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Question Number 150549 by mathdanisur last updated on 13/Aug/21
Prove that:  ∀n∈N  Π_(k=1) ^n k! ∙ k^(n−k+1)  ≤ (((n+2)/3))^(n∙(n+1))
$$\mathrm{Prove}\:\mathrm{that}:\:\:\forall\mathrm{n}\in\mathbb{N} \\ $$$$\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\prod}}\mathrm{k}!\:\centerdot\:\mathrm{k}^{\boldsymbol{\mathrm{n}}−\boldsymbol{\mathrm{k}}+\mathrm{1}} \:\leqslant\:\left(\frac{\mathrm{n}+\mathrm{2}}{\mathrm{3}}\right)^{\boldsymbol{\mathrm{n}}\centerdot\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)} \\ $$

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