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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Congruence for Bernoulli numbers

It appears that for every odd prime $p$, the following congruence holds for Bernoulli numbers: $$ 2pB_{p-1}-pB_{2p-2}\equiv p-1\mod p^2\mathbb{Z}_{(p)}. $$ The weaker statement that ...
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20 views

Fix a typo involving the Lobachevsky function in Thurston's notes

I believe that there is a typo in these great notes Thurston's Three-Dimensional Geometry and Topology, Volume 1 (Princeton University Press, 1997), Chapter 7 that is provide us by MSRI, in the ...
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1answer
39 views

Particular values of the Riemann zeta function.

On the wikipedia, near the bottom of the "Specific Values" section, there is a statement that bothers me. $$\zeta(-13)=\zeta(-1)$$ Firstly, it is well noted that the summations must be evaluated ...
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3answers
344 views

Help understanding the weak law of large numbers with respect to statistics

I'm trying to do some self-studying to upgrade my statistics knowledge, and came across this term in a section discussing the weak law of large numbers and Bernoulli's theorem: $$\sum_{k=0}^n ...
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21 views

A new formula relating the factorial and Riemann Zeta function resp. Bernoulli numbers?

I proved the following identities involving the factorial and Riemann's Zeta function respectively the Bernoulli numbers: $$\sum _{k=1}^{i}-{\frac {{\pi }^{-2\,k}\zeta \left( 2\,k \right)\left( -1 ...
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1answer
55 views

Approximation of product of Bernoulli with different proportions

I want to update a variable $Y$ with Beta (uniform for simplicity, $Y \sim U(0, 1)$) distribution, with Bernoulli information each period... But each period the proportion parameter of the Bernoulli ...
3
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1answer
135 views

Find the sum of $1^{n}-2^{n}+3^{n}-4^{n}+\cdots+m^{n}$

After seing this question I started wondering about a generalization of a similar sum. The sum is $$ S(m,n)=\sum_{r=1}^{m}(-1)^{r-1}\;r^{n} $$ I gave this to WA to crunch and it gave $$ S(m,n)= ...
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1answer
29 views

What is the MLE of Bernoulli distribution? or how do you derive it?

Derive the MLE for Bernoulli distribution? Just need help studying for a test tomorrow and this is a question on the practice exam. I think I will get a question on this just with a different ...
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1answer
23 views

Number of success runs of length T or more in N coin tosses

We flip a coin, with success probability $p$, $N$ times. What is the distribution of the number of success runs of length at least $T$?
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1answer
37 views

Manipulations with Bernoulli numbers

Is there a simple way to rewrite the expression $$ c(g):= \sum_{k=0}^{2g} \frac{B_k B_{2g-k}}{k! (2g-k)!} (-1)^k $$ involving a sum of Bernoulli numbers, as a product of just one or two Bernoulli ...
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2answers
66 views

Bernoulli trial: probability of even times of sum of $7$.

We throw a pair of dice unlimited number of times. For any $n\in \Bbb N$, let $$E_n=\text{"at the first n trials, the number of time we get sum of $7$ is even"}$$ Also let $P_n=P(E_n)$. We need ...
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120 views
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How to evaluate this limit about Bernoulli number?

First,we define $\displaystyle I_{1}\left ( x \right )=\frac{\sin x}{x}$, then $\displaystyle \lim_{x\rightarrow 0^+}I_{1}\left ( x \right )=1$, also we have \begin{align*} I_2\left ( x \right ...
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0answers
26 views

Baker Campbell Hausdorff formula and bernoulli numbers

The BCH formula states that the product of two exponentials of non commuting operators can be combined into a single exponential involving commutators of these operators. One may write that $\ln(e^A ...
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1answer
32 views

Importance of Bernoulli Numbers

I'm writing a research paper on the foundations of computing. Supposedly Ada Lovelace wrote an algorithm to find Bernoulli numbers. It sounds cool, but it won't mean anything to my history teacher. ...
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25 views

Delta Method: Estimate the Variance of $T$

Let $X = (X_1,\ldots,X_n)$ be a random sample, where $X_1 \sim \mathrm{Bern}(p)$. Let $\lambda = e^p$. Question: By law of large numbers, $T=e^{(\bar{X})}$ is a consistent estimator for $\lambda$, ...
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2answers
47 views

Bernoulli Number Congruence

In a paper by L. Carlitz entitled A Property of the Bernoulli Numbers, it is written The Bernoulli numbers bay be defined by the symbolic relation $$(B+1)^n-B^n=0 $$ with $n>1$ and $B_0=1$, ...
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15 views

