论文标题
调整Ramanujan的N近似！
Tweaking Ramanujan's Approximation of n!
论文作者
论文摘要
1730年，詹姆斯·斯特林（James Stirling）建立在亚伯拉罕·德·莫伊夫（Abraham de Moivre）的作品的基础上，出版了所谓的斯特林（Stirling）的近似值$ n！$。他给出了一个很好的配方，该配方是渐近的$ n！$。从那以后，数百篇论文提供了他的结果的替代证明，并得到了改进，包括著名的伯赛德，戈斯珀和莫蒂奇。但是，Srinivasa Ramanujan给出了一个更好的渐近公式。赫希霍恩（Hirschhorn）和维拉里诺（Villarino）很好地证明了Ramanujan的结果以及近似值的错误估计。近年来，斯特林的公式已经有了一些改进，包括尼姆斯，温斯基特尔和陈。在这里显示（i）所有这些渐近结果如何易于验证； （ii）Hirschhorn和Villarino的论点如何允许对Ramanujan的结果进行调整，以提供更好的近似值； （iii）可以通过进一步调整Ramanujan的结果来获得新的渐近公式； （iv）陈的渐近公式比此处提到的其他公式好，而新的渐近公式与陈的公式相当。
In 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling's approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave a nice proof of Ramanujan's result and an error estimate for the approximation. In recent years there have been several improvements of Stirling's formula including by Nemes, Windschitl, and Chen. Here it is shown (i) how all these asymptotic results can be easily verified; (ii) how Hirschhorn and Villarino's argument allows a tweaking of Ramanujan's result to give a better approximation; (iii) that a new asymptotic formula can be obtained by further tweaking of Ramanujan's result; (iv) that Chen's asymptotic formula is better than the others mentioned here, and the new asymptotic formula is comparable with Chen's.
