Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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154 votes
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Sorting of prime gaps

Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we rearrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$, if the gaps are arranged from smallest to largest, we have a new ...
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44 votes
0 answers
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Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$ in exact form.

$$\large \text{Introduction:}$$ We know that: $$\sum_{x=0}^\infty \frac{1}{x!}=e$$ But what if we replaced $x!$ with $!x$ also called the subfactorial function also called the $x$th derangement number?...
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41 votes
1 answer
1k views

$2^n$th decimal place of $\sqrt{2}.$

Someone on Art of Problem Solving claims to know how to calculate the $2^{2020}$th decimal place of $\sqrt{2},$ and will tell us if everyone gives up. Brute force will not work, nor will a BBP style ...
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37 votes
0 answers
1k views

Are there infinitely many primes of the form $12345678901234567890\dots$

Related to this question, What is the smallest prime number made of sequential number? are there infinitely many primes of the following form (OEIS A057137)? $1, 12, 123, 1234, 12345, 123456, ...
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31 votes
0 answers
602 views

Grasshopper jumping on circles

Can we characterize the grasshopper sequence? Let $n\in\mathbb N$ be the number of stones $s\in\{0,1,2\dots,n-1\}=S$ on a circle that the grasshopper can jump on. Let $v(s)$ be the number of times ...
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30 votes
0 answers
567 views

The sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms

PROBLEM. Show that the sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms. It suffices to show that the terms of the sequence $$\,b_n=\mathrm{e}^...
28 votes
0 answers
839 views

Have I discovered an analytic function allowing quick factorization?

So I have this apparently smooth, parametrized function: The function has a single parameter $ m $ and approaches infinity at every $x$ that divides $m$. It is then defined for real $x$ apart from ...
20 votes
0 answers
2k views

Are these generalizations known in the literature?

By using $$\int_0^\infty\frac{\ln^{2n}(x)}{1+x^2}dx=|E_{2n}|\left(\frac{\pi}{2}\right)^{2n+1}\tag{a}$$ and $$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(...
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19 votes
0 answers
391 views

A Polynomial Formed from the Roots of Another Polynomial ad infinitum

Let $P(x)$ be a monic polynomial of degree $d$ with complex coefficients. Let $r_1(P),r_2(P),\dots, r_d(P)$ denote the set of roots, ordered so that $|r_1(P)| \leq |r_2(P)|\leq\dots\leq |r_d(P)|$. ...
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19 votes
0 answers
449 views

Convergence acceleration technique for $\zeta(4)$ (or $\eta(4)$) via creative telescoping?

Question Is it already known whether the $\zeta(4):=\sum_{n=1}^{\infty}1/n^4$ accelerated convergence series $(1)$, proved for instance in [1, Corollaire 5.3], could be obtained by a similar ...
19 votes
1 answer
288 views

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
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18 votes
0 answers
703 views

When does $x^{x^{x^{...^x}}}$ diverge but $x^{x^{x^{...^c}}}$ converge?

Let us define these two sequences as follows: $a_0=1$, $b_0=c$ $a_{n+1}=x^{a_n}$, $b_{n+1}=x^{b_n}$ $b_{n+1}\ne b_n$ for any $n$. $x,c\in\mathbb C$ Is it possible for $a_n$ to diverge but $b_n$ ...
16 votes
0 answers
201 views

Have "groupy" numbers been studied before?

In number theory, a positive integer $n$ is called highly composite if it has more divisors than any smaller positive integer. This notion has been studied by several notable mathematicians; for ...
16 votes
0 answers
678 views

Asymptotic related to the infinite product of sine

The amount is somewhat complicated ($x$ is a constant): $$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$ I want to enrich my handy powerful ...
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15 votes
0 answers
307 views

Number as the sum of digits of some degree

We will say that the measure of a number is equal to the maximum degree in which it is possible to represent a number in the form of a sum of digits copied (You can not rearrange the numbers). For ...
15 votes
0 answers
699 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
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14 votes
0 answers
104 views

How do permutations of $\Bbb N$ affect series?

