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- Theorem 1
For all integers,
,
![{\displaystyle \int _{[0,1]^{n}}{\frac {1}{1-\prod _{i=1}^{n}x_{i}}}\prod _{i=1}^{n}\mathrm {d} x_{i}=\zeta (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03f96bb99e1ad286146ebc78131de9c745dbbbb0)
- Proof
Using the power series of
,
![{\displaystyle \int _{[0,1]^{n}}{\frac {1}{1-\prod _{i=1}^{n}x_{i}}}\prod _{i=1}^{n}\mathrm {d} x_{i}=\int _{[0,1]^{n}}\sum _{j=0}^{\infty }\left(\prod _{i=1}^{n}x_{i}\right)^{j}\prod _{i=1}^{n}\mathrm {d} x_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79ad879c872d3dc8bb21e765f8ed0c5bf5faa8d1)
Evaluating,
![{\displaystyle =\int _{[0,1]^{n}}\sum _{j=0}^{\infty }\prod _{i=1}^{n}x_{i}^{j}\mathrm {d} x_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93d60704735705d64944cd99407eef2dec52109f)
![{\displaystyle =\sum _{j=0}^{\infty }\prod _{i=1}^{n}\int _{0}^{1}x_{i}^{j}\mathrm {d} x_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/029d05a42fdf6bac5119f1fd3fbe7a52e23d2257)
Evaluating the integral,
![{\displaystyle =\sum _{j=0}^{\infty }\prod _{i=1}^{n}{\frac {1}{j+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25c9a24348c7b4c33d458fc1687c6145dd5bf4da)
Evaluating the product,
![{\displaystyle =\sum _{j=1}^{\infty }\prod _{i=1}^{n}{\frac {1}{j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e26d2b75b6f0801d78ffd4bd46291d6a023b90e2)
![{\displaystyle =\sum _{j=1}^{\infty }{\frac {1}{j^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/966d64157e17e537beacced79bdde70757bad270)
Using the definition of the zeta function that holds only for
,
for all integers
![{\displaystyle \blacksquare }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8733090f2d787d03101c3e16dc3f6404f0e7dd4c)
- Note
It can be noted that,
![{\displaystyle \int _{0}^{1}{\frac {1}{1-x}}\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/968e87ceeefbb5eff06df49e39154e9a0dddb1a3)
fails to converge, as
.