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-2
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0answers
34 views

Find a parameter for which sum converges [on hold]

I have $\sum\limits_{n=0}^{\infty} e^{na}=2$ and need to find $a$ for which this sum conv. to 2. How can I approach this assuming I only have a knowledge of a Taylor series and don't know about ...
6
votes
2answers
567 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}...
1
vote
1answer
82 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 $\...
0
votes
0answers
48 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
votes
1answer
91 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 $C_2$ ...
-4
votes
0answers
31 views

Finding Laplace inverse transformation of a product series [closed]

compute the inverse Laplace transformation of the following equation. \begin{align*} f(s)&=\frac{A}{\prod_{i=1}^{L}(s+a_i)^m} \\ &=\frac{A}{(s+a_1)^m\,(s+a_2)^m\cdots (s+a_L)^m}. \end{align*}...
2
votes
1answer
67 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 ...
2
votes
1answer
87 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 ...
24
votes
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
votes
2answers
60 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-\...
0
votes
1answer
111 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
222 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
votes
2answers
247 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
134 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
votes
0answers
137 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
397 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}...
50
votes
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
293 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&...
0
votes
0answers
34 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
70 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
144 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
votes
1answer
102 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
56 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
votes
0answers
64 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
121 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
336 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 ...
-1
votes
1answer
64 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
vote
0answers
29 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
vote
0answers
108 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
115 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
732 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, ...
5
votes
0answers
235 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
130 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
192 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
230 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 ...
24
votes
0answers
591 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
146 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 ...
1
vote
0answers
111 views

Determining a Closed Formula for the Positive Zeroes of the $n^{th}$ Derivatives of the Function $x↦x^{-x}$

The derivatives of the function $ f(x)=x^{-x}$ have interesting properties, especially when looking at their roots. I am interested in studying the behavior of the roots of the derivatives as the ...
5
votes
2answers
190 views

Asymptotic rate for $\sum\binom{n}k^{-1}$

This MO question prompted me to ask: What is the second order asymptotic growth/decay rate for the sum $$\sum_{k=0}^n\frac1{\binom{n}k}$$ as $n\rightarrow\infty$?
8
votes
1answer
938 views

What is the value of this double sum in closed form?

I encountered the following double sum which requires an evaluation. Is there a closed form for this? $$\sum_{n=0}^{\infty}\frac{\sum_{k=0}^n\binom{n}k^{-1}}{(n+1)(n+2)}.$$ Incidentally, it ...
0
votes
1answer
168 views

An interesting series converging to a constant

Let $K>0$ be a constant. Suppose $\{z_n\}_{n=1}^\infty$ is a non-decreasing positive sequence. Then the series $$\sum_{n=1}^\infty\frac{z_n}{(K+z_1)(K+z_2)\cdots(K+z_n)}K^n=K$$ This is a quite ...
4
votes
1answer
578 views

Are all the numbers $\pi(n^2)/n^2\ (n=1,2,3,\ldots)$ pairwise distinct?

For $x>0$ let $\pi(x)$ denote the number of primes not exceeding $x$. A well-known conjecture of Legendre states that $\pi(n^2)<\pi((n+1)^2)$ for any positive integer $n$. Here I ask the ...
5
votes
2answers
573 views

Is the value of $\sum\limits_{k=1}^{\infty}\frac1{(C_k)^n}$ known?

I posted the question https://math.stackexchange.com/questions/2799068/is-the-value-of-sum-limits-k-1%e2%88%9e-frac1c-kn-known before on mathstackexchange but realised that it might be more ...
3
votes
1answer
245 views

Bounding a series of nested integrals

Consider the following matrix function $$ f(t) = \cos(\omega_1t) A_1 + \cos(\omega_2t) A_2, \quad t\ge 0, $$ where $A_1$, $A_2$ are real square matrices and $\omega_1$, $\omega_2$ positive numbers. ...
2
votes
0answers
225 views

Conjectured initial values of Inkeri's primality test for Fermat numbers

This is a repost of this question . Can you provide a proof or a counterexample to the claim given below ? First , we shall give a definition of the Inkeri's primality test for Fermat numbers : ...
1
vote
1answer
198 views

Is there a procedure for extracting first integer $q_0$ from $\sum\limits_{k=0}^{\infty}\frac{1}{q_k^z}$, all $0<q_0<q_1<…$ integers, $z$ complex?

Take $q_0<q_1<...<q_k<q_{k+1}<...$ positive integers, $z$ complex From $$T(z)=\sum\limits_{k=0}^{\infty}\frac{1}{q_k^z}$$ I would need to extract the first coefficient $q_0$ It is ...
1
vote
1answer
88 views

Proving a sum to be sublinear in growth

Suppose that $\alpha \in (0,1)$. The goal is to prove that the following sum is of $o(T)$ (or, if possible, give a more accurate growth rate, e.g. $O(T^{1-\alpha})$ or something like that): $$ \sum_{t=...
2
votes
1answer
223 views

Is this the correct closed form for a series similar to $\zeta(2)$?

I hope this question is well received. I don't have a computer that can calculate very many terms for the infinite series: $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n-1)^{2}},$$ but is it going to equal ...
1
vote
0answers
253 views

approaching the border between absolute convergence and divergence of series

Let us consider absolute convergent series $\ell^{1^+}$ ordered under eventual dominance (mod finite) $<^*$. T. Bartoszynski proved that unbounded number ${\frak b}(\ell^{1^+}, <^*)$ equals ...
1
vote
1answer
69 views

Maximum of the periodogram of a truncated sequence

Suppose that $Z,Z_1,Z_2,\ldots$ are iid random variables such that $\operatorname EZ=0$, $\operatorname EZ^2=1$ and $\operatorname E|Z|^s<\infty$ with some $s>2$. Let $\tilde Z_t=Z_tI_{\{|Z_t|\...