Telescoping continued fractions for the error term in Stirling’s formula
نویسندگان
چکیده
In this paper, we introduce telescoping continued fractions to find lower bounds for the error term rn in Stirling’s approximation n!=2πnn+1/2e−nern. This improves given earlier by Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is terms of a fraction, together with an algorithm successive fraction. technique allows us experimentally obtain upper sequence convergents fraction difference two fractions.
منابع مشابه
Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions
In this paper, we consider a general tridiagonal matrix and give the explicit formula for the elements of its inverse. For this purpose, considering usual continued fraction, we define backward continued fraction for a real number and give some basic results on backward continued fraction. We give the relationships between the usual and backward continued fractions. Then we reobtain the LU fact...
متن کاملGeneralized Continued Logarithms and Related Continued Fractions
We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base b. We show that all of our so-called type III continued logarithms converge and all rational numbers have fin...
متن کاملContinued Logarithms and Associated Continued Fractions
We investigate some of the connections between continued fractions and continued logarithms. We study the binary continued logarithms as introduced by Bill Gosper and explore two generalizations of the continued logarithm to base b. We show convergence for them using equivalent forms of their corresponding continued fractions. Through numerical experimentation we discover that, for one such for...
متن کاملPeriodic Continued Fractions And
We investigate when an algebraic function of the form φ(λ) = −B(λ)+ √ R(λ) A(λ) , where R(λ) is a polynomial of odd degree N = 2g + 1 with coefficients in C, can be written as a periodic α-fraction of the form
متن کاملPalindromic continued fractions
An old problem adressed by Khintchin [15] deals with the behaviour of the continued fraction expansion of algebraic real numbers of degree at least three. In particular, it is asked whether such numbers have or not arbitrarily large partial quotients in their continued fraction expansion. Although almost nothing has been proved yet in this direction, some more general speculations are due to La...
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 Approximation Theory
سال: 2023
ISSN: ['0021-9045', '1096-0430']
DOI: https://doi.org/10.1016/j.jat.2023.105943