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In addition to the MO question The Ramanujan Problems. , I would like to ask the following.

Problem 754 from the list of the Ramanujan's problems ( http://www.imsc.res.in/~rao/ramanujan/collectedpapers/question/q754.htm ) asks to show that $$e^xx^{-x}\pi^{-1/2}\Gamma(1+x)=(8x^3+4x^2+x+E)^{1/6},$$ where E lies between $\frac{1}{100}$ and $\frac{1}{30}$ for all positive values of $x$. According to http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.9228 (The Problems Submitted by Ramanujan to the Journal of the Indian Mathematical Society, by Bruce C. Berndt, Youn--seo Choi and Soon-Yi Kang) this problem was still open in 1997. What is the status of this problem at present?

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Perhaps I should elevate my comment to an answer.

The problem is solved in Ekatherina A. Karatsuba, On the asymptotic representation of the Euler gamma function by Ramanujan, J. Comput. Appl. Math. 135 (2001), no. 2, 225–240, MR1850542 (2002i:33004).

A version of this paper is available here.

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