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
136 views

Did anyone ever propose the distinction between “divergent to infinity” as opposed to “divergent but with finite average”?

There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences. Still, in my view there is fundamental difference between divergent ...
3
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0answers
180 views

Spherical harmonic expansion of a power function

Let $f$ be an even continuous function on the sphere $S^{n-1}$. Find a relation of the spherical harmonic expansion between coefficients of $f^n$ and those of $f$.
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40 views

An inequality for a recursively defined sequence of numbers

Consider an arbitrary sequence $(x_n)_{n \in \mathbb{N}} \subseteq \mathbb{R}$ and $r \in \mathbb{R}$ with $r > 2$. Set $y_0 = 1$ and $z_0 = 0$ and for $n \in \mathbb{N}$ recursively define $$y_n = ...
2
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1answer
137 views

Truncated Euler products, Dirichlet eta function, and convergence issues

Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as $$W(\sigma,...
3
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1answer
201 views

Formulas for $\sum_{x=1}^{n}\Big\{\frac{x^q}{n}\Big\}$

Consider sum: \begin{equation} S_q(n) = \sum_{x=1}^{n}\Big\{ \frac{x^q}{n} \Big\} \end{equation} where $\{x\}$ is fractional part of $x$. It's easy to see that $S_{1}(n) = \frac{1}{2}(n-1)$, but ...
3
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0answers
126 views

The sequence $a(n)=(2^n \bmod p)^{p-1} \bmod p^2$

Related to this question. Let $p$ be prime and $n$ positive integer. Define $a(n)=(2^n \bmod p)^{p-1} \bmod p^2$ Let $D(n)$ be the base $2$ discrete logarithm of $a(n)$, i.e. given $p,a(n)$ we have $2^...
0
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1answer
121 views

If $P_n \rightrightarrows P$ in $\mathbb{R}$ and $P_n$ are polynomials proof that $P$ is polynomial [closed]

I know that if $P_n$ are continuous functions and $P_n \rightrightarrows P$, $P$ is also continuous function. But I can't see in which direction I should dig to prove that $P$ is polynomial. I will ...
4
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1answer
185 views

A problem on rate of decay of fill distance?

Let $X$ be a random variable with values in a closed compact $\Omega \subset \mathbb{R}^m$. Assume $\Omega$ is has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and ...
4
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0answers
212 views

Is the arithmetic-geometric mean of 1 and 2 rational?

It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
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38 views

Can I write this series in a recursive way?

I would like to know given the following definition of the function X(n) if it is possible to express two consecutive values of X(n) (example: X(1) and X(2) ) to obtain a recursive expression. What I'...
4
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105 views

Positivity conjecture for Somos sequences

Let $\{s_n\}$ be the Somos-$4$ sequence, which is defined by $$s_{n+4}s_n=\alpha s_{n+3}s_{n+1}+\beta s_n^2.$$ It is known that $s_n$ is a Laurent polynomial: $s_n\in\mathbb{Z}[s_1^{\pm1}, \ldots, s_4^...
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48 views

A closed form or a good approximation of an infinite series related to the negative binomial distribution

Does anyone know a closed form for this expression: $$\sum_{r =1}^{\infty}{{\alpha + 2r - 1}\choose{ r - 1}}(1 - p)^{\alpha + r}p^{r},$$ where $\alpha \geq 1$ and $0<p<1$. A good approximation ...
-5
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1answer
40 views

What is the pattern in this sequence of fractions? [closed]

1/2, 1/2, 5/8, 5/7, 17/22, 13/16,... I notice the top numbers are all primes but could not find how that helps. At first I thought maybe it is similar to a Fibonnaci type sequence because of the first ...
10
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2answers
675 views

What can one say about $\sum\limits_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$?

Denoting by $p_i$ the $i$-th prime, is it known that $\displaystyle \sum_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$ converges? Can one compute a few digits based on euristic considerations or plausible ...
14
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1answer
1k views

Calculation of a series

It seems that we have: $$\sum_{n\geq 1} \frac{2^n}{3^{2^{n-1}}+1}=1.$$ Please, how can one prove it?
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114 views

What is the regularized numerocity of prime numbers?

Basically, I want to find this sum: $$\lim_{s\to0}\sum_{n=1}^\infty \frac{p(n)}{n^s}$$ where $p(n)$ is the membership function of the set of primes. That is, $p(n)=1$ if $n$ is prime and $0$ otherwise....
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0answers
49 views

Tail asymptotics of Durfee square identity

This post is related to the problem Asymptotics of a combinatorial series According to the Durfee square identity: $$\sum_{k \ge 0} \frac{q^{k^2}}{(q;q)_k^2} (q;q)_{\infty} = 1,$$ where $(q;q)_k$ is ...
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0answers
17 views

Convergence mode with inputs and functions varying in tandem

Given a sequence $(f_n)$ of functions between metric spaces, let's say that $f_n$ "converges flexibly" to $f$ if, whenever $x_n \to x$ is a convergent sequence of inputs, it follows that $...
1
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1answer
189 views

