Bernoulli numbers: comparison to factorials

Question

I am trying to understand the behaviour of the Bernoulli numbers with respect to factorials, specifically I'd like to know whether it is true that, for all $n \in N$ with $n \ge 2$ we have $$ \left|\frac{2B_{2n}}{(2n)!}\right| < \frac{1}{n!} $$

Answer 1

The asymptotic of the Bernoulli numbers and of the central binomial coefficients is well known : $$|B_{2n}|\sim 4\sqrt{\pi\,n}\,\left(\frac n{\pi\,e}\right)^{2n},\qquad\binom{2n}{n}\sim\frac{4^n}{\sqrt{\pi\,n}}$$

This implies that \begin{align} \frac{2|B_{2n}|}{n!\;\binom{2n}{n}}&\sim \frac{8\sqrt{\pi\,n}}{n!}\,\left(\frac n{\pi\,e}\right)^{2n}\frac{\sqrt{\pi\,n}}{2^{2n}}\\ &\sim \frac{8\;{\pi\,n}}{\sqrt{2\,\pi\, n}}\,\left(\frac en\right)^{n}\,\left(\frac {n^2}{4\,\pi^2\,e^2}\right)^n\\ &\sim 4\sqrt{2\,\pi\, n}\,\left(\frac {n}{4\,\pi^2\,e}\right)^n\\ \end{align} This asymptotic goes clearly to infinity and will become larger than $1$ for $n$ a little smaller than $4\,\pi^2\,e\approx 107$, more exactly for $n=103$ as indicated by Old John.

Answer 2

According to pari/gp on my laptop, we have:

$$\frac{2B_{206}.103!}{206!} = 1.488\dots,$$

or, in pari/gp notation:

$$2*bernreal(2*103)*factorial(103)/factorial(2*103) = 1.488\dots,$$

which seems to indicate that your proposed result fails at $n=103$, and probably for all $n>103$.

Answer 3

To expand my comment: as Euler proved, $\zeta(2n)=|(2\pi)^{2n}B_{2n}/(2\times (2n)!)|$; since $\zeta(2k)>1$, we get $|2B_{2n}/(2n)!|>4/(2\pi)^{2n}$. But $1/n!$ goes to $0$ much faster.

Answer 4

The even-indexed Bernoulli numbers $B_{2k}$ satisfy the double inequality \begin{equation}\label{Bernoulli-ineq} \frac{2(2k)!}{(2\pi)^{2k}} \frac{1}{1-2^{\alpha -2k}} \le |B_{2k}| \le \frac{2(2k)!}{(2\pi)^{2k}}\frac{1}{1-2^{\beta -2k}}, \quad k\in\mathbb{N}, \end{equation} where $\alpha=0$ and \begin{equation*} \beta=2+\frac{\ln(1-6/\pi^2)}{\ln2}=0.6491\dotsc \end{equation*} are the best possible in the sense that they cannot be replaced respectively by any bigger and smaller constants.

References

1. H. Alzer, Sharp bounds for the Bernoulli numbers, Arch. Math. (Basel) 74 (2000), no. 3, 207--211; available online at https://doi.org/10.1007/s000130050432.
2. Feng Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, Journal of Computational and Applied Mathematics 351 (2019), 1--5; available online at https://doi.org/10.1016/j.cam.2018.10.049.
3. Ye Shuang, Bai-Ni Guo, and Feng Qi, Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios, Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A Matematicas 115 (2021), no. 3, Paper No. 135, 12 pages; available online at https://doi.org/10.1007/s13398-021-01071-x.