Questions tagged [sequences-and-series]

for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.

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2
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0answers
264 views

For which values of $ x$ we have $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges?

The copy of this question is posted here I have tried to to determine values of real $x$ for which $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges but I can't , Presumably the set of values $\{...
2
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0answers
131 views

Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?

Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
2
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0answers
61 views

(Dis)continuity of periodic functions with non-summable Fourier series

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$. We assume moreover that the square-summable Fourier coefficients of $f$, ...
4
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0answers
162 views

Square root of a sequence given by a linear recurrence relation

This question is closely related to this one but in a sense a converse of it. Let us concentrate for simplicity on a third order relation $x_{n+3}=ax_{n+2}+bx_{n+1}+cx_n$ with given intial terms $x_1,...
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0answers
31 views

Classification of the behaviours of the logistic map

On this this wikipedia page, it is claimed that the iterative sequence $x_{n+1}=rx_n(1-x_n)$ (the logistic map) starting at a point $[0,1]$ and where $r$ ranges in $[0,4]$ behaves differently ...
4
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1answer
117 views

A binomial coefficient identity involving two parameters

In a recent calculation I obtain a result involving the following expression depending on two integers $n,m\geq 0$: $$S(n,m):=\frac{(n+m+1)!}{n!m!}\sum_{l=0}^{n+m}\frac{1}{n+m-l+1}\sum_{\substack{j+k=...
13
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2answers
808 views

Is there a known condition for partial sums of a decreasing positive sequence to take all values up to the total sum?

Let $a_0>a_1>\cdots>0$ have the property that, for each positive $a<\sum_{n\in\Bbb N}a_n$ (admitting $\infty$ for the sum), there is $A\subset\Bbb N$ such that $a=\sum_{n\in A}a_n$ . Are ...
2
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0answers
111 views

Are there any extensive treatments on rational zeta series?

I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very ...
4
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4answers
207 views

On Köthe sequence spaces

I asked this a week ago at math.stackexchange, but without success. As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his "...
2
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1answer
97 views

What is the collection of series that amount to $\gamma$ deduced by Ramanujan?

On p. 20 of an article by Borwein et al., it is stated that Ramanujan could generalize the following formula due to Glaisher $$\gamma = 2 - 2\log2 -2\sum_{n=3, \text{ odd}} \frac{\zeta(n)-1}{n(n+1)} $$...
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55 views

Convergence of a nonlinear iterative sequence

I have the following iterative sequence: \begin{eqnarray*} a_{t+1} &=& (1+\alpha-\beta)^2a_{t} - 2\alpha(1+\alpha-\beta)b_{t} +\alpha^2a_{t-1}+\frac{L}{a_{t}}, \\ b_{t+1} &=& (1+\alpha-...
1
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1answer
57 views

Kolmogorov tightness criterion for stochastic processes

I am searching for the criterion stated above and also here: The question about Kolmogorov tightness criterion. It should state the following: If a sequence of stochastic processes $(X^n)$ fulfills: ...
0
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1answer
117 views

Spherical harmonics expansion

In the context of $L^2$ space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{...
0
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1answer
42 views

Is asymptotic growth bound on a sequence equivalent to an asymptotic growth bound on its partial sum?

The following question was asked at https://mathoverflow.net/questions/360053/asymptotic-growth-bound-on-a-sequence-equivalent-to-an-asymptotic-growth-bound-o, but then deleted by the user: I ...
0
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1answer
77 views

Analyze a complicated double summation

Let $f(x)$ be a real-valued twice continuously differentiable function, and considered the below double sum $$F(t,f(x)):=\dfrac{1}{t}\Big(\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}f(x+(k-m)/\sqrt{n})\...
8
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2answers
457 views

Less fundamental applications of Zeta regularization:

As we all know, zeta regularization is used in Quantum field theory and calculations regarding the Casimir effect. Are there less fundamental applications of zeta function regularization? By "less ...
1
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1answer
109 views

Does this series, related to the Hasse/Ser series for $\zeta(s)$, converge for all $s \in \mathbb{C}$?

I have asked this question at math stack exchange, however it did not get any traction. Still curious about the answer though. Numerical evidence suggests that: $$\lim_{N \to +\infty} \sum_{n=1}^N\...
11
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3answers
606 views

Series and sequences in physical systems & closed form expressions

I gave a colloquium a while ago about physics inspiring recent developments in mathematics and as is almost borderline cliche in such talks, I mentioned the Fibonacci sequence with closed form ...
2
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2answers
487 views

Monotonic and bounded sequences throughout mathematics [closed]

When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure ...
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1answer
65 views

Maximizing the length of a sequence under constraints

Fix $\{w_n\}_n$ a sequence of positive real numbers, fix positive integers $N,K$, and fix $\eta>1$. I'm looking for a sequence of integers $\{k_n\}_n$ optimizing the following problem: $$ \begin{...
11
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2answers
468 views

Do infinitely nested radicals have any applications?

