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8
votes
1answer
362 views

Do the complex zeros of the sum/difference of these series all reside on the line $\Re(s)=\frac12$?

The following series seems convergent for all $s\in \mathbb{C}$: $$\displaystyle f(s):=\sum_{n=1}^\infty \frac{(-1)^n}{(n+s)^{n+s}}$$ The function itself does not appear to have any real or complex ...
2
votes
1answer
96 views

Asymptotic upper bound for recursive function $f(x) = f(x-1) + 2f\left(\lceil\frac{x}{2}\rceil \right)+2$

I'm looking for an asymptotic upper bound for the function f(x), which is recursively defined as follows: $$f(x) = f(x-1) + 2f\left(\lceil\frac{x}{2}\rceil \right)+2$$ with $f(1)=1$. I am pretty ...
1
vote
1answer
156 views

A term for sequences whose mean is defined?

This may be an extremely stupid and elementary question, but is there a name for sequences $\{a_i\}$ such that $\lim_{n\to\infty} \left( \frac{1}{n}\sum_{i=1}^{n} a_i\right)$ exists? This seems to be ...
0
votes
1answer
157 views

How to solve a complex recursive relation

Before I get started, let me say for complete disclosure this question came up while I was solving a problem from https://projecteuler.net/. I've been trying to find a non-recursive representation of ...
3
votes
2answers
294 views

Asymptotics of the least common multiple of the first natural numbers

What is $$ \limsup_{n \to \infty} \frac{\log(\mathrm{lcm}(1,2, \dots, n))}{n} \ \ ?$$
8
votes
0answers
530 views

Is this 2x2 determinant sequence positive and increasing?

Let $X_1,X_2,X_3$ be a three discrete (integer and non-negative valued) random variables with local probabilities $a_k:=\mathbb{P}(X_1=k)$, $b_k:=\mathbb{P}(X_2=k)$, $c_k:=\mathbb{P}(X_3=k)$ and ...
7
votes
3answers
266 views

Upper bound on length of addition chain

An addition chain for $n$ is a finite sequence of integers starting at 1 and ending at $n$, such that each element is a sum of two previous elements. A short addition chain for $n$ can be used, for ...
8
votes
1answer
318 views

The sum of a series

Let $0< \alpha <1$ and $q>1.$ Consider the (alternating) series: $$ \sum_{k=1}^\infty (-1)^k \frac{q^k (q^k-1)^\alpha}{(q^k-1)\dots (q-1)}.$$ Denote its sum by $f(q,\alpha).$ Prove (or ...
5
votes
0answers
125 views

Has unconditional convergence ever been proved other than by deducing it from absolute convergence?

Nobody's answering this question so I'll try it here. This is really a reference request: Has a certain kind of proof ever been used? A series $\displaystyle\sum_n a_n$ converges absolutely if ...
7
votes
1answer
250 views

Sums of reciprocals involving divisor sums

This question was asked at MSE but never received an answer. Let $A\subset\mathbb{N}$ be a subset of the natural numbers, and let $\sigma(n)$ denote the sum of divisors of $n$. Recall that we have ...
-1
votes
1answer
167 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
vote
1answer
102 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 ...
1
vote
0answers
124 views

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

I am 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 ...
0
votes
0answers
144 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; ...
26
votes
5answers
998 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 ...
3
votes
1answer
151 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 ...
6
votes
1answer
284 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, ...
18
votes
1answer
480 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 ...
15
votes
5answers
598 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 ...
3
votes
1answer
99 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
115 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 ...
6
votes
1answer
119 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 $a(n,k)$ ...
2
votes
1answer
492 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 $1$?
4
votes
0answers
358 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 ...
6
votes
2answers
284 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 ...
9
votes
1answer
437 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 of 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. Šalát's paper: ...
2
votes
1answer
101 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
340 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 ...
4
votes
4answers
494 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 ...
0
votes
0answers
95 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
108 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
92 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 ...
4
votes
1answer
431 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 ...
5
votes
1answer
167 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 ...
2
votes
1answer
78 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
171 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 $k=3$ ,$n=2$ , ...
1
vote
0answers
154 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 ...
15
votes
7answers
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 ...
0
votes
0answers
50 views

Multivariate recurrence relation

Consider the recurrence relation \begin{equation} \mathbb{I}(m<M)[- k_{\rm on} c_{m,n} + (m+1) k_{\rm off} c_{m+1,n}] + \mathbb{I}(m>0)[-k_{\rm off} m \, c_{m,n} + k_{\rm on} c_{m-1,n} ] + ...
1
vote
0answers
123 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 ...
4
votes
1answer
233 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. ...
1
vote
1answer
116 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
103 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
46 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 ...
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
101 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 ...
5
votes
1answer
161 views

On one class of Somos-like sequences

This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer? Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence ...
6
votes
0answers
197 views

Asymptotic behavior of a sequence of functions

For $n\in\mathbb{N}$ and $q\in(0,1)$, define $$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$ ...
1
vote
1answer
117 views

Local Uniform Convergence

Suppose $f(x)$ is a positive continuous function on $[0,\infty)$ and that $f(x+u)-f(x)\to 0$ as $x\to\infty$ for every given $u\in[0,\infty)$. Prove that, given any $a>0$, $f(x+u)-f(x)\to 0$, as ...
0
votes
0answers
61 views

Fractal in discrete time series/discrete time sequence

Consider a time series of real number $x_1, x_2,\dots,...x_n$. How one can define fractal dimension of this series? I would like to know famous formula $F+H=2$ where H is Hurst exponent and F is ...