies

MalleusScientiarum
You can make a Taylor expansion of the denominator since it is convergent for every point in the domain of integration.
 
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Alright, a friend showed me how to do this for m=3. Does this work?

[tex]\sum_{n=1}^{\infty}\frac{1}{n^3}=\sum_{n=0}^{\infty}\frac{1}{(n+1)^3}[/tex]

and since [tex]\int_{0}^1x^kdx=\frac{1}{k+1}[/tex], the sum can be rewritten as follows:

[tex]\sum_{n=0}^{\infty}\frac{1}{(n+1)^3}=\sum_{n=0}^{\infty}\int_{0}^1\int_{0}^1\int_{0}^1(xyz)^kdxdydz=\int_{0}^1\int_{0}^1\int_{0}^1\frac{1}{1-xyz}dxdydz[/tex].

This works? Or, is the proof more in-depth? I would like to know whether the needs to be a justification for evaluating the outside sum first.

Thanks again.
 

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