Given $$S=\sum_{N=1}^\infty\left(\dfrac{1}{\Gamma(N+1)^2}\right)^{\dfrac{1}{N}}$$ Using the Carleman inequality, I got for S: $$S\le\dfrac{1}{6}e\pi^2$$ Using numerical calculation I suppose that the value of S is less than $3.2$. Is it possible to have a better vaule or eventually a better iniquality for S?
2026-03-26 11:18:13.1774523893
Numerical evaluation of $\sum_{N=1}^\infty\left(\frac{1}{\Gamma(N+1)^2}\right)^{\frac{1}{N}}$
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