Bernoulli numbers and $b_m$

Find a formula for $b_m$ by evaluating both sides for $f(x)=e^{\lambda x}$ where $\lambda$ is a parameter. The formula is $\int_0^1 f(x)dx=1/2(f(0)+f(1))+\sum_{m=1}^\inf ...
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16 views

Estimating expected value of a missing parameter in data

I'm trying to run an EM algorithm on a Bernoulli dataset with missing values, and am unsure how to tabulate rows with missing data. Would a missing value count towards both the probability of the ...
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1answer
46 views

Prove/Argue that X is a binomial random variable.

assume that an experiment is conducted and that the outcome is considered o be either success or a failure. Let p denote the probability of success. Define X to be ...
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3answers
88 views

Asymptotic formula for $\prod_{k=1}^{\infty}\zeta (2kn)$

Suppose $n\geq 1$ is a positive integer. Can we find an asymptotic formula for this product below. $$\prod_{k=1}^{\infty}\zeta (2kn)=\zeta (2n)\zeta (4n)\zeta (6n) \cdots$$ I tried to use $\zeta ...
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1answer
19 views

Show that $ ix+\frac{2ix}{e^{2ix}-1} = 1+\sum_{j=1}^\infty \frac{(-1)^jB_{2j}}{2j!}(2x)^{2j}$

I proved that $x\cot x = ix+2ix/(e^{2ix}-1)$, now I need to show that $ ix+\frac{2ix}{e^{2ix}-1} =x\cot x = 1+\sum_{j=1}^\infty \frac{(-1)^jB_{2j}}{2j!}(2x)^{2j}$. I know that ...
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2answers
33 views

Show that $f(x)= x/(e^x -1)+x/2$ is even.

Show that $f(x)= x/(e^x -1)+x/2$ is even. So an even function is such that $f(-x)=f(x)$. So I need to show that $f(x)= (-x)/(e^{-x} -1)+(-x)/2=x/(e^x -1)+x/2=f(x)$. I also know that ...
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2answers
44 views

Can you get a closed-form for $\sum_{j=0}^{\infty}\frac{2^{2j-1}B_{2j}}{(2j)!}$?

Let $B_{k}$ the kth Bernoulli number, then using their asymptotic I can justify the absolute convergence of this series $$\sum_{j=0}^{\infty}\frac{2^{2j-1}B_{2j}}{(2j)!},$$ since, if there are no ...
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1answer
35 views

Solving a differential equation with Bernoulli's Method

What approach do you take to solve the differential equation $ y' + (6y/x) = (y^3)/ x^5\ $ through the use of Bernoulli's method? I've assumed u = y^(-2) for substitution, but I don't know where to ...
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1answer
56 views

An Identity Involving Bernoulli Numbers and Stirling Numbers

I am trying to prove the following identity involving the Bernoulli numbers $B_n$: $$\sum_{i=0}^m\sum_{t=0}^{m-i}B_{2t}2^{2t}{4m+4\choose 2t,2i+1,4m-2t-2i+3}=(2m+2)\left(2^{4m+2}-{4m+2\choose ...
2
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1answer
52 views

Probability with flipping the coins

I flip a coin for $N$ times. I stop the flipping until I get 4 consecutive heads. Let $X=P(N\leq6)$. On the other hand, I flip the coin for exactly 6 times. Once I finish all the flips, I check ...
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1answer
21 views

product of Bernoulli and Categorical distribution

I have random variable, which is the product of two random variables, derived such that. $Z = X_i*Y$, where $X_i\sim Ber(p_i)$ and $Y \sim Categorical(i,\frac{1}{n}) $, here $n$ is the number ...
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Can this ratio of Euler Maclaurin Summation terms be simplified somehow?

Here's the equation I'm working with: $$ { (1-z){\sum_{k=0}^\infty}{\binom{z}{k}B_k n^{z-k}}\over{{ z\sum_{k=0}^\infty}{\binom{1-z}{k} B_k n^{1-z-k}}}}$$ Here $B_k$ are the Bernoulli numbers, with ...
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1answer
149 views

On $e^{5x}+e^{4x}+e^{3x}+e^{2x}+e^{x}+1$

Define the following, $$F_2(x) := \frac{1}{2}+\frac{(2x)}{1!} B_2\Big(\tfrac{1}{2}\Big)+\frac{(2x)^2}{2!}B_3\Big(\tfrac{1}{2}\Big)+\frac{(2x)^3}{3!}B_4\Big(\tfrac{1}{2}\Big)+\dots $$ ...
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1answer
42 views

A recursion similar to the one for Bernoulli numbers

For the Bernoulli numbers $B_m$, there is a recursion: $B_0=1$ and $\sum_{j=0}^{m-1}\binom{m+1}{j}B_j=-(m+1)B_m $ for $m\ge 1$. It is known that $B_{m}=0$ when $m\gt 1$ is odd. Now, ...
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Does this self-conjured RV converge almost surely?

