# Bernoulli numbers tend to infinity

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?

• One way to prove it is to find a different formula for the coefficients via a partial fraction decomposition. The partial fraction decomposition of $\pi \cot (\pi z)$ is pretty famous and closely related. Nov 26, 2019 at 14:04
• Check out a classical result of Euler relating the zeta values at $2n$ with Bernoulli numbers $B_{2n}$, which up to sign is your "$B_n$". Nov 27, 2019 at 20:53
• @DanielFischer I derived that $\pi \cot \pi z= 1/z - \sum _{k=1} ^\infty B_k (2 \pi )^{2k} z^{2k-1} / (2k)!$ but I can't see nothing.. Nov 27, 2019 at 21:26
• The partial fraction decomposition of $\pi \cot (\pi z)$ is $$\frac{1}{z} + \sum_{n \in \mathbb{Z}\setminus \{0\}} \biggl(\frac{1}{z-n} + \frac{1}{n}\biggr) = \frac{1}{z} + 2z \sum_{n = 1}^{\infty} \frac{1}{z^2 - n^2}\,.$$ In the last sum, for $\lvert z\rvert < 1$, expand $\frac{1}{z^2-n^2}$ into a geometric series, then change the order of summation. You get a form of the Laurent series in which the coefficients are given in terms of $\zeta(2k)$ and not in terms of the $B_k$, and thus a relation between $B_k$ and $\zeta(2k)$. (That's the classical result of Euler that Pythagoras refers to.) Nov 27, 2019 at 21:36
• And that you already found. Rewrite that relation so that $B_n$ stands alone on one side. The other side can easily be shown to tend to $\infty$ (if you know Stirling, you can use that, but much weaker estimates suffice). Nov 27, 2019 at 21:39

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.

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.

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 $$$$\label{Bernoulli-ineq} \frac{2(2k)!}{(2\pi)^{2k}} \frac{1}{1-2^{\alpha -2k}} \le |B_{2k}| \le \frac{2(2k)!}{(2\pi)^{2k}}\frac{1}{1-2^{\beta -2k}}, \quad k\in\mathbb{N},$$$$ where $$\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.