Sequences over finite fields

Let's we have finite field $F_q$ for some prime $q=2^M-1$. I am looking for special sequence {$a_{i}$, $i \in {1,..,q-1}$}, ($\{a_{1},...,a_{q-1}\}=F_q/\{0\}$) with the following properties: $r_{1}=...
3
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0answers
98 views

Asymptotics of a combinatorial series

I am interested in the exact asymptotics of the following combinatorial series (which arises from the study of a Markov chain): $$F(q):=\sum_{k \ge 1} \frac{q^{k^2}}{(q;q)_k^2}\quad \mbox{as } q \to 1^...
2
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2answers
238 views

sum of odious numbers to the power of k

In number theory, an odious number is a positive integer that has an odd number of $1$s in its binary expansion. The first odious numbers are: $1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, ...
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194 views

Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$

Definitions For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows: $$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
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0answers
86 views

Is this pair of coupled sequences known, and what are their properties?

I was examining the following pair of 'coupled' sequences (I don't know the correct terminology): $a_{n+1}=a_n+b_n+\frac{a_n}{b_n}$ $b_{n+1}=b_n\left(1+\frac{b_n}{a_n}\right)$ Both sequences grow ...
2
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0answers
79 views

Banach limit with added properties

Let $c=\{a:\mathbb{N}\rightarrow \mathbb{C}: \exists \alpha\in \mathbb{C} \textrm{ so that } \lim\limits_{n\rightarrow \infty} a(n)=\alpha=:L_c(a)\}\subset \ell^\infty$, where $\ell^\infty$ is the ...
7
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0answers
170 views

Fraction of elements in $\mathbb{Z}_n$ satisfying a certain equation

From a question arising in Game Theory, I want to calculate the sequence $$ a_n = \max_{f_A, f_B : \mathbb{Z}_n \to \mathbb{Z}_n} \frac{\# \left\{ (x,y) | f_A(x) - f_B(y) = xy \mod n \right\}}{n^2} $$...
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1answer
43 views

Comparing growth of sequences in weighted spaces

I would like to ask a follow-up question on a previous question of mine here whose proof does not seem to carry over to this case in an obvious way: We define the function $$F_{\varepsilon}(x) = \sum_{...
7
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1answer
162 views

Comparing divergent and convergent sums

Let $(x_n)$ be a monotonically decreasing sequence of positive real numbers that is also summable. Let $(y_n)$ be a sequence of positive real numbers such that $\sum_n x_n y_n$ converges. Let $(z_n)$ ...
5
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1answer
420 views

Is the harmonic series worse than any summable series?

It is well-known that the harmonic series is not summable. In some sense this means that it takes a lot of rather large values. We define the operator $F_{\varepsilon}: \ell^{\infty}(\mathbb N) \...
4
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0answers
189 views

Is there a conjecture about the bounds (constant or a function) of $\sum_{n \le x} \mu(n)/\sqrt{n}$

Here $\mu(n)$ is Möbius function and $M(x)$ is Mertens function. The computations show that the partial sums $\sum_{n \le x} \frac{\mu(n)}{\sqrt{n}}$ stays between $-0.2$ and $-1.2$ when $10^1<x<...
10
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0answers
279 views

When exactly is the principal AGM equal to the optimal AGM?

Definitions Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
1
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1answer
255 views

A dig at Ramanujan's: $\sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,~ p>0$

Ramanujan's claim on page 98 in the book Ramanujan's note book part 1 by Bruce C. Berndt, is that $$ \sum_{k=1}^{\infty} (−1)^{k−1}\frac{x^{pk}}{k(k!)^p}∼p\ln (x)+p\gamma,\quad p>0 \label{1}\tag{1} ...
3
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2answers
251 views

Finding the summation formula for the recurrence relation $T_n=T_{n-1}\;+(n-1)\cdot T_{n-2}\;$

The exponential generating function of this recurrence relation, $T_n=T_{n-1}\;+(n-1)\cdot T_{n-2}\;$, is $$f(x)=e^{x + \frac{x^2}{2}}$$ Multiplying the exponential generating functions for each term, ...
2
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1answer
73 views

Logistic sequence convergence

1) How can we prove that the logistic sequence $$x_{n+1}=rx_n(1-x_n),\ x_1=a\in (0,1)$$ converges to $\frac{r-1}{r}$, for $r\in [1,3]$? 2) Also I wonder how can we prove that the sequence $(x_n)_{n\in\...
4
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0answers
54 views

Minimal growth condition for a rearrangement

Let $\sigma: \mathbb{N}\to\mathbb{N}$ be bijective such that there is a sequence $(n_k)_{k\ge 0}$ in $\mathbb{N}$ satisfying $|\sigma(n_k)−n_k|\to\infty$ for $k\to\infty$. Question: Is there a (...
1
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0answers
77 views

Log-convexity of Lassalle's sequence

Lassalle's sequence is defined by the recurrence $A_1:=1$ and for $n\geq2$, $$A_n=(-1)^{n-1}C_n + (-1)^{n- 1}\sum_{j=1}^{n-1}(-1)^j\binom{2n - 1}{2j - 1}A_jC_{n - j}$$ where $C_k=\frac1{k+1}\binom{2k}...
1
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0answers
94 views

Organization Program of Random Integers

Suppose you have an infinite list of distinct positive integers, $x_1, x_2, \ldots$ and you don't know what each $x_i$ is, for any $i$. However, you are allowed to "compare" any two of the ...
9
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3answers
690 views

Method to evaluate an infinite sum of ratio of Gamma functions (how does Mathematica do it?)

