Bernoulli numbers tend to infinity

Question

Recall that the Laurent series of $\displaystyle\frac{1}{e^z-1}$ near $z=0$ is given as

$$\frac{1}{e^z-1}=\frac1z-\frac12+\sum_{k=1}^\infty \frac{(-1)^{k+1}}{(2k)!}B_k z^{2k-1},$$

where the $B_k$ are the Bernoulli numbers. (This definition of the Bernoulli numbers is slightly different with that from the Wikipedia, but this definition is just the nonzero terms with all positive sign.)

I want to prove that $\displaystyle\lim_{k \to \infty} B_k = \infty$, but I have no idea. How do I have to prove these kinds of statements?

Answer 1

To avoid confusion, it might be a better idea to keep a more conventional way to define the Bernoulli numbers, namely by the generating function $$\frac z{e^z-1}=\sum_{n=0}^\infty B_n\frac {z^n}{n!}.$$ The function $\frac z{e^z-1}-1+\frac 12z$ being even, one has $$\frac z{e^z-1}=1-\frac 12z+\sum_{k=1}^\infty \frac{B_{2k}}{(2k)!}z^{2k}.$$ Then your question boils down to asserting that $$\lim_{k\rightarrow\infty}(-1)^{k+1}B_{2k}=\infty.$$ This is just a classical result of Euler relating the zeta values to Bernoulli numbers:

Theorem $B_{2n}=\frac {(-1)^{n+1}2(2n)!}{(2\pi)^{2n}}\zeta(2n)$

For the formula, you may look up proof or use the other suggested comments. Granting this, note that $\zeta(2n)>1$, so one gets that $$|B_{2n}|>\frac {2(2n)!}{(2\pi)^{2n}},$$ which goes to $\infty$ by trivial comparison test.

Answer 2

Idea, too long for a commentary: $$f(z) = \frac{1}{e^z - 1} - \frac1z + \frac12$$ has a removable singularity at $z = 0$. Defining $f(0) = 0$, you can apply the Cauchy integral formulas (Cauchy's differentiation formula), writing $f^{(n)}(0)$ as a line integral along a small circle $|z| = \epsilon$ and estimating a lower bound of the absolute value of the integral.

Answer 3

The Bernoulli numbers $B_n$ can be generated by \begin{equation*} \frac{z}{e^z-1}=\sum_{n=0}^\infty B_n\frac{z^n}{n!}=1-\frac{z}2+\sum_{k=1}^\infty B_{2k}\frac{z^{2k}}{(2k)!}, \quad \vert z\vert<2\pi. \end{equation*} Because the function $\frac{x}{e^x-1}-1+\frac{x}2$ is even in $x\in\mathbb{R}$, all of the Bernoulli numbers $B_{2k+1}$ for $k\in\mathbb{N}$ equal $0$.

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. Consequently, we acquire $\lim_{k\to\infty}B_{2k}=\infty$.

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.