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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0
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1answer
28 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 ...
3
votes
1answer
58 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
35 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
votes
3answers
143 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
41 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
42 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
votes
1answer
21 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
25 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 ...
0
votes
0answers
19 views

$P(X\le z)$ for $z$ between $z\in(1,\infty)$ for binomial distribution

If I have a binomial distribution of $X$ (a random variable), where $X=\{I:X_1= \dots X_{i-1}=0, X_i = 1\}$, how do I find an expression $P(X\le z)$ for $z\in(1,\infty)$? Any help appreciated!
3
votes
1answer
23 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
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0answers
18 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
28 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
31 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 ...
6
votes
2answers
110 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 - ...
7
votes
2answers
38 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, ...
0
votes
2answers
17 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: $c(9,3)\cdot (\frac{4}{5})^3\cdot (\frac{1}{5})^6=.002753$ ...
0
votes
2answers
27 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 ...
2
votes
2answers
303 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 ...
0
votes
2answers
28 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
27 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 ...
1
vote
0answers
28 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
votes
1answer
31 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 ...
0
votes
1answer
82 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
votes
0answers
47 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 ...
4
votes
1answer
92 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 $$ ...
2
votes
1answer
67 views

Stirling-like sum equal to zero when $k>n$

I need to prove that $$\sum_{r=0}^k\binom{k}{r}(-1)^r r^n=0$$ when $n<k$. I know that the formula above can be easily transformed into the Stirling number of the Second kind formula, which is ...
6
votes
2answers
105 views

Sum Involving Bernoulli Numbers : $\sum_{r=1}^n \binom{2n}{2r-1}\frac{B_{2r}}{r}=\frac{2n-1}{2n+1}$

How can we prove that $$\sum_{r=1}^n \binom{2n}{2r-1}\frac{B_{2r}}{r}=\frac{2n-1}{2n+1}$$ where $B_{2r}$ are the Bernoulli numbers? $$\begin{array}{c|c|c|} n & \frac{2n-1}{2n+1} & ...
0
votes
0answers
12 views

conditional mean of geometric RV

Say, there are three nodes: $S$, $R$, $D$. $S$ transmits to $R$, $R$ stores the packets, and later transmits to $D$. At any time, either $S$ or $R$ is selected to transmit according to some random ...
1
vote
1answer
82 views

Bernoulli numbers closed form.

I found this nice explicit formula for the Bernoulli numbers: $$B_n = \sum_{k \mathop = 0}^n \sum_{i \mathop = 0}^k (-1)^i \binom k i \frac {i^n} {k + 1}$$ I can't find a proof though. I want to ...
1
vote
1answer
77 views

Why do the even Bernoulli numbers grow so fast?

Question is in the title. We have: $$B_{2n} \sim (-1)^{n-1} 4 \sqrt {\pi n} \left( \frac n {\pi e} \right)^{2n}$$
1
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0answers
64 views

Ergodic Theory, Bernoulli Measure, Cylinder Set

Let $N\geq2$ be an integer and consider the probability space ($\Sigma^+$,B, $\mu_p$) where $\mu_p$ is the Bernoulli measure with respect to probability vector $\ p = (p_1,...,p_N)$ . Show that for ...
0
votes
1answer
20 views

Functional Choice for p in a Bernoulli Distribution

Why is the functional choice $p = \exp(x)/(1+\exp(x))$ to model $p$ a good one in a Bernoulli distribution? Is it because it is limited at $0$ as $x$ approaches $0$ and $1$ as $x$ approaches ...
1
vote
1answer
62 views

{Probability}: choosing keys from a pool without replacement

The OP is trying to understand the following question. The OP understand that if you can always write out the term $$P(X=k) \implies (1-\frac{1}{N})(1-\frac{1}{N-1})\cdots(1-\frac{1}{N-k+1}),$$ ...
0
votes
1answer
58 views

Probability: deviation from the mean

I am having trouble to understand the following. If $S_n=X_1+X_2+......+X_n$, where X_1,X_2 are Bernouli (p). I don't understand this. So you get an intermediate point Constant* sqrt(n). To the ...
2
votes
1answer
56 views

Probability exercise Bernoulli. [closed]

Probability random signals. Im late I have no idea to start and this is for tomorrow. I was on training and have no break to do this work. I do this.You are an Internet savvy and enjoy watching video ...
5
votes
1answer
79 views

Question about $ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) \forall s\in \Bbb N$

what I found from messing around was $$ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) $$ $$ s\in \mathbb{N} $$ when the partial sum is changed to an equivalent polynomial using Faulhaber's formula. ...
3
votes
4answers
282 views

How is the Bernoulli numbers? For example, as against $B_2$?

