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Question Number 132091 by rs4089 last updated on 11/Feb/21

$$evaluate$$$${\int }_0^1{\int }_0^1{\int }_0^1{\int }_0^1{\int }_0^1\frac{da.db.dc.dd.df}{1-abcdf}$$

Answered by Dwaipayan Shikari last updated on 11/Feb/21

$$\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{1}{n^5}=\zeta \left(5\right)$$

Answered by Dwaipayan Shikari last updated on 11/Feb/21

$$\underset{n=0}{\sum ^{\mathrm{\infty }}}{\int }_0^1{\int }_0^1{\int }_0^1{\int }_0^1{\int }_0^1{\left(abcdf\right)}^{n}dadbdcdddf$$$$=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{1}{{(n+1)}^5}=\zeta \left(5\right)$$

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