Let $G$ be the group of all permutations of $\mathbb{N}$ and $\sum a_n$ a conditionally convergent series of reals. What do we know about how $G$ "acts" on this series? We can partition $G$ ...
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14 votes
0 answers
393 views

Properties of the remainders from division into primes

This is a question that has bothered me for almost 6 years now on and off, and I still don't really know enough to tackle it. To phrase it somewhat formally: Let $P$ be the series of prime numbers ...
14 votes
1 answer
376 views

Is there an analytic solution for such problem?

Given function $$f_n(x) = \cos x - (\cos \cos x) + (\cos \cos \cos x) - (\cos \cos \cos \cos x) + \dots + (-1)^{n-1} \underbrace{ \cos \cos \dots \cos }_n x,$$ where $n \in \mathbb{N}$ and $\...
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14 votes
0 answers
409 views

Combinatorial Proofs of Real Analysis Identity

In this question, a proof using real analysis is given of the following identity: $$ \sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^2}$$ Is there a ...
14 votes
0 answers
238 views

How to prove that my sum coincides with a sequence on the Online Encyclopedia of Integer Sequences?

I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be $$\small \dim \tilde{H}_t(X; {\mathbb{Z}}_2) = \...
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13 votes
0 answers
403 views

Prove $ \int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n}$

Nice little generalization: $$\int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n},\quad a=0,1,2,...$$ The point of this post is to save us some calculations ...
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13 votes
0 answers
449 views

Odd values for Dirichlet beta function

Hello there I want to find a proof for the generating formula for odd values of Dirichlet beta function given by wikipedia: link I searched MSE and didnt find something similar. My try was to start ...
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13 votes
1 answer
1k views

Intuition about the Bernstein polynomials proof of the Weierstrass approximation theorem

The Weierstrass approximation theorem can be stated as follows: Let $f\in C([a,b])$. There exists a sequence $(p_n)_{n\in \mathbb{N}}$ of polynomials in $[a,b]$ such that $(p_n)$ converges uniformly ...
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13 votes
0 answers
250 views

Which Fourier series are "legal"?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$ f(x)=\sum_{n\geq 0}...
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12 votes
0 answers
305 views

Evaluate $\sum_{n=1}^{\infty} \frac{(-1)^n \, \psi^{(1)} (\sqrt{2}n)}{n}=-\frac{3\zeta(3)}{8}+\int_{0}^{1}\frac{\ln (x) \ln(1+x^{\sqrt{2}})}{x-1}\,dx$

Is it possible to evaluate, in closed-form, the following sum: $$\sum_{n=1}^{\infty} \frac{(-1)^n \, \psi^{(1)} (\sqrt{2}n)}{n}$$ or perhaps more generally $$f(x):=\sum_{n=1}^{\infty} \frac{(-1)^n \, ...
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12 votes
0 answers
205 views

Asymptotic behavior of recurrence $x_{n+1}=\mbox{Stdev}(x_1,\dots,x_n)$

Here $x_1>0$ is the initial condition and $x_{n+1}$ is defined by $$x_{n+1}=\Big[\frac{1}{n}\sum_{k=1}^n x_k^2 -\frac{1}{n^2}\Big(\sum_{k=1}^n x_k\Big)^2 \Big]^{1/2}. $$ Clearly, $x_n=\lambda_n \...
12 votes
3 answers
716 views

Solving the functional equation $f(x) = f(\frac{x}{\phi}) f(\frac{x}{\phi^2} - 1)$

I'm trying to find a function $f(x)$ such that the spacing between consecutive roots looks like the infinite Fibonacci word: $$1, \phi^{-1}, 1, 1, \phi^{-1}, 1, \phi^{-1}, 1, 1, \phi^{-1}, 1, 1, \phi^{...
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11 votes
0 answers
383 views

Is the closed form of $\int_0^1\frac{\text{Li}_{2a+1}(x)}{1+x^2}dx$ known in the literature?

