Abstract
Let \(R_{n}={\sum }_{k=1}^{n}\frac {1}{k}-\ln \left (n+\frac {1}{2}\right )\). DeTemple proved the following inequality:
\( \frac {1}{24(n+1)^{2}}<R_{n}-\gamma <\frac {1}{24n^{2}} \)
for all integers n ≥ 1, where γ denotes the Euler–Mascheroni constant. In this paper, we give a pair of recurrence relations for determining the constants a ℓ and b ℓ such that
\( R_{n}-\gamma \sim \sum\limits_{\ell =1}^{\infty }\frac {a_{\ell }}{(n^{2}+n+b_{\ell })^{2\ell -1}},\qquad n\to \infty . \)
Based on this expansion, we establish some inequalities for the Euler–Mascheroni constant.
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Chen, CP. Inequalities and asymptotics for the Euler–Mascheroni constant based on DeTemple’s result. Numer Algor 73, 761–774 (2016). https://doi.org/10.1007/s11075-016-0116-9
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DOI: https://doi.org/10.1007/s11075-016-0116-9