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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1answer
18 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
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1answer
70 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
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0answers
24 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 ...
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0answers
28 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 ...
3
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2answers
103 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
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1answer
100 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 $B_0=1$ $\forall n\geq0, B_{n+1}'(X)=(n+1)B_n(X)$ ...
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2answers
36 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
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1answer
16 views

Fourier series coefficient miscaluculation

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
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2answers
49 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
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1answer
27 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} ...
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0answers
29 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
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1answer
77 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
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1answer
26 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
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1answer
67 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
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1answer
32 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
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4answers
136 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 ...
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0answers
41 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 ...
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3answers
66 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
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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
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1answer
52 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
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1answer
45 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 ...
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0answers
25 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!) - ...
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0answers
34 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
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1answer
105 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
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1answer
31 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
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1answer
140 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
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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
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1answer
75 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
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1answer
58 views

exponential generating function for bernoulli numbers [closed]

How I can find exponential generating function for this sequence $(2^n − 1) B_n,$ where $B_n$ is Bernoulli numbers
7
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3answers
182 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
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2answers
130 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
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1answer
70 views

Proving of the multiplication theorem for Bernoulli polynomial

How the expression below can be proven: $$B_n(mx) = m^{n−1} \sum\limits_{k=0}^{m-1}B_n\left(x+\frac{k}{m}\right)$$ Where $B_n(x)$ is Bernoulli polynomial I know it is already proved by Joseph ...
2
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1answer
33 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
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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
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1answer
29 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 ...
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0answers
45 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
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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 ...
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1answer
42 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
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3answers
171 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 - ...
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2answers
52 views

Proving $\int_0^1 B_n(x) dx=0$ for Bernoulli polynomials

The Bernoulli polynomials $B_k(.)$ are given by $$ \frac{t\:e^{xt}}{e^t-1}=\sum_{k=0}^\infty B_n(x)\frac{t^n}{n!}, \quad |t|<2\pi. \tag{1} $$ I would like to prove that $$ \int_0^1 B_n(x) dx=0, ...
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2answers
20 views

Bernoulli Trial Help!

Assume that $n = 9$, and $p = \frac{4}{5}$ . Find the probability of at least 3 successes and at least 2 failures. What I have so far: $c(9,3)\cdot (\frac{4}{5})^3\cdot (\frac{1}{5})^6=.002753$ ...
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2answers
36 views

Bernoulli Process Help!

The professor who sometimes forgets to bring her briefcase to the office, but assume that, each day, the probability that she forgets the briefcase is 1 /8 . Assume that her forgetting is a Bernoulli ...
3
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2answers
338 views

Bernoulli Numbers

I've read that Bernoulli Numbers are defined by the series $$ \frac{z}{e^z-1}\equiv \sum\limits_{n=0}^{\infty}B_n\frac{z^n}{n!},$$ So if I start with $0$ I get $$ B_0\frac{1}{1}=B_0{1}. $$ My ...
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2answers
39 views

Almost Sure Convergence for Sample Mean of Bernoullis

Let {$B_i$} be a sequence of Bernoulli($\mu$) variables and $X_n$ its sample mean $X_n=\frac1n\sum_i^nBi$. Because of the Strong Law of Large Numbers, we know that $X_n$ converges almost surely to ...
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0answers
28 views

Prove the congruence $pB_{p-1} \equiv -1 \pmod p$ for Bernoulli numbers. [duplicate]

I need to prove that: If $p$ is prime greater than or equal to five, then $pB_{p-1}$ belongs to the p-integers and more over: $$pB_{p-1} \equiv -1 \pmod p$$ Hint:Put $N=p$ in the Faulhaber´s ...
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0answers
43 views

Prove the von Staudt-Clausen congruence of the Bernoulli numbers

I need to prove that: If $p$ is prime greater than or equal to five, then $pB_{p-1}$ belongs to the p-integers and more over: $$pB_{p-1} \equiv -1 \pmod p$$ Hint:Put $N=p$ in the Faulhaber´s ...
2
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1answer
33 views

Meaning of congruence notation for Bernoulli Numbers

I am studying Theorem 4(von Staudt's Theorem) in Borevich-Shafarevich's Number Theory(1966)(page 384) which states: Let $p$ be a prime and $m$ an even integer. If $(p-1)\nmid m$, then $B_m$ is ...
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1answer
93 views

Calculating Laurent Series of Complex Function

How does one alternate the Bernoulli number series expansion $$\frac x{e^x - 1}=\sum_{n=0}^{\infty}\frac{B_nx^n}{n!}$$ To calculate the Laurent Series centered at 0 in the annulus of convergence of ...
2
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0answers
52 views

Bernoulli Conjecture on $B_{2^n}$

So in a recent question I was trying to prove that $2^n-1$ will never be a Carmichael number (Can a Mersenne number ever be a Carmichael number?), I was going to prove it true as long as a certain ...
5
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1answer
160 views

Analytic Continuation of Zeta Function using Bernoulli Numbers

In my complex analysis textbook by Stein and Shakarchi, as an exercise, I am supposed to extend $\zeta(s)$ to the entire complex plane using Bernoulli numbers, but I am stuck. I can prove that $$ ...