if $\alpha ,\beta$ belong to $\mathbb{R}$, $\beta$ is not equal to zero, $n$ belong to $\mathbb{N}$ and $$\lim_{n\to\infty} \frac{((1!)(2!)(3!).....(n!))^{1/n^2}}{n^{\alpha}} = \beta$$ then please help me. Any help will be appreciated. Thanks I think this can be solved using Squeeze theorem but i m not able to apply it .... how to apply and solve the problem how to apply and solve the problem i tried using different technique but fail
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Take logs, then approximate the sum by an integral – Empy2 May 08 '21 at 08:14
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Does this answer your question? What is the closed form approximation of the asymptotic growth rate of the superfactorial function? – Parcly Taxel May 08 '21 at 08:51
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Also this product factorial is associated with the Barnes G function which is completely new to me and the limit seems to go to 0 or diverge unless n is small, but not too small:demo – Тyma Gaidash May 08 '21 at 12:37
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Please update this question so that we can find out this amazing limit and reopen it again! – Тyma Gaidash May 08 '21 at 12:47