The sequences-and-series tag has no usage guidance.

**-2**

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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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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**

votes

**0**answers

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

**3**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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 ...

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vote

**0**answers

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**

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**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**3**answers

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

**3**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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 ...

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**0**answers

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 ...

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votes

**1**answer

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 ...

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**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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 ...

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**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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|\...