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

learn more… | top users | synonyms

-2
votes
0answers
85 views

Non-standard numbers and exponential form of Zeta function

Basic idea For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...
1
vote
1answer
29 views

Are the Bernoulli denominators always divisible by these corresponding primes?

I was wondering whether it has been proven/disproven yet or at least conjectured that the bernoulli denominator of $B_{2n}$ is divisible by $2n+1$ if and only if $2n+1$ is prime? If not, must the ...
2
votes
2answers
40 views

What is a “Contour Integral” and how do I evaluate one?

A very general question, I apologize, but as you read this, hopefully you get what I'm asking. Recently, Bernoulli Numbers have caught my eye, for I am studying infinite series' and it is a part of ...
0
votes
1answer
54 views

Coin Toss Experiment

I conducted an experiment where I tossed a coin $n=100$ times. I am assuming that the coin flips heads with a probability $p=0.5$. So that the coin is fair with a level of significance of $5%$, I want ...
1
vote
1answer
27 views

Bernoulli-like generating function

What are the coefficients of the series for: $$\frac x{e^x+1}$$ It looks similar to the Bernoulli generating function, but the $+$ sign is throwing me off. I already found the series for its ...
2
votes
1answer
43 views

Bernoulli Numbers generating function and Riemann Zeta function

I've been studying Bernoulli numbers and I came across this summation: $$ \sum_{n=1}^{\infty}\frac{B_n x^n}{n!} = \sum_{n=1}^{\infty}\frac{-n \zeta(1-n) x^n}{n!} = -\sum_{n=1}^{\infty}\frac{\zeta(1-n) ...
1
vote
4answers
50 views

solving limit from 2nd bernoulli number

I'm having trouble solving the following limit: $$ \lim_{x \to 0}\frac{xe^{2x}+xe^{x}-2e^{2x}+2e^{x}}{(e^{x}-1)^{3}} $$ substitution gives a 0/0 indeterminate, and we can get around it with de ...
0
votes
0answers
17 views

Generating bernoulli variables for different lambda's

In order to generate $M$ paths of length $N$ I have to generate Bernoulli variables. In Matlab I used: Q=binornd(1,lambda,L,N); Now I want to generate this for a sequence of values lf lambda, but I ...
9
votes
2answers
124 views

Integral of binomial coefficients

Let the integral in question be given by \begin{align} f_{n}(x) = \int_{1}^{x} \binom{t-1}{n} \, dt. \end{align} The integral can also be seen in the form \begin{align} f_{n}(x) = \frac{1}{n!} \, ...
4
votes
2answers
88 views

Ways to prove Eulers formula for $\zeta(2n)$

I recently, out of interest, tried to prove Euler's formula $\zeta{(2n)}=(-1)^{n-1}\frac{(2\pi)^{2n}}{2(2n)!}B_{2n}$ for all $n\in\mathbb{N}$. I adapted Euler's original proof for ...
3
votes
2answers
73 views

Tangent numbers are divisible by $2^{n}$

Let us consider a $$\tan(z) = \sum_{n=1}^{\infty}{T_{2n-1} \cdot \frac{z^{2n-1}}{(2n-1)!}}$$. So, it can be shown that $$T_{2n+1}=\frac{(-1)^{n} 4^{n+1}(4^{n+1}-1) B_{2n+2}}{2n+2} $$ where $B_{2n+2}$ ...
0
votes
1answer
20 views

Bernoulli odd numbers are 0 $B_{2n+1}=0,\;n>0$

I left the Maclaurin expansion of the function $f(x)=x/(e^x-1)$ $$\frac{x}{e^x-1}=\sum_{k=1}^{\infty} B_k\frac{x^k}{k!}=B_0\frac{x^0}{0!}+B_1\frac{x^1}{1!}+\sum_{k=2}^{\infty}B_k\frac{x^k}{k!}$$ $$ ...
5
votes
1answer
87 views

Cauchy-Ramanujan Formula $ \displaystyle \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} $

Cauchy and Ramanujan both gave the formula: $$ \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} = (2\pi)^{4p+3}\sum_{k=0}^{2p+2} (-1)^{k+1} ...
3
votes
0answers
29 views

Connections between Lambert ProductLog and Bernoulli numbers

The Bernoulli numbers have many (as demonstrated here). Here is one property/characteristation: $$\frac{t}{e^t-1}=\sum_{k=0}^\infty \frac{B_k}{k!} t^k$$ Conspicuously missing from the MathOverflow ...
2
votes
1answer
54 views

Explicit function for Bernoulli numbers

Is there any general explicit formula for Bernoulli numbers ? Something like: $$f(x)=B_x$$ Where $B_x$ is the $x$-th Bernoulli number ? Searching the internet I only found the so-called "generating ...
4
votes
2answers
130 views

An identity involving Bernoulli and Stirling numbers

I was playing with some combinatorial sums and made an observation that I didn't know how to prove: $$\forall n\in\mathbb N,\hspace{10px}\sum_{k=1}^n\frac{B_k\ ...
5
votes
1answer
104 views

