Some sharp continued fraction inequalities for the Euler-Mascheroni constant
نویسندگان
چکیده
منابع مشابه
Inequalities for the Euler-Mascheroni constant
Let Rn = n k=1 1 k −log n + 1 2 , H(n) = n 2 (Rn −γ), n = 1, 2,. . ., where γ is the Euler-Mascheroni constant. We prove that for all integers n ≥ 1, H(n) and [(n + 1/2)/n] 2 H(n) are strictly increasing, while [(n + 1)/n] 2 H(n) is strictly decreasing. For all integers n ≥ 1, 1 24(n + a) 2 ≤ Rn − γ < 1 24(n + b) 2 with the best possible constants a = 1 24[−γ + 1 − log(3/2)] − 1 = 0.55106. .. a...
متن کاملLimits and inequalities associated with the Euler-Mascheroni constant
(i) We present several limits associated with the Euler-Mascheroni constant. (ii) Let γ = 0.577215 . . . be the Euler-Mascheroni constant, and let Tn = ∑n k=1 1 k − ln ( n+ 1 2 + 1 24n ) and Pn = ∑n k=1 2 2k−1 − ln(4n). We determine the best possible constants α, β, a and b such that the inequalities 1 48(n+ α)3 ≤ γ − Tn < 1 48(n+ β)3 and 1 24(n+ a)2 ≤ Pn − γ < 1 24(n+ b)2 are valid for all int...
متن کاملContinued Fraction Digit Averages and Maclaurin's Inequalities
A classical result of Khinchin says that for almost all real numbers α, the geometric mean of the first n digits ai(α) in the continued fraction expansion of α converges to a number K ≈ 2.6854520 . . . (Khinchin’s constant) as n → ∞. On the other hand, for almost all α, the arithmetic mean of the first n continued fraction digits ai(α) approaches infinity as n → ∞. There is a sequence of refine...
متن کاملOn The Khintchine Constant For Centred Continued Fraction Expansions∗
In this note, we consider a classical constant that arises in number theory, namely the Khintchine constant. This constant is closely related to the growth of partial quotients that appear in continued fraction expansions of reals. It equals the limit of the geometric mean of the partial quotient which is proved to be the same for almost all real numbers. We provide several expressions for this...
متن کاملInequalities for the Normal Integral including a New Continued Fraction
for R(t) which is rapidly convergent for small values of t and which incidentally provides a new set of inequalities. The rapidity of convergence is compared with a series for R(t) and with the Laplace C.F. for R(t). This assessment is similar to recent work by Teichroew (1952) on the comparative rapidity of convergence of series ando.F.'s for the elementary function e?, hi (1 + x) and arc tan ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Inequalities and Applications
سال: 2015
ISSN: 1029-242X
DOI: 10.1186/s13660-015-0834-x