Bernoulli numbers: comparison to factorials 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
$$
\left|\frac{2B_{2n}}{(2n)!}\right| < \frac{1}{n!}  
$$
 A: 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.
A: 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 $n>103$.
A: 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.
A: 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

*

*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.

*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.

*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.