This question arose from Amdeberhan's question, the evaluation of a double integral, which can be reduced to the evaluation of this series: $$\sum _{n=0}^{\infty } \frac{\Gamma \left(n+\frac{1}{2}\...
1
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0answers
46 views

Convergence of numerical method [closed]

I would like to prove that the following sequence : $S^{n+1} =1 - I +\frac{β}{α}\ln(S^{n})$, where $\alpha,\,\beta,\,I$ are constants and $S^{0} = 1$ converges as long as $\alpha\cdot S<\beta.$
10
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3answers
415 views

Asymptotic analysis of $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2$

Problem: Let $x_1 = 1$ and $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2, \ n\ge 1$. Find the third term in the asymptotic expansion of $x_n$. I have posted it in MSE six months ago without ...
0
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0answers
49 views

Applying Tannery's theorem to generalised hypergeometric functions

I am thinking about applying Tannery's theorem to some generalised hypergeometric functions, which seems to be a standard method to derive various formulæ. For example, \begin{eqnarray} \lim_{n\to+\...
2
votes
1answer
168 views

Evaluations of three series involving binomial coefficients

Question. How to prove the following three identities? \begin{align}\sum_{k=1}^\infty\frac1{k(-2)^k\binom{2k}k}\left(\frac1{k+1}+\ldots+\frac1{2k}\right)=\frac{\log^22}3-\frac{\pi^2}{36},\tag{1} \end{...
2
votes
1answer
281 views

Find better than $ 4^n\prod_{k=1}^{n-1}\cos^2(k)\sim e^{o(n)}$

Let $$u_n=\prod_{k=1}^{n-1}\cos^2(k)$$ then $$\frac1n \ln(u_n) = \frac1n\sum_{k=0}^{n-1} \ln(\cos^2(k)) \underset{n\to\infty}\longrightarrow \frac1{2\pi} \int_0^{2\pi} \ln(\cos^2(x))\,{\rm d}x = -\ln(...
1
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0answers
119 views

Prove that: $\sum _{c=1}^n \sum _{b=1}^n \sum _{a=1}^n \left(\left([b|c][b|a]\frac{\mu(b)b}{a}\right)-\frac{1}{a b\sqrt{c}}\right)<H_n+n$

In the OEIS there is the quote from Lowell Schoenfeld that the Riemann hypothesis is equivalent to: $$|\psi(n) - n| < \sqrt{n} \log^2(n)$$ From the Euler Maclaurin formula one gets: $$\sum _{c=1}^n ...
2
votes
2answers
258 views

About $\lim_{n\to +\infty} n\prod_{k=1}^{n-1}\cos^2(k)$

In my research work, I need to show that the sequence $(nu_n)$ tends to 0 where $ (u_n)$ is defined by $$u_{n+1}=u_{n} \cos^{2}(n),\quad u_{0}=1$$ $(u_n)$ is a positive and decreasing sequence. My ...
2
votes
1answer
68 views

Are there theorems dealing with the “amount of oscillatory divergence” of series?

Are there a set of theorems dealing with "amount of divergence" series? Let me explain by example. The Dirchlet $\eta$ series $\sum_n (-1)^{n-1} n^{-x}$ converges when $x > 0$. We may say ...
9
votes
1answer
283 views

Two-term recurrence relation

We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$ $$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n $$ ...
2
votes
0answers
137 views

How can collections of rational zeta series that are equal to $\sum_{n=2}^{\infty} (\zeta(n) - 1)^{p} $ be obtained?

It has been discovered long ago that $$\sum_{n=2}^{\infty} \big(\zeta(n) - 1\big) = 1. \label{1} \tag{1} $$ More recently, a generalization of this result with binomial coefficients has been obtained: ...
11
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1answer
538 views

Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$

Consider a sequence $(x_n)$ satisfying $x_{n+1}=x_n +\lambda \sin x_n$. You would expect the sequence $x_n$ to depend on $x_0$ and to exhibit a chaotic, Brownian-type behavior, and indeed it does ...
12
votes
1answer
363 views

Ordered Bell numbers

The ordered Bell numbers (also known as Fubini numbers, sequence A000670 in OEIS) count the number of ordered partitions of an n-element set. Experimentally I have found the following expression for ...
3
votes
1answer
135 views

Integrality of ratios of $\ell$-sequences

The so-called $\ell$-sequences are defined by $a_0=0, a_1=1$ and $a_n=\ell\,a_{n-1}-a_{n-2}$. The Generalized Lecture Hall Theorem (due to Mireille BousquetMelou and Kimmo Eriksson) depends on a ...

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