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22 views

Does Fibonacci series 3, 6, and 9 all work?

There are terms of the Fibonacci series. Of these clauses, I want to know if the number of terms satisfying a specific condition is finite or infinite. The special conditions that I am talking about ...
4
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0answers
95 views

A relation concerning the “sum of squares” counting function $r_2(n)$

This is a re-post from MSE as I did not get any response there. Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here ...
0
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0answers
46 views

Looking for prior work on mapping sequences to $\mathbb{R}^3$ with vector cross product [on hold]

To give some context into this post, I am a first year undergraduate in mathematics, and I was told I should post here by one of my math professors. I recently discovered an interesting phenomena in ...
6
votes
1answer
218 views

Asymptotic behavior of a certain trigonometric partial sum

Let $a<0$ and $b>0$ be real numbers such that $a<-2b$. Let $n>1$ be a positive integer and consider the following partial sum: $$ f(n) = \frac{1}{(n+1)^2}\sum_{i=1}^{n}\sum_{j=1}^{n} (-1)^{...
5
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0answers
185 views

A problem of Erdős on convergence of $\sum (-1)^nn/p_n$ and equidistribution of $\pi(n)$ modulo 2

Erdős asked1 whether the series $$\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$$ converges. Here, $p_n$ denotes the n-th prime. I can show that this series converges simultaneously with the series $\sum_{...
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0answers
93 views

When will the series be zero

Let $$S_n=a_n-\frac{1}{a_{n-1}-\frac{1}{a_{n-2}-\frac{1}{a_{n-3}-\frac{1}{\ddots \\ a_2-\frac{1}{a_1}}}}}$$ where $a_1,\dots, a_n$ are real numbers and $a_1$ is nonzero. It can be seen that if $a_1,\...
6
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3answers
398 views

Randomly picking $k$ members of $\{1,\ldots,n\}$

Every day, I randomly pick a sample consisting of $k$ members of $\{1,\ldots,n\}$ where $k\leq n$. I stop as soon as every number of $\{1,\ldots,n\}$ has been picked at least once. Let $S$ be the ...
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0answers
314 views

Why do some mathematicians believe that the notation $(x_n)_{n\in \omega}$ is better than $(x_n)_{n=0}^\infty$ or $(x_n)_{n\in \mathbb N}$ [closed]

It seems that there is a trend among certain set-theory-oriented mathematicians to prefer the notation $n\in \omega$ to $n\in \mathbb N$ even when dealing with things totally unrelated to ordinal ...
1
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1answer
102 views

$\sum_{n=0}^{N-1}\exp(-i\frac{\pi}{N}n^2)=\sqrt{N}\exp(-i\frac{\pi}{4})$

The question is: N is an even positive integer, then $\sum_{n=0}^{N-1}\exp(-i\frac{\pi}{N}n^2)=\sqrt{N}\exp(-i\frac{\pi}{4})$. I thought the terms on the left are the solutions set of some polynomial ...
3
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1answer
230 views

Constant “periodization” of a function

Let $w$ be a rapidly decaying function on $\mathbb{R}$ such that $$ \sum_{n \in \mathbb{Z}} w(x+n) = 0$$ for all $x \in \mathbb{R}$. Does that imply that $w$ is identically zero? What if we assume ...
1
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1answer
117 views

Finding closed form of recurrence relation in two variables [closed]

I have a recurrence, $$F(n, m) = F(n-1, m) + F(n, m-1) + F(n-1,m-1) $$ $$F(n,1) = 0$$ $$F(1,n) = 2*(n-1)$$ I would like to compute $F(N,M)$ in terms of $N$ and $M$. The system is defined for $1 \...
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3answers
1k views

Adventure with infinite series, a curiosity

It is easily verifiable that $$\sum_{k\geq0}\binom{2k}k\frac1{2^{3k}}=\sqrt{2}.$$ It is not that difficult to get $$\sum_{k\geq0}\binom{4k}{2k}\frac1{2^{5k}}=\frac{\sqrt{2-\sqrt2}+\sqrt{2+\sqrt2}}2.$$ ...
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1answer
127 views

Approximate the following series on the euclidean grid

I had trouble in finding a closed form solution for the following series, so now I am trying to find a good approximation for it. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the ...
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0answers
40 views

Is there a formula for computing the parity of a sequence with discrete alphabet?

Suppose we have a sequence of number, whose alphabet is chosen from a discrete set such as (0,1,2). An example of such sequence is 0210122. Now I would like to determine if it is an odd permutation ...
3
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2answers
222 views

Infinite sum of reciprocals of squares of lengths of tangents from origin to the curve $y=\sin x$

This question is actually from MSE. I had to post it here due to the lack of response there even after placing a bounty. Here goes the question Let tangents be drawn to the curve $y=\sin x$ from ...
5
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2answers
597 views

Searching for a proof for a series identity

The below identity I have found experimentally. Question. Is this true? If so, may you provide a "slick" (or any) proof. $$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
2
votes
1answer
88 views

Quotient with positive second derivative in the limit?