There is a simple necessary and sufficient condition for a continued radical of the form $\sqrt{a_1 + \sqrt{a_2 + \dotsc}}$ to converge (where all terms $a_1, a_2$ etc. are nonnegative). Namely, that ...
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0answers
34 views

Prove the supremum of Mittag-Leffer function

I find two interesting limits : \begin{align*} \frac{1}{2}& =\lim_{s\to 1^-}\sum_{n=0}^{\infty}\left(-1\right)^n\frac{\Gamma(1+ns)}{\Gamma(1+n)}\\ & =\lim_{s\to 1^+}\sum_{n=0}^{\infty}\left(-1\...
25
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4answers
1k views

For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded? I feel that it is not easy to treat every irrational $x$. I have asked in S.E. ...
2
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1answer
34 views

Subdividing a sequence such that sum is somewhat equally distributed

I have a sequence ( n, n-1, n-2,...,1). I need to find numbers in this sequence in this order that somewhat approximately divide it into M parts- within each M subgroup the sum is somewhat the same. ...
0
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1answer
98 views

What are some interesting relationships between pi and phi? [closed]

Phi is the golden mean solution to the 1/x=1+x and pi the transcendental number relating the radius of the circle to its area. A side note: while there are really interesting series converging to pi, ...
9
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2answers
963 views

An inequality involving square roots and sums

I've been trying to prove (maybe even disprove) the following inequality: $$ \sum_{n=1}^{N} \frac{a_n}{\sqrt{\sum_{i=1}^{n}a_i}} \leq C \sqrt{\sum_{n=1}^{N}a_n} $$ Where $ a_1,...,a_N\geq 0 $ are ...
3
votes
1answer
251 views

Efficient computation of $\sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor$

I need to compute efficiently the sum $$ \sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor. $$ We can do this in $O({\sqrt{n}})$ but I need a faster algorithm: for example, it ...
0
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0answers
50 views

Sequence estimate for slow variation to study the wandering rate of a set

Consider the two matrices $$ M_1 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix} \quad \text{and} \quad M_{2} = \begin{pmatrix} 0 & 0 &...
0
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0answers
72 views

Ranking graph's nodes by score propagation

Problem I have the following directed tripartite graph $G(E\cup V\cup P, A)$, where there is a many-to-one symmetric relationship between the subsets V and E - $e\in E,v\in V,[e, v]\in A \iff [v, e]\...
1
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1answer
190 views

Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$

(Question is short and straight-forward. ) What is/are "nice and non-trivial" series representation/s of $\log(|\zeta(\frac{1}{2}+it)|)$ ?? By "nice and non-trivial" I mean contains no ...
0
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2answers
146 views

Sequence $(a_n)$ for which $ \prod_{n=1}^{\infty} \left ( 1- \dfrac{m}{a_n +m} \right ) =r.$ [closed]

I am trying to solve the following question . Let $r\in (0,1)$ and $m\in \mathrm{N}$. Then there exists a sequence $(a_n)_{n\in \mathrm{N}}$ in $\mathrm{N}$ such that $$ \prod_{n=1}^{\infty} \left ( 1-...
0
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0answers
44 views

Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?

Suppose I have an equation in some reflexive separable Banach space $X$: $$Au=f$$ for given data $f \in X^*$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin ...
2
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1answer
112 views

Proof of a discrete isoperimetric inequality

The following inequality appears in the proof of certain isoperimetric-type inequalities for analytic functions in two dimensions: $$\sum_{m=0}^{\infty}\frac{|c_m|^2}{m+1} \leq \pi \left(\sum_{m=0}^{...
2
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0answers
75 views

Sequence of least prime-multiples with smallest Hamming weight

It is known that Almost all primes have a multiple of small Hamming weight, which makes me wonder what is known about the least multiples of primes that have least Hamming weight. Questions: what ...
0
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0answers
52 views

Mean and logarithmic values for arithmetic function

Define the mean value for function $f$ as $\lim \limits_{x \to \infty} \frac{1}{x} \sum \limits_{n \leq x} f(n)$ if the limit exists denoted as $M_f$ Define the logarithmic value for function $f$ as $...
3
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1answer
96 views

