The tag has no usage guidance.

learn more… | top users | synonyms

11
votes
1answer
843 views

Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum? $$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$ Additional information: Since $$ ...
0
votes
1answer
41 views

For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
-1
votes
0answers
19 views

Statistics, the deviation and expection of a number sequence [on hold]

There is a sequence of number $a_{0},a_{1},...,a_{n}$, $(0 < a_{i} < 1)$ Define $b_{t} = \frac{ \sum_{i=0}^{t}{w^{t-i}a_{i}} }{ \sum_{i=0}^{t}{w^{t-i}} }$ where $w \in (0, 1)$. Can we proof ...
0
votes
0answers
95 views

The maximum lengthed sequence of prime numbers with certain conditions (denizens)

Definition - Denizen A sequence $\lbrace a_k \rbrace$ is a denizen if all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P} $; and it satisfies the following condition; ...
1
vote
1answer
89 views

Number of binary sequences in which the number of $(1, 1)$ and $(0, 0)$ is prespecified

Consider a binary sequence $\mathbf{a}_n$ consisting of 1s and 0s. Let us denote by $f(\mathbf{a}_n)$ the number of $(1, 1)$ and $(0, 0)$ in $\mathbf{a}_n$; I am not sure whether there is a formal ...
40
votes
1answer
790 views

Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series $$f_{\sigma}(z) = ...
0
votes
0answers
44 views

Generating a series representation for the inverse of the operator $f(f)$

I was considering the following problem: Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...
3
votes
1answer
476 views

How to find the coefficients of a poor-converging series?

I have the series $\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$ and the boundary conditions $\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi ...
21
votes
5answers
723 views

How many rearrangements must fail to alter the value of a sum before you conclude that none do?

This will not be altogether unrelated to this earlier question. For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of ...
6
votes
1answer
726 views

Asymptotic behaviour of a sequence

Hello, I am interested in some kind of sequence that are "not finitely recurrent". Let $a_i$ be a sequence taking values in $\{0,1\}$. Consider the sequence $(u_i)$ such that $u_0=1$, and for any ...
18
votes
1answer
403 views

Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...
1
vote
1answer
286 views

Expression for infinite sum of two Bessel functions and a power

I'm searching for an expression of the following sum (all indices are integers and all variables are positive real numbers): $$\sum_{\lambda=-\infty}^{+\infty} A^{|l-\lambda|} B^{|\lambda|} ...
2
votes
1answer
56 views

Closed Form Expression for Nested Series Summation?

Just wandering if there are any criteria that can decide whether a finite series summation has closed form or not. for example, In the following nested summation, $n$ is some even integer that will be ...
-1
votes
0answers
19 views

A problem upon function series [migrated]

Function series $\sum_{n=1 }^{ \infty} u_{n}(x)$ converges to $S(x)$ in bounded interval $[a,b]$, if every $u_n(x)$ is non-negative and continuous in $[a,b]$. prove that $S(x)$ attains its infimum in ...
8
votes
3answers
244 views

Asymptotics of a recurrence relation

The sequence $(a_n)_{n \ge 0}$ satisfies, $a_0 = a_1 = 1$ and the recursion relation: $$a_n = \sum\limits_{k=0}^{[n/2]} \frac{a_k}{(n-2k)!}$$ where, $[x]$ is the nearest integer to $x$ not exceeding ...
6
votes
1answer
230 views

Conway's subprime Fibonacci sequences

I want to be certain I have the latest information on Conway's subprime Fibonacci sequences, arXiv-posted a year ago; I am referencing the status in a review. To wit, starting with $(0,1)$:1 $$ 0, 1, ...
5
votes
1answer
96 views

Asymptotics of a Bivariate Generating Function

I have the following generating function, $$G(x,y)=\sum_{n,k \geq 0}a(n,k)x^ny^k = \frac{(y^2-y)x+1}{(y-y^3)x^2-(y+1)x+1}$$ and I am interested in obtaining an asymptotic for the sequence $a(n,k)$ ...
6
votes
2answers
301 views

Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$

I'm interested in an asymptotic expansion of the following Riemann zeta-type function $$ \begin{align} \displaystyle \zeta(s \mid a,b) := \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}, \quad \Re a ...
6
votes
0answers
243 views

Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$

We encountered polynomials defined by the recursive relations for the coefficients $b_k>0$ as defined below: $$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$ ...
4
votes
3answers
301 views

What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...
4
votes
0answers
333 views

Elementary treatment of elementary functions in constructive math

I would appreciate a reference to constructive math literature with elementary proofs that elementary functions are locally non-constant (i. e. densely apart from any real in any interval with ...
3
votes
1answer
84 views

Any formula for the partial sum of a remainder series?

Let $N \ge 1$ be an integer, and there is a series $ \{ N \mod 1, N \mod 2, ... , N \mod i, ... \}$. Obviously when $i \gt N+1$, the series will become $\{N, N, N, ..., \}$. So only take $i \le N$ ...
-2
votes
1answer
93 views

Recursion, Common Term, Combinatorics [closed]

May we find the common term for recursive sequence? if yes that how to find the common term of recursive sequence such: 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 6 ... in a ...
2
votes
1answer
420 views

What is the rate of convergence? [closed]

How quickly does the series defined by $$x_0 = 0, \ x_{n+1} = \frac{x_n^2+1}{2}$$ converge to $1$?
6
votes
2answers
261 views

Determining when combinatorial sums are zero

To keep things simple with a specific example, we ask: Prove that $\displaystyle\ a_n:=\frac{1}{n!}\sum_{k=0}^n \binom{n}{k} \frac{1}{k!} (-1)^{n-k}$ is zero if and only if $n=1$. (Or find a ...
7
votes
1answer
281 views

Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?

It is a result from additive-theory folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. ...
2
votes
1answer
85 views

Asymptotic expansion of a sequence given by an integral with reciprocal Gamma function

I would like to know the asymptotic expansion of the sequence of positive numbers given by $$I_{n}:=-\int_{0}^{1}\frac{n^{x-1}}{\Gamma(x-1)}dx,$$ for $n\rightarrow\infty$. One can easily derive an ...
-1
votes
2answers
322 views

What conditions imply that a function over $\mathbb{Z}$ is a polynomial? [closed]

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can ...
0
votes
0answers
83 views

Differentiating and integrating an infinite series arising from a PDE

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $(\varphi_k, \lambda_k)$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Given $u \in H^{\frac ...
1
vote
0answers
97 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
2
votes
1answer
75 views

Is this series involving hyperbolic functions uniformly convergent?

Suppose that $\mu_k$ is an increasing sequence of numbers such that $0 < \mu_1 \leq \mu_2 \leq ..$ with $\mu_k \to \infty$ as $k \to \infty$ $\sum_{k=1}^\infty |u_k|^2 < \infty$ and ...
3
votes
1answer
370 views

How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$ if it's not open problem?

I proposed this question on MSE but some comments affirmed that is unsolved problem and no answer. I would like to see what MO say about it. How do I evaluate this sum ...
4
votes
1answer
127 views

Motzkin polynomials and enumeration of chord diagrams

On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of ...
7
votes
2answers
519 views

Newton series and Fourier transform - is there an analogy?

Fourier expansion for a function: $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$ Newton series expansion of a function: ...
2
votes
1answer
72 views

On finding the region $R$ for which the multi-variable sequence converges [closed]

Find the region $(x,y) \in R$ for which the following sequence converges $$\lim_{n \to \infty} \; \;\left| e^n\frac{(\sqrt{y}-\sqrt{x})^{2n}}{x^n} \right| = 0$$ I am currently doing number theory ...
2
votes
1answer
149 views