Using $$\text{Li}_{2a+1}(x)-\text{Li}_{2a+1}(1/x)=\frac{i\,\pi\ln^{2a}(x)}{(2a)!}+2\sum_{k=0}^a \frac{\zeta(2a-2k)}{(2k+1)!}\ln^{2k+1}(x)\tag{1}$$ and $$\int_0^1x^{n-1}\operatorname{Li}_a(x)\mathrm{d}...
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11 votes
0 answers
167 views

Rearranging series and "placid" permutations

This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. Let $...
11 votes
0 answers
119 views

Does there exist any $p >0$ such that $\frac{1}{n^p \sin(n)} \to 0 \;,n\to+\infty$?

Does there exist any $p >0$ such that \begin{equation*} \frac{1}{n^p \sin(n)} \to 0 \;,n\to+\infty \;? \end{equation*} If there is one, what's the infimum of those $p$? Is it also a minimum? I ...
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11 votes
0 answers
285 views

Triple sum $\sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} \sum\limits_{c=1}^{\infty} \frac{\cos a \cos b \cos c}{a^2 + b^2 + c^2}$

We have poor water heating system in our countryside house (currently it takes 4 hours to warm up the water), and my father has decided to improve it; he bought a water tank and placed it up in the ...
11 votes
1 answer
512 views

A non-composite sequences

Can you provide a counterexample for a claim given below? Inspired by Puzzle 937 I have formulated the following claim: For any $n > 0$ let $B = p_1 \cdot p_2 \cdot .... \cdot p_n$ be the ...
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11 votes
0 answers
477 views

A conjecture about the connection between a Penrose tiling and the Fibonacci word fractal

Consider the Penrose tiling $P3$, inflated up to $6$ generations: We draw a line passing through the center of the tiling (red dot) and the outer vertex of the rightmost starting tile (black dot). ...
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11 votes
1 answer
321 views

Calculus of variation with discontinuous solutions

I'm thinking of the following question: Consider a function $f: U\rightarrow\mathbb{R}$ where $U=[0,L_1)\cup(L_1,L]$, and an energy functional $$F=\int_{U}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\...
11 votes
0 answers
248 views

A curious coincidence in the series representation of $\zeta(7)$

Let $\zeta(n)$ denote the Riemann Zeta function defined for positive integers $n$ as usual by: $$ \zeta(n)=\sum_{m=1}^{\infty} \frac{1}{m^n}. $$ It is currently unknown whether there exists a series ...
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11 votes
0 answers
395 views

Does the sequence $x_0=12$ , $x_{n+1}=x_n^2+1$ contain a prime?

I wonder whether the sequence defined by $$x_0=12$$ $$x_{n+1}=x_n^2+1$$ for all non-negative integers $n$ contains a prime number. The following table shows from left to right : The index $n$ , the ...
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11 votes
1 answer
250 views

Assume the limit $\lim_{n\to\infty}\frac{a_1+a_2+\cdots+a_n}{n}$ exists. Prove that $\lim_{n\to \infty}\frac{a_n}{n}=0$.

Does the following proof make sense? Proof: Let $\lim_{n\to\infty}\frac{a_1+a_2+\cdots+a_n}{n}=A$. By the operation on limits, we have $$ \lim \frac{a_1+\cdots+a_{n-1}}{n-1}\cdot\lim\frac{n-1}{n}=\lim\...
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10 votes
1 answer
148 views

On the convergence of $\sum_n 1 /(n\sin(2^nx))$

Find all values of $x$ such that $\displaystyle\sum_{n=1}^\infty\frac{1}{n\sin(2^nx)}$ converges. I've been attempting to solve this problem without much success. Firstly, it is defined for all $x\in\...
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10 votes
0 answers
349 views