Comparison of product topology and colimit topology in sequence spaces

In Munkres Theorem 20.4 it is shown that the (relative) uniform topology induced by: $$ d(x,y)\triangleq \sup_{n \in \mathbb{N}} d(x_n,y_n) $$ is strictly finer than the product topology on $\prod_{n \...
-1
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1answer
83 views

Sum of $\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}$ [closed]

I want to calculute or estimate of order $O(n^{2-\varepsilon})$, where $\varepsilon>0$, of the following sum for $0<\alpha<1$ $$\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}.$$
4
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0answers
56 views

Logconcavity of height of Dyck paths

A finite sequence $a_i$ is called logconvace in case $a_i^2 \geq a_{i-1} a_{i+1}$. Question : For a fixed $n$, is the sequence $a_{n,k}$ giving the number of Dyck paths of semilength $n$ having ...
2
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1answer
67 views

O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval

O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval. Using $\{x\}$ to denote the fraction part of $x$ we can define for any $I\subset [0,1]$, $$E(n,\theta, I) ={ ...
3
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0answers
47 views

Properties of sequences associated to Nakayama algebras

Assume Nakayama algebras are connected and given by quiver and relations. Note that such Nakayama algebras have global dimension at most $2n-2$ in case it is finite and the algebra has $n$ simples. ...
1
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0answers
110 views

About the distribution of Fibonacci numbers that are primes

Let's consider the Fibonacci sequence, that is the sequence of naturals defined by: $F_1=F_2=1$ $F_{n+1}=F_{n}+F_{n-1}$ It is an open problem whether the sequence contains an infinite number of ...
5
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1answer
112 views

How to prove the relationship between Stern's diatomic series and Lucas sequence $U_n(x,1)$ over the field GF(2)?

I found the bit count of Lucas sequence $U_n(x,1)$ over the field GF(2) is Stern's diatomic series, I want to know the reason? https://oeis.org/A002487 : Stern's diatomic series https://oeis.org/...
3
votes
1answer
152 views

About the sum $S(p_n)=\sum_{1\le k\lt n}\,p_n\mod\;p_k$

For $\,p_n\gt2\,$ let's define the sum $\,S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$, where $\,p_k\,$ represents the $\,k$-th prime. The first terms of the sequence $\,S(p_n)\,$ (OEIS A033955 - sum of ...
1
vote
1answer
78 views

Is there a rectangular tiling based on the Padovan sequence? [closed]

I'm thinking of developing a rectangular tiling based on the Padovan sequence (think of the Fibonacci mosaic). It seems like something that should exist, but I can't find anything in the literature. ...
3
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0answers
52 views

Boundedness of $\chi_{\{f_n=0\}}$ in the BV norm

Let $f_n \in H^2(\Omega) \cap C^0(\bar \Omega)$ be a sequence of functions that are uniformly bounded in $H^2(\Omega) \cap C^0(\bar \Omega)$ on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ ...
3
votes
1answer
150 views

Find the closed-form expression for $c_n$ of this recursive sequence [closed]

$$c_{n + 1} = 2\cdot|c_n| - \sqrt{c_n^2 + 16}$$ with $c_0 = 3$. My question on math stackexchange was closed because lack of details. Let me clarify: I'm an amateur novelist, I don't know anything ...
-3
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1answer
64 views

On the series of the product of the terms of two sequences whose respective series are one convergent and the other not [closed]

Let us consider two sequences of real numbers $a_n$ and $b_n$, about which we only know that: $$\sum_{1}^{\infty}a_n = 0$$ and that all $b_n > 0$, with $b_{n+1} > b_n$. Can it be proved that ...
2
votes
2answers
116 views

Positions in the Wythoff array

Suppose that $x$ and $y$ are positive integers. How can the position of $x+y$ in the Wythoff array (A035513) be predicted from the positions of $x$ and $y$? Background. The Wythoff array begins with ...
-1
votes
1answer
266 views

Is it possible to determine whether the sequence $\,a_0=p,\;a_{n+1}=(a_n-2)\cdot a_n+2\,$ will reach another prime number?

Given a prime $\,p\,$ let's consider the following sequence: $a_0=p$ $a_{n+1}=(a_n-2)\cdot a_n+2$ Is it possible to determine whether the sequence $\,a_n\,$ will reach, sooner or later, another ...
0
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0answers
62 views

About counting the number of graphs by the maximum degree $D$

This is another way of attacking the problem that I posted on the link: What is the number of connected graphs with $n$ vertices of max. degree up to $D$? Leaving $F(x) = x + x^2 + 2x^3 + 6x^4 + 21x^5 ...

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