- #1

amcavoy

- 665

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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.

[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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