# Questions tagged [bernoulli-numbers]

Questions on Bernoulli numbers, a special sequence of rational numbers that arise as the coefficients in the power series expansions of certain elementary functions.

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### Show that $\sum_{k=1}^n{2^{2k-1}\binom{2n+1}{2k}B_{2k}(0)}=n$

Lately, I've been working on a proof (whose context is not necessary to discuss) and I only need one last thing in order to finish it. To be more specific, for completeness it would suffice to show ...
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### Sum of reciprocal Bernoulli numbers

What is sum of the Bernoulli numbers? discusses the sum of the Bernoulli numbers, using divergent sum methods since the Bernoulli numbers grow exponentially. This exponential growth makes it so that ...
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### How the modified Bernoulli numbers relate to the ordinary Bernoulli numbers

The modified Bernoulli numbers are defined as the numbers $b_k$ whose generating series is $$\frac 1 2\log\left(\frac{\sinh \frac t 2}{\frac t 2}\right) = \sum_k b_k t^k.$$ (I use a slightly different ...
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### About the $n$th derivative of the Riemann zeta function on positive even integers

I know there exist a formula for the Riemann zeta function on positive even integers involving Bernoulli numbers. Do there exist any closed form for the $n$th derivative of the Riemann zeta function ...
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### Bernoulli numbers alternate signs

Michael Spivak in Calculus, Fourth ed., Chapter 27, Problem 16 (page 572) defines the Bernoulli numbers based on $$\frac{z}{e^z-1} = \sum_{n=0}^\infty\frac{B_nz^n}{n!}.$$ He asks the reader to ...
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1 vote
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### How do you apply the digit-extraction algorithm from Plouffe (2022)?

I recently became interested in programs that calculate pi, and I was reading about various spigot algorithms when I came across this Wolfram MathWorld site: https://mathworld.wolfram.com/Digit-...
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### The relation of the Bernoulli numbers to the Catalan numbers

The Bernoulli numbers $B_n$ are the backbone of calculus, and according to B. Mazur, they "act as a unifying force, holding together seemingly disparate fields of mathematics." The Catalan ...
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### How is possible that non independent events have the same probability?

There's a question in the book "An Introduction to Mathematical Statistics and Its Applications" that I can't understand: An urn contains $r$ red balls and w white balls. A sample of $n$ ...
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