论文标题
望远镜的远程分数是斯特林公式中的错误术语
Telescoping continued fractions for the error term in Stirling's formula
论文作者
论文摘要
在本文中，我们介绍了望远镜的持续分数，以在Stirling的近似值$ \ displayStyle n中找到错误项$ r_n $的下限！ = \ sqrt {2π} n^{n+1/2} e^{ - n} e^{r_n}。$这改善了Cesàro（1922），Robbins（1955），Robbins（1955），Nanjundiah（Nanjundiah（1959），Maria（Maria（1965）（1965）和Popov（2017）（2017年））。该表达方式是持续的部分，以及算法，以找到该持续分数的连续术语。我们引入的技术使我们能够以两个持续分数的差异来实验获得一系列持续分数的收敛序列的上限和下限。
In this paper, we introduce telescoping continued fractions to find lower bounds for the error term $r_n$ in Stirling's approximation $\displaystyle n! = \sqrt{2π}n^{n+1/2}e^{-n}e^{r_n}.$ This improves lower bounds given earlier by Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.
