Summary
Fairly sharp bounds (lower and upper) of the quantities log Γ(x+1), log\(\left( {1 \pm x} \right)\sum\limits_{i = 1}^p {1/(x + i)} \) and\(\sum\limits_{i = 1}^p {1/(x + i)^2 } \) are given by evaluating the corresponding series of inverse factorials. These results are useful in the asymptotic theory of order statistics and record value statistics and also in the elementary analytic number theory, with which the quantities frequently concerned.
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References
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Matsunawa, T. Some inequalities based on inverse factorial series. Ann Inst Stat Math 28, 291–305 (1976). https://doi.org/10.1007/BF02504747
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DOI: https://doi.org/10.1007/BF02504747