Power series related to Bernoulli numbers

I'm reading Tenenbaum's Introduction to analytic number theory. He defines Bernoulli polynomials as the unique sequence $B_n$ such that $B_0=1$ $\forall n\geq0, B_{n+1}'(X)=(n+1)B_n(X)$ ...
1
vote
2answers
42 views

Limit in combination with an infinite series

How would I go about showing the following limits that involve infinite series $$ \lim_{x \to 0^{+}} \sum_{n=1}^{\infty} \frac{(-1)^n}{n^{2k+1}} \sin (2\pi n(x - \frac{1}{2})) = 0 \text{ with } k \in ...
0
votes
1answer
22 views

Fourier series coefficient miscalculation

In a nice introductory paper about Bernoulli numbers that I found, the following claim is made (p. 5, theorem 4.3) The Fourier series of $x$ is given by $b_n = \dots$ (not important, it is wrong in ...
2
votes
2answers
50 views

Serie involving Bernoulli's numbers

I need to find the exact sum of this serie which involves Bernoulli's numbers: $$\sum_{k=1}^\infty {{B_{2k}(k-1)!\over (2k)!}}$$ It converges very quickly but I'm knew to this kind of problems so I ...
3
votes
1answer
35 views

Bounding Riemann zeta function by Euler product formula for finite $N$

In a paper concerning a quick calculation of Bernoulli numbers the following inequality is presented (page 3), only referring to it as "not hard to see" that it holds. $$ \sum_{n \leq N}^{} n^{-s} ...
0
votes
0answers
32 views

Is there a name for this stochastic process?

Let $(\Omega,\mathscr{F},P)$ be a probability space and $\{X_n\}_{n\geq 1}$ be a stochastic process. Assume each $X_n$ only takes two values $0$ or $1$, i.e., $X_n:\Omega\rightarrow \{0,1\}$. Of ...
2
votes
1answer
82 views

Extracting Bernoulli polynomials from their generating function

The generating function for Bernoulli polynomials is $$ \frac{te^{tx}}{e^t-1} = \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}$$ The only way that I know of to get the coefficients out of this is to use ...
0
votes
1answer
28 views

bernuoulli random variable possibility

Last season Ryan hit a homerun in about $7\%$ of his bats. Suppose we model at bat as the outcome of Bernoulli random variable. In a typical week, Ryan takes $25$ at bats. consider the following ...
1
vote
1answer
73 views

What are Bernoulli numbers?

In my calculus class, my teacher said that if one was to try to calculate the maclaurin or taylor series of $\tan x$ by strictly using the definition , then you would run into many problems and your ...
1
vote
1answer
36 views

Probability to get the same cards in a card game two times

I want to calculate the probability to get a set of cards two times in a card game, respectively how many two times I would have to play to get it with a specific probability. The card game is ...
9
votes
4answers
141 views

Bernoulli Number analog using Cosine

I know that Bernoulli Numbers can be found with the generating function $$\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$$ I was wondering if any work has been done using a similar equation ...
0
votes
0answers
44 views

Bernoulli Numbers -Identity?

I have been searching for an identity that would help me simplify an equation. Let, $B_m(x)$ be the Bernoulli Polynomial. What are min and maximum bounds on $B_m(0)$? (essentially the last term in ...
-1
votes
3answers
70 views

Is there a closed-form of $\sum_{n=0}^{\infty }\frac{|B_n|}{n!}=??$ [closed]

Is there a closed-form of $$\sum_{n=0}^{\infty }\frac{|B_n|}{n!}=??$$ where $B_n$ Bernoulli number Thanks
0
votes
0answers
31 views

Is this formula correct?

Is this formula correct? $$\frac{1/x+\sum_{0}^{\infty }B_{2n}(x)^{-n}}{\sum_{0}^{\infty }B_{2n}(x-1)^{-n}}=\frac{x}{x-1},$$ where $B_{2n}$ is a Bernoulli Number, and $x >1$. I tried to ...
3
votes
1answer
57 views

Proving an identity for Bernoulli polynomials

Consider the Bernoulli polynomials $B_n(x)$ given by the expansion $$\frac{te^{xt}}{e^t-1} = \sum\limits_{n=0}^{\infty}B_n(x)\frac{t^n}{n!}.$$ I want to prove the identity $$B_n(1-x)=(-1)^nB_n(x).$$ ...
0
votes
1answer
50 views

Identity of Bernoulli Numbers and Bernoulli Polynomials

Consider the Bernoulli Polynomials $B_n\in\mathbb{R}$ given as the coefficients of the series: $$\frac{t}{e^t-1}=\sum\limits_{n=0}^{\infty}B_n\frac{t^n}{n!}$$ and the Bernoulli polynomials gven by ...
0
votes
0answers
28 views

Trying to understand how the trapezoidal rule applies to a derivation of Stirling's Approximation

I am reading through the wikipedia article on how to derive the Stirling's Approximation. The article applies the Trapezoidal Rule to get the following: $$\begin{align} \ln (n!) - ...
0
votes
0answers
38 views

