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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26 views

Can you define a Cauchy product from this identity?

I can write vague expessions for $\zeta(3)$, the Apéry's constant, for example when I multipliy by $\frac{1}{n^4}$ the recursion relation (2) in page 2 here, and after I take the sum ...
2
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2answers
38 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 ...
2
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1answer
26 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 ...
1
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1answer
50 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
46 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 ...
1
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1answer
14 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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0answers
6 views

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 ...
6
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1answer
146 views

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

Define the following, $$F_2(x) := \frac{1}{2}+\frac{(2x)}{1!} B_2\Big(\tfrac{1}{2}\Big)+\frac{(2x)^2}{2!}B_3\Big(\tfrac{1}{2}\Big)+\frac{(2x)^3}{3!}B_4\Big(\tfrac{1}{2}\Big)+\dots $$ ...
1
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1answer
39 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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0answers
7 views

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 ...
1
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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 ...
1
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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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0answers
34 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
40 views

Where in the theory of higher dimensions do Bernoulli numbers arise? [closed]

In the theory of metric spaces of higher dimensions, where the Bernoulli numbers arise? Are there any formulas for instance for volumes of manifolds involving Bernoulli numbers?
2
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1answer
69 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 ...
1
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2answers
31 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 ...
1
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1answer
156 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 ...
2
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1answer
45 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. ...
6
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3answers
157 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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0answers
21 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
48 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 ...
1
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1answer
42 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
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3answers
95 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
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1answer
61 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
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1answer
34 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
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1answer
60 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
51 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
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0answers
32 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
131 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!} \, ...
8
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2answers
139 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
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2answers
78 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 $B_{2n+2}$ ...
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1answer
24 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
94 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
38 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
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1answer
71 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
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2answers
153 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
113 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
57 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
31 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
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2answers
59 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
43 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
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
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1answer
91 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
30 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 ...
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1answer
82 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
42 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
150 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
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0answers
51 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
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3answers
74 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
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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 ...