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Question Number 159045 by amin96 last updated on 12/Nov/21

Σ_(n=0) ^∞ (1/((3n+1)^3 ))=?

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{3}{n}+\mathrm{1}\right)^{\mathrm{3}} }=? \\ $$

Answered by mindispower last updated on 16/Nov/21

=Σ_(n≥0) .(1/(27))(1/((n+(1/3))^3 ))  Ψ^((2)) (z)=2Σ_(n≥0) (1/((n+z)^3 ))  we get (1/(54))Ψ^((2)) ((1/3))

$$=\underset{{n}\geqslant\mathrm{0}} {\sum}.\frac{\mathrm{1}}{\mathrm{27}}\frac{\mathrm{1}}{\left({n}+\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{3}} } \\ $$$$\Psi^{\left(\mathrm{2}\right)} \left({z}\right)=\mathrm{2}\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{1}}{\left({n}+{z}\right)^{\mathrm{3}} } \\ $$$${we}\:{get}\:\frac{\mathrm{1}}{\mathrm{54}}\Psi^{\left(\mathrm{2}\right)} \left(\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$

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