On the density of a particular subset of integers

Given a positive integer $n$ in the standard form $$n=\prod_k p_k^{\alpha_k}$$ and the arithmetic function $$f(n)=\sum_k \alpha_k p_k$$ let's define the subset $F$ of positive integers $$F=\Big\{n\in ...
10 votes
0 answers
641 views

$\int_0^\infty \log\left(1 - \frac{x}{\sinh x}\right)\,dx$ and generalisations

I'm interested in the value of $$ \theta(2) := -\frac{1}{2\pi^2}\int_0^\infty \log\left(1 - \frac{x}{\sinh x}\right)\,dx. $$ I have some hopes this admits a closed-form solution. Expanding the ...
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10 votes
0 answers
154 views

If $a_n \rightarrow a$ Then $\frac{1}{n} \sum_{k=1}^{n} a_k \rightarrow a$

If $a_n \rightarrow a$ as $n \rightarrow \infty$, then can we say that $\frac{1}{n} \sum_{k=1}^{n} a_k \rightarrow a$ as $n \rightarrow \infty$? If we consider $\frac{1}{n} \sum_{k=1}^{n} a_k$ as ...
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10 votes
1 answer
244 views

Simplifying $\prod\limits_{k=0}^{n-1}\left(\sin\frac\pi{2^{k+3}}+\frac1{\sqrt{2}}\right)$

I have recently stumbled upon the sequence $\left( u_n \right)_{n \in \mathbb{N}}$ defined as follows : $$\forall n \in \mathbb{N}, ~ u_n = \prod\limits_{k=0}^{n-1} \left[ \: \sin \left( \dfrac{\pi}{2^...
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10 votes
1 answer
135 views

Proving $\sin(\tanh x) \ge \tanh(\sin x)$, for $x \in [0,\pi/2]$

Earlier, a very interesting proof of an inequality has been proposed at MSE: How prove this inequality $\tan{(\sin{x})}>\sin{(\tan{x})}$ Here the question is: How to prove that $$\sin(\tanh x) \ge ...
10 votes
0 answers
247 views

Finite analog of these two infinite series with inverse tangents?

The identity $$ \sum_{n=1}^{\infty}\chi(n)\arctan e^{-\alpha n}+\sum_{n=1}^{\infty}\chi(n)\arctan e^{-\beta n}=\frac{\pi}{8}, \qquad \alpha\beta=\frac{\pi^2}{4},\tag{1} $$ where $\chi(n)=\sin\frac{\...
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10 votes
0 answers
221 views

Find radius of convergence for a complicated series for $f'(f(x)) = f(f'(x))$

Several months ago, I answered this question asking for solutions to the functional equation $f'(f(x)) = f(f'(x))$ by expanding as a formal Taylor series around some arbitrary fixed point of $f$. This ...
10 votes
0 answers
169 views

"Taylor series" is to "Volterra series" as "Padé approximant" is to _________?

Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it. Volterra ...
10 votes
1 answer
103 views

A topology over $\Bbb N$ based on convergence of series.

Define $\tau=\{U\subseteq \Bbb N:U\in\{\Bbb N,\emptyset\}\vee\sum_{n\notin U}n^{-1}<\infty\}$. In other words, a subset of $\Bbb N$ is closed iff it is $\Bbb N$ or the sum of the inverses of its ...
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10 votes
0 answers
408 views

Generating function of the sequence $\binom{2n}{n}^3H_n$

Generating functions of the sequences $\binom{2n}{n}^2H_n$ and $\binom{2n}{n}^2H_{2n}$, where $H_n$ is $n$-th harmonic number, are known in terms of elliptic integrals $$ \sum_{n=1}^\infty\binom{2n}{n}...
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10 votes
0 answers
203 views

Can the sum of powers of the first primes be a square?

Let $p$ be a prime and $u\ge 1$ be a positive integer. Define $$\begin{align} S(p,u) &:= \sum_{q\text{ prime, }q \le p} q^u \\ &= 2^u+3^u+\cdots +p^u\end{align}$$ I wonder whether $S(p,u)$ ...
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