I am studying the quotient of $$f(\varepsilon) = \sum_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$ and $$g(\varepsilon) = \sum_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$ for some $\...
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0answers
49 views

Coordinates of Spheres Sequence in Sphere packing

I'm trying to achieve a "sphere packing" by organizing the spheres in a "sphere like form". So basically, I want to put together more spheres and compose a bigger sphere. The spheres don't have to ...
0
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1answer
93 views

May Champernowne constants $C_m$ be related to other numbers than $m$?

[This question is related to another question concerning normal numbers I asked at Math SE.] Has it ever been found worth to ask the question if the Champernowne constants $C_m$, especially $C_2$ ...
2
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2answers
116 views

How to estimate a recursive inequality with an upper bound

The below is a simplification of part of a proof I'm working on, in numerical analysis. It is similar to a paper that I studied some months ago, for which I got some advice here on MathOverflow. I ...
3
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1answer
92 views

Matching two sequences between each other

Given the sequence of symbols $A$ (contains ~10,000 symbols) and sequence of blocks $B$ (contains ~3,000 blocks, ~30 symbols inside each block) I need to exclude some blocks from sequence $B$ so that ...
26
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1answer
2k views

A remarkable almost-identity

OEIS sequence A210247 gives the signs of $\text{li}(-n,-1/3) = \sum_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$. Mikhail Kurkov noticed that it ...
2
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2answers
62 views

Divergence rate of geometric sum of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that $$ 0<\lim_{\beta\rightarrow 1}(1-\...
2
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1answer
168 views

Guess (or upper bound) the general formula for a double sequence

Let $t,s \geq 0$ be integers. We have the following recursive formula: $$f(t+1,s) = f(t,s) + f(t,s-1) + \sum_{0\leq a,b,c \leq h(t):\\a+b+c = s-1}f(t,a)f(t,b)f(t,c),$$ where $$h(t) = \frac{1}{2}3^t -\...
7
votes
1answer
226 views

Descartes' rule of signs for infinite series

Consider the function given by $$f(x)=1-a_1x-a_2x^2-a_3x^3-\cdots$$ where each $a_k\geq0$ and some $a_j>0$. If $f(x)$ is a polynomial then Descartes' Rule of signs tells us there is exactly one ...
10
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2answers
257 views

Denominators of certain Laurent polynomials

Consider the following somos-like sequence $$x_n=\frac{x_{n-1}^2+x_{n-2}^2}{x_{n-3}}.$$ It's known that $x_n$ is a Laurent polynomial in $x_0, x_1$ and $x_2$. I got interested in the denominators of ...
1
vote
1answer
135 views

About a Dirichlet series [closed]

I would like to know if the following assertion is true: Let consider a real decreasing sequence $(t_n)$ of positive numbers with limit zero, if the series $\sum\limits_{n=1}^\infty(t_n)^a$ is ...
8
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0answers
143 views

Infinite series identities in search of a proof

This comes in relation to the Fishburn numbers. I stumbled on the following relation for which I ask a proof if true. Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then $$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
16
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0answers
420 views

Does every real function have this weak derivation property?

After this question : Does every real function have this weak continuity property? Natrualy there are an other (more difficult) : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}...
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3answers
3k views

Does every real function have this weak continuity property?

In my research I came across the following question : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x_n)_n$, taking infinitely many values, ...
-1
votes
2answers
301 views

Are there integral solutions for $(2a-1)(2^{(b+c)}-3^c )=2^b-1$?

Can anyone prove this assertion? Or at least suggest a method of attack? It has come up in my research. There do not exist $a,b$ and $c$ such that$$ (2a-1)(2^{(b+c)}-3^c )=2^b-1 $$where $a>0,b&...
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0answers
35 views

Order 2 linear/geometric recurrent sequence

Let $(u_n)_{n\geq 0}$ a sequence of reals satisfying the following recurrence relation: $$ \forall n>1, \qquad u_{n+2} = r^{n+2}\big( Au_{n+1}+ Bu_n\big) $$ for fixed non zero constants $A,B,r$. ...
4
votes
1answer
71 views

Convergence of a real sequence (stochastic approximation)

Let $(\gamma_n)_{n \geq 1}$ be a sequence of positive real numbers satisfying $$\sum_{n \in \mathbb{N}}\gamma_n = + \infty \text{ and }\sum_{n \in \mathbb{N}}\gamma_n^2 < + \infty$$ I would like ...
3
votes
1answer
146 views

What is the shortest length of an Egyptian fraction expansion for a given $p/q$?

