Abstract
The aim of this paper is to establish new inequalities for the Euler-Mascheroni constant by the continued fraction method.MSC:11Y60, 41A25, 41A20.
Highlights
The Euler-Mascheroni constant was first introduced by Leonhard Euler ( - ) in as the limit of the sequence n γ (n) := – ln n. ( . ) m m=There are many famous unsolved problems about the nature of this constant
1 Introduction The Euler-Mascheroni constant was first introduced by Leonhard Euler ( - ) in as the limit of the sequence n
A good part of its mystery comes from the fact that the known algorithms converging to γ are not very fast, at least, when they are compared to similar algorithms for π and e
Summary
The Euler-Mascheroni constant was first introduced by Leonhard Euler ( - ) in as the limit of the sequence n γ (n) :=. There are many famous unsolved problems about the nature of this constant (see, e.g., the survey papers or books of Brent and Zimmermann [ ], Dence and Dence [ ], Havil [ ] and Lagarias [ ]). The rate of convergence of the sequence (ν(n))n∈N is n–. Theorem For the Euler-Mascheroni constant, we have the following convergent sequence: r(n). Remark Theorem implies that r (n) is a strictly increasing function of n, whereas r (n) is a strictly decreasing function of n It has similar inequalities for rk(n) ( ≤ k ≤ ), we leave these for readers to verify. Lemma If the sequence (xn)n∈N is convergent to zero and there exists the limit lim n→+∞. From Lemma , we see that the rate of convergence of the sequence (r (n) – γ )n∈N is even higher than the value s satisfying
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