Sum Of n numbers taken $k$ at a time, where numbers are of form $r\choose k$

I have an array of numbers lets call it $p$ , where $p[r]={k+r-1\choose k-1}$ I want to find the sum of all the elements of $p$ taken $n$ at a time . $0\le r\le k$ For instance, for $k=3$ ,$n=2$ , ...
8
votes
2answers
3k views

Sums of arctangents

$$ \begin{align} \arctan(x) & = \arctan(1) + \arctan\left(\frac{x-1}{2}\right) \\ & {} - \arctan\left(\frac{(x-1)^2}{4} \right) + \arctan\left(\frac{(x-1)^3}{8}\right) - \cdots \end{align} $$ ...
9
votes
6answers
2k views

Is this a rational function?

Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$ In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...
1
vote
0answers
152 views

Can one show that the terms of prime index of a certain recursive sequence are not divisible by their index infinitely often?

Let $(a_n)$ be the sequence defined by $$a_{n+1}=2na_n-n^2a_{n-1}$$ and $a_0=0$ and $a_1=1$. I would like to prove that there exist infinitely many primes $p$ such that $p$ does not divide $a_p$. Any ...
1
vote
1answer
272 views

“Unbalanced” maximum length sequences?

Maximum length sequences (MLS) are a type of pseudorandom binary sequences with specific properties (see Wikipedia: Maximum length sequence, or m linear feedback shift register. Properties that hold ...
1
vote
0answers
105 views

Four kinds of generalized hypergeometric formulas for $\pi$

Given, $$\begin{array}{|c|c|c|c|} \hline n&a_n&b_n&c_n\\ \hline 1 & 6541681608 & 163096908 & -640320^3\\ \hline 2 & 85840 & 4492 & -14112^2\\ \hline 3 & 28302 ...
7
votes
1answer
1k views

Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...
4
votes
1answer
159 views

how to evaluate the following double summation to infinity without using integration method?

The expression is as follows: $\sum_{x=0}^{\infty}\sum_{y=0}^{\infty} \exp(-\sqrt{x^2+y^2})$ I have thought about using Taylor approximation to get started but it doesn't seem to get me anywhere. ...
6
votes
2answers
333 views

expression for infinite series with powers of factorial in denominator

The series $$\sum_{k=0}^\infty \frac{\exp(c k \beta)}{(k!)^\beta} $$ has come up when I'm trying to apply the methodology in this paper (http://www.ism.ac.jp/~eguchi/pdf/Robustify_MLE.pdf) to Poisson ...
1
vote
1answer
100 views

Probability of sub-sequence of exact length to occur

Let's suppose that I have a sequence of length $L$ of uniformly distributed random numbers on interval $(a,b)$. How can I calculate probability that increasing sub-sequence of length $M,M <L, $ ...
3
votes
0answers
68 views

Maximizing the discrepancy in Jensen's inequality for a certain function

Let $\underline{b}=\{b_1,\dots,b_n\}$ be a fixed sequence of positive numbers, and let $a>0$ be a parameter. Define $$ D(a;\underline{b}):=\frac{1}{\frac{1}{na}+\frac{1}{\sum_{i=1}^n b_i}} ...
1
vote
0answers
39 views

Simplifying closed form for Meta Operator

I was consider the set of linear operators: $$O_{a,k} = \frac{f(ax^k) - f(x)}{ax^k - x} $$' Particularly I am looking for the closed forms of the eigenfunctions of this operator, that is the ...
12
votes
5answers
3k views

Upper bounds for the sum of primes <= n

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...
3
votes
0answers
57 views

Pointwise (a.e) evaluation of $\sum_{n \geq 0}(u,w_n)_{L^2}w_n$ and equalities in $L^2$

Let $w_n$ be a orthonormal basis of $L^2(\Omega)$. Given $u \in L^2$ we can write $$u=\sum_{n \geq 0}(u,w_n)_{L^2}w_n.$$ Suppose $w_n$ are the eigenfunctions of the Neumann Laplacian. We can write ...
4
votes
1answer
86 views

Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible. Let $a_k >0$ be an increasing sequence ...