I thought of this example in hopes of helping me understand almost sure convergence a little better. So, if you could add any additional (relevant) details in your response I would greatly appreciate ...
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1answer
16 views

How to prove the equality $ B_n(sx) =s^{n - 1}\sum_{j = 0}^{s - 1} B_n (x + \frac{j}{s}) $

Given the following equality: $$ B_n(sx) =s^{n - 1}\sum_{j = 0}^{s - 1} B_n (x + \frac{j}{s}) $$ where $B_n(x)$ - Bernoulli polynomial How to prove the equality? I tried to use generating ...
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1answer
46 views

Prove $\left(1 + 1/\sqrt{n}\right)^n > \sqrt{n}$ for all natural $n$

$$\left(1 + 1/\sqrt{n}\right)^n > \sqrt{n}$$ I'm trying to use Bernoulli's inequality So $\left(1 + 1/\sqrt{n}\right)^n \ge 1 + n/\sqrt{n}$, but I'm not sure what to do from there. Could I say ...
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38 views

Problem inside the derivation of $\zeta(-k)= -\frac{B_{k+1}}{k+1}$

I am having trouble with one step for the derivation of $\zeta(-k)= -\frac{B_{k+1}}{k+1}$ found here. In the below steps, how do we get from $$\frac{1}{\pi{i}}\Bigl(G(z)-2G(2z)\Bigr) = -F(z)+F(-z)$$ ...
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1answer
76 views

On asymptotic properties of the sum of consecutive powers

For positive integers $p, n$, the sum of consecutive $p$-th powers is $$ S_p(n) := \sum_{k=1}^n k^p. $$ From Asymptotic behaviour of sums of consecutive powers we have that $S_p(n)/n^{p+1}$ is ...
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2answers
36 views

The order of sum of powers?

For example, the sum of n is n(n+1)/2, the dominating term is n square(let say this is order 2). For the sum of n^2, the order is 3. Then for the sum of n^k, is the order k+1? I been searching ...
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1answer
158 views

Generating function of a polynomial sequence

Using Wolfram Alpha, I find that the first 6 members $p_j(x)$, $0\leq j\leq 5$, of the polynomial sequence happen to be the first 6 non-zero coefficients of the Maclaurin series of ...
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48 views

Find all integer solutions for $ \frac{B_{2m}}m =\frac{B_{2n}}n$.

For Bernoulli number $ B_n$, prove (or disprove) that the only integer solution for $\dfrac{B_{2m}}m= \dfrac{B_{2n}}n $ is $ (m,n) = (1,7) $ for $ 1\leq m <n $. I have no clue how to prove this. ...
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164 views

How to sum up this series? $\sum_{n=1}^\infty\frac{(-1)^{n-1} B_n}{n}$

I wonder what is the sum of this series? $$\sum_{n=1}^\infty\frac{(-1)^{n-1} B_n}{n}$$ where $B_n$ are Bernoulli numbers. Wolfram Alpha does not help. P.S. As this series diverges I am interested ...
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24 views

Series expansions of trigonometric functions using cosecant numbers

There is a lot of places where one can find series expansions of trigonometric functions using Bernoulli numbers. But I am looking for similar expansions using cosecant numbers, $B_n(1/2)$. Seems ...
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1answer
51 views

How to get formula for sums of powers?

Assuming I have Bernoulli numbers: $B = [\frac{1}{1},\frac{1}{2},\frac{1}{6},\frac{0}{1},-\frac{1}{30}, \frac{0}{1}, \frac{1}{42}, ...]$ How can I get the coefficients of the sums of powers ...
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1answer
49 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 ...
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3answers
154 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 ...
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1answer
64 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 ...
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1answer
39 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 ...
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1answer
67 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) ...
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4answers
52 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 ...
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0answers
48 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 ...
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2answers
141 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!} \, ...
9
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2answers
175 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 ...