Link bernoulli numbers and bernoulli polynomials

I got a little question regarding the Bernoulli polynomials/numbers. Basically, I want to show that $$B_n(0) = -\frac{B_n(2\pi i)^n}{n!}$$ Where $B_n(x)$ the $n$-th Bernoulli polynomial and $B_n$ the ...
4
votes
1answer
109 views

Proper Bernoulli Function Generating Function [duplicate]

Consider the function $$\frac{t}{e^t - 1} = \sum_{i=0}^{\infty}\frac{B_i}{i!}t^i$$ This has been one of the famous generating functions for the bernoulli numbers. What about the function associated ...
0
votes
1answer
38 views

Bernoulli Trials with dependent events

There are 3 urns. The first contains 3 red and 2 green balls; the second 2 red and 1 green ball; the third 2 red and 4 green balls. A fair die is rolled and the number appearing on top is noted. If ...
7
votes
1answer
197 views

Finding taylor expansion for $\tanh(x)$

I am a high school student and am trying to find the taylor expansion of $\tanh(x)$ in terms of a summation form. I have gotten this far, and am aware it might get complicated very quickly. If someone ...
1
vote
1answer
32 views

$\frac{-1}{2}$ Zero of odd powers sum polynomials?

Consider the polynomial $S_k(x) \in \mathbb{Q}[x]$ such that $S_k(n)=\sum_{i=1}^{n}i^k, \forall n \in \mathbb{N}$. Now if i recall correctly the definition it should be that ...
4
votes
1answer
90 views

Congruence for the sums of odd powers of integers [duplicate]

Does someone know how to prove ***EDIT by induction**** that for all integers $n\ge1$, $k\gt0$ $$\sum\limits_{i=1}^{n} {i^{2k+1}}\equiv 0\ \ \ \pmod{\frac{n(n+1)}{2}}$$ I thought this should be a ...
1
vote
1answer
60 views

exponential generating function for bernoulli numbers [closed]

How I can find exponential generating function for this sequence $(2^n āˆ’ 1) B_n,$ where $B_n$ is Bernoulli numbers
7
votes
3answers
185 views

Asymptotic for sum

How can I find formula for $\displaystyle{\sqrt[3]1 + \sqrt[3]2 + \sqrt[3]3 + \cdots + \sqrt[3]n}$ with an accuracy ${\rm O}\left(\, 1 \over \vphantom{\LARGE A}n^{5}\,\right)$ Is here we should use ...
0
votes
2answers
141 views

Bernoulli Numbers and radius of convergence

consider the function $f(x)=\frac{x}{e^x-1}$. Since the function $\frac{1}{f(x)}=\frac{e^x-^1}{x}=\sum\limits_{k=0}^{\infty} \frac{x^k}{(k+1)!}$ has a taylor expansion with $\frac{1}{f(0)}\neq 0$ we ...
4
votes
1answer
81 views

Proving of the multiplication theorem for Bernoulli polynomial

How the expression below can be proven: $$B_n(mx) = m^{nāˆ’1} \sum\limits_{k=0}^{m-1}B_n\left(x+\frac{k}{m}\right)$$ Where $B_n(x)$ is Bernoulli polynomial I know it is already proved by Joseph ...
2
votes
1answer
36 views

Exponential generating function of product

It needs to find an exponential generating function for the next sequence: $(2^n-1)B_n$. Where $B_n$ is the n-th number of Bernoulli. I found that exponential generating function for sequence of $B_n$ ...
0
votes
1answer
26 views

Probability function and random variables

Given a Bernoulli r.v., W, which is derived from r.v. T(Poisson) (a)if T=0 then W=1 and b) if T>0 then W=0). One has to show that the sample mean (the proportion of 0s in the sample), is an ...
3
votes
1answer
32 views

Is the Pattern in the Number of Digits in the Bernoulli Numbers Showing Something Significant

For the first couple of powers of $10$, the number of digits in these show a certain pattern, is this a coincidence or is their a reasonable explanation. Specifically if we look at $$ \lfloor ...
0
votes
0answers
65 views

Dependent Bernoulli trials when probability of success depends on last failure

Assume you have a series of $n$ Bernoulli trials $B_1,\ldots, B_n$ each with unconditional probability $p_i$, and these are dependent in the following way: $$\mathcal P(B_i=1 | \mathcal F_{i-1}) ...
0
votes
1answer
34 views

Probability - Random viarbles

A notepad manufacturer requires that 90% of the production is of sufficient quality. To check this, 12 computers are chosen at random every day and tested thoroughly. The day's production is deemed ...
1
vote
1answer
48 views

Probability involing percentages (Bernoulli?)

Assume that about 56% of population belong to group type of O. A) What is the probability that it will need to take a blood test from exactly three individuals in order to find a person with O-type ...
7
votes
3answers
196 views

Exact result of a series using Euler-Maclaurin expansion.

This is a variant of Exercise 64 in Chapter 9 of concrete mathematics. Prove the following identity \begin{equation} \sum_{n = -\infty}^{\infty}' \frac{1 - \cos( 2\pi n k )}{n^2 } = 2 \pi^2 ( k - ...