An Egyptian fraction expansion is a sum of reciprocals of integers, for example: $$\frac{4}{17} = \frac{1}{5} + \frac{1}{29} + \frac{1}{1233} + \frac{1}{3039345}$$ Every positive rational number $p/...
7
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1answer
111 views

Is anything known about this class of series involving the divisor function?

I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it! Let $N\in\mathbb{N}$, let $q$ be a point in the open ...
1
vote
0answers
57 views

Show that the norm's bound is an exponent

Let $f$ be a continuous function on $S^2$. Consider $g\in C^{\infty}(R)$, such that $g(x)=1$ for $|x|\leq 1$ and for $|x|\geq 2$. Let $h(x)=g(x)-g(2x)$. The notation $proj_k$ denotes the orthogonal ...
3
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0answers
67 views

Gray Code for Combinations

Question: are Gray Codes known for enumerating all fixed-size subsets of a given finite set? Background of the question is trying to find the lightest k-clique in a symmetric TSP instance of size n ...
1
vote
1answer
127 views

How many points appear in the plane when the chain of n-gons is close?

Let $A_{11}A_{12}\cdots A_{1n}$ be a regular $n$ polygon, we call $A_{11}A_{12}\cdots A_{1n}$ is the $1st-n-gons$. Now we construct the $2nd-n-gon$ based two condition as follows: $2nd-n-gons$ is ...
3
votes
2answers
339 views

Is the exponential version of Catalan-Dickson conjecture true?

The aliquot sum function $s:\mathbb{N}\rightarrow \mathbb{N}$ assigns to any natural number $n$ the sum of its proper divisors. Perfect numbers are fixed points of this function. The open conjecture ...
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1answer
65 views

Create approximations of finite integer sequence

Given a function of real numbers f(x), I can create approximations to arbitrary precision using Taylor polynomials. Is there something equivalent in the discrete case when I have a sequence of ...
1
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0answers
30 views

linear difference inequality with error

Let $(\alpha,\gamma,\beta)$ real number such that $\alpha+\gamma+\beta=1$, let $(e_n)_{n\in\mathbb{N}}$ be a summable non-negative sequence and let $(u_n)$ be a non-negative sequence such that $\...
1
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0answers
111 views

Calculus of variation with discontinuous solutions?

I'm thinking of the following question: Consider a function $f: [0,L]\rightarrow\mathbb{R}$ and an energy functional $$F=\int_{0}^{L}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\mathrm{d}x.$$ The ...
5
votes
3answers
132 views

Continuity and sequential continuity of a linear functional

Let $E = C_c^0(\mathbb{R}^n;\mathbb{R}^m)$ be the space of compactly supported continuous functions on $\mathbb{R}^n$ with values on $\mathbb{R}^m$. There is a natural norm on this space: given $\...
15
votes
3answers
739 views

Infinite series with signed sums

This was asked earlier at MSE. Let $A = \{a_0, a_1, a_2, \dotsc\}$ denote a weakly decreasing sequence of positive terms whose sum converges. Next introduce plus minus signs in every possible way, ...
4
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0answers
236 views

Convergence of $a_n=(1-\frac12)^{(\frac12-\frac13)^{…^{(\frac{1}{n}-\frac{1}{n+1})}}}$ [closed]

I'm interesting to see the opinion of MO about my question which I posted here in SE, Answers I received have not convinced me, And no clear proof posted there only numerical computation are provided. ...
4
votes
1answer
138 views

An inequality involving $k$-generalized Fibonacci numbers

I have worked on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, to solve completely the equation I have one complicated case and I proved ...
3
votes
1answer
217 views

Is $\frac{\pi}{4}L_0(z) = \sum\limits_{n=1}^{+\infty} (-1)^{n+1} \frac{I_{2n-1}(z)}{2n-1}$ between Bessel and Struve known?

Based of the detailed attempt to solve the integral $\int e^{\sin(x)} dx$ I stumbled upon a connection between modified Struve and modified Bessel function of the first kind. But, I cannot find a ...
2
votes
1answer
232 views

Does this sequence contain a nonnegative number?

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
25
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0answers
638 views

How to prove the identity $L(2,(\frac{\cdot}3))=\frac2{15}\sum\limits_{k=1}^\infty\frac{48^k}{k(2k-1)\binom{4k}{2k}\binom{2k}k}$?

For the Dirichlet character $\chi(a)=(\frac a3)$ (which is the Legendre symbol), we have $$L(2,\chi)=\sum_{n=1}^\infty\frac{(\frac n3)}{n^2}=0.781302412896486296867187429624\ldots.$$ Note that this ...
5
votes
1answer
150 views

Show a sequence of sums involving Catalan Numbers converges

Let $C_n$ be the $n$-th Catalan Number and let $\mathcal{O}_{s,j} = {{2s-j-1}\choose{j}} C_{s-j}^2$. Then we want to consider $\mathcal{E}_s = \sum_{j=0}^{s-1} (-1)^j\mathcal{O}_{s,j}$. We want to ...