How is the Bernoulli numbers? For example, as against $B_2$? For example, found that in internet $$\sum_{n=0}^{\infty} \frac{B_n\;x^n}{n!}=\frac{x}{e^x-1}$$ but if I want to find $B_2$ then ...
1
vote
1answer
33 views

dual formula to Bernoulli polynomials

$$ \tilde{B}_n(x) = \frac{(-1)^{n + 1}}{n!} \left( \delta^{(n - 1)}(x - 1) - \delta^{(n - 1)}(x) \right) $$ Wikipedia says this formulae is DUAL to the Bernoulli POlynomials but dual in what sense ?? ...
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votes
0answers
25 views

Showing that Faulhaber's formula is a polynomial of degree $m+1$ [duplicate]

The Faulhaber polynomials $S_{m}(n)$ are defined as $$S_{m}(n)=\sum_{k=1}^{n}k^{m}$$ It is the case that $S_{m}(n)$ is a polynomial in $n$ of degree $m+1$; I've seen a proof of this before, but I ...
2
votes
1answer
50 views

Two-term Binomial-Bernoulli Transform

The binomial transform states that if one has two real sequences $\{a_k \}$ and $\{b_k \}$ satisfying $b_n = \sum_{k = 0}^{n} \binom{n}{k} a_k$, then $a_n = \sum_{k = 0}^{n} (-1)^{n-k} \binom{n}{k} ...
0
votes
0answers
43 views

Find the closed form of $S(m, n)=\sum_{j=1}^{m}\sum_{i=1}^{n}i^j$

I know that the closed form of the sum $\sum\limits_{i=1}^{n}i^j$ can be written using Bernoulli numbers $B_k$. It is the famous Faulhaber's formula: $$\sum\limits_{i=1}^{n}i^j =\frac{1}{j+1} ...
0
votes
2answers
101 views

Finding a Correlation between Bernoulli Variables?

Let X and Y be Bernoulli random variables. We don't assume independence or identical distribution, but we do assume that all 4 of the following probabilities are nonzero. Let a := P[X = 1, Y = 1], b ...
2
votes
2answers
71 views

How does one derive this formula $\zeta(-n) =\frac{B_{n+1}}{n+1}$?

$$\zeta(-n) =-\frac{B_{n+1}}{n+1}$$ What is the motivation or the derivation of this formula?
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vote
0answers
104 views

Is there a closed form formula for the Bernoulli numbers?

A while ago I found this algorithm. Today I read in wikipedia that Euler zig zag numbers can be used for computing the Bernoulli numbers. This Mathematica program computes the Euler zig zag numbers ...
7
votes
1answer
312 views

Convergence of $\sum_\lambda \frac{1}{1-\lambda x}$ where $p(\lambda)=0$ for a certain polynomial $p$

The powers of the roots $\lambda$ of these polynomials $$p_n(x):=\sum_{k=1}^{n-1}\frac{n!}{(n-k)!}x^{k-1}$$ (compare with the $p_n$ here) sum to these values $$\sum_\lambda ...
3
votes
0answers
58 views

References mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers?

Is there anyone knows where is some official reference mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers ...
4
votes
1answer
169 views

Roots of some modified Bernoulli polynomials

Update The polynomials are generated as follows: Where $B_n(x) = \sum_{k=0}^n {n \choose k} b_{n-k} x^k$ is used to generate standard Bernoulli polynomials, top plot is generated as follows: ...
39
votes
5answers
831 views

Generalizing the sum of consecutive cubes $\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$ to other odd powers

We have, $$\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$$ $$2\sum_{k=1}^n k^5 = -\Big(\sum_{k=1}^n k\Big)^2+3\Big(\sum_{k=1}^n k^2\Big)^2$$ $$2\sum_{k=1}^n k^7 = \Big(\sum_{k=1}^n ...
5
votes
1answer
152 views

Is $\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$ a rational number for $m,n\ge 2\in\mathbb N$?

Question : Is $$\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$$ a rational number for $m,n\ge 2\in\mathbb N$ where $\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$? Motivation : We know that $$\zeta ...
5
votes
2answers
104 views

Verifying a relation involving Bernoulli polynomials

I would appreciate help, please, as to how to verify this relation from Kato's "Fermat's Dream" p.96. He say: By the definition of $B_n(x)$, the Bernoulli polynomial, we have ...