Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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72
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Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
60
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0answers
2k views

Arithmetic-geometric mean of 3 numbers

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{a_n+b_n}2,...
57
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0answers
708 views

Conjectured formula for the Fabius function

The Fabius function is the unique function ${\bf F}:\mathbb R\to[-1, 1]$ satisfying the following conditions: a functional–integral equation$\require{action} \require{enclose}{^{\texttip{\dagger}{a ...
52
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1answer
1k views

Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?

This question is inspired by my answer to the question "How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?". The sums $f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ (for positive integer $...
38
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710 views

Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$

Is there a way to assess the convergence of the following series? $$\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$$ From numerical estimations it seems to be convergent but I don't know how to prove it.
22
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537 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, ...
21
votes
1answer
540 views

An interesting formula for $\pi$

Looking through some old notebooks I found this monster of a formula: For any integer $r>1$, we have $$\pi=(-1)^{\left\lfloor\frac{r}{2}\right\rfloor-\left\lfloor\frac{2r-1}{4}\right\rfloor}\...
21
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0answers
669 views

A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. I'm looking at the (gap-)series $$ s(1/2,2) = (1/2)^1+(1/2)^{4}+(1/2)^{9}+(1/2)^{16}+(1/2)^{25}+... $$ ...
20
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0answers
515 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 ...
16
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0answers
374 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 ...
16
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0answers
749 views

Convergence/Divergence of infinite series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{1+\left|{\cos n}\right|}}$

It is well known that $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$ is divergent while $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\epsilon}}$ is convergent for a fixed positive value of $\...
16
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1answer
244 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 ...
15
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252 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}^...
14
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162 views

Prove that $\lim_{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{k!} = e$

Define $e, e'$ by $$e: =\lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{n} \quad \text{and} \quad e' := \lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{k!}$$ Prove that $e' \in \...
14
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558 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 ...
14
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0answers
637 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 ...
13
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0answers
138 views

How much can we rearrange a series?

There's a well-known result that if $\sum a_n$ is conditionally convergent, then for any real $c$ there exists a permutation $\pi:\mathbb{N} \to \mathbb{N}$ such that $\sum a_{\pi(n)} = c$. A ...
13
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0answers
246 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 ...
12
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0answers
131 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 ...
12
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2answers
397 views

show that $\sum_{k=1}^{n}(1-a_{k})<\frac{2}{3}$

Let $a_{1}=\dfrac{1}{2}$, and such $a_{n+1}=a_{n}-a_{n}\ln{a_{n}}$,show that $$\sum_{k=1}^{n}(1-a_{k})<\dfrac{2}{3}$$ My attemp: let $1-a_{n}=b_{n}$,then we have $$b_{n+1}=b_{n}+(1-b_{n})\ln{(1-b_{...
11
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0answers
159 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 $\...
11
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0answers
240 views

The famous root of $x(1 - x^{12})^2(1 - x^{24})^2 = (1 - x^6)^7(1 - x^8)^4$

I. Let $n=2$ and $x = \frac1\phi$ with golden ratio $\phi$. Then it satisfies, $$x(1 - x^{12})^2(1 - x^{24})^2 = (1 - x^6)^7(1 - x^8)^4\tag1$$ $$x(1 - x^{12})^2 = (1 - x^4)^\color{blue}2(1 - x^6)^3\...
11
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0answers
204 views

Can (linear) differential equations of infinite order be recast into equations of first order?

In most analysis courses one sees that differential equations of order $n$ are basically a subset of higher dimensional differential equations of order $1$, for example the equation: $$f^{(n)}(t)=F\...
11
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0answers
206 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}...
11
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0answers
349 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 ...
11
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0answers
211 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) = \...
11
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3answers
554 views

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

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^...
10
votes
1answer
62 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 ...
10
votes
1answer
472 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 ...
10
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0answers
198 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). ...
10
votes
1answer
196 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\...
10
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0answers
262 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 ...
10
votes
1answer
226 views

Considering the equation, $6 + (2k+1)\sum_{n=1}^{2k+1}p_n^{ \ \ 3}(-1)^{n+1} = x^2$.

I noticed that, $$\begin{align}3(2^3 - 3^3 + 5^3) + 6 &= 18^2 \\ \text{and } \qquad 5(2^3 - 3^3 + 5^3 - 7^3 + 11^3) + 6 &= 74^2.\end{align}$$ These equations are of the form, $$6 + (2k+1)\sum_{...
10
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1answer
369 views

Pi series that converges arbitrarily fast.

The old series for $\pi$ is this alternating series: $$\pi = 4 \sum_{i=0}^{\infty}\frac{(-1)^i}{2i+1}$$ Now, as already noticed, the series is alternating: adding one term overshoots $\pi$ every ...
10
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0answers
343 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 ...
10
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0answers
182 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 $S(p,u)$ ...
10
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0answers
619 views

Calculating growth rate of a population of Minecraft chickens

I have a rather strange question (for this Stack Exchange anyway). It felt too mathematical to ask elsewhere. If this is out of place here, please let me know. A chicken in Minecraft lays eggs; ...
10
votes
1answer
144 views

Is it true that $\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb N^+$ such that $a^2-b=k^2 $?

This is a curiosity question: Question Given two positive integers $a$ and $b$ do we have the following equivalence: $$\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb N^+\...
10
votes
1answer
696 views

Arithmetic and geometric sequences: where does their name come from?

Where does the name of these two famous types of sequences come from? The article Geometric progression of Wikipedia says that the geometric sequence is called like this because every term is the ...
10
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0answers
434 views

Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would like to extend the idea for ...
9
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0answers
106 views

Series of product of Bessel functions

The Christoffel-Darboux formula applied to Bessel functions states that $$\sum\limits_{j=0}^{+\infty}J_{j+n}(t)J_{j+m}(t)=\frac{t}{2(m-n)}\left(J_{m-1}(t)J_n(t)-J_m(t)J_{n-1}(t) \right)$$ See for ...
9
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0answers
170 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 ...
9
votes
0answers
329 views

How many primes does this sequence find?

The sequence in question is: $$S=\left\{\int_0^1\pi(x)\pi(1-x)dx,\int_0^2\pi(x)\pi(2-x)dx,...\right\},$$ where $\pi(x)$ is the prime counting function. I don't know how to check this for an ...
9
votes
1answer
173 views

Recurrence $a_{n}=a_{\lfloor 2n/3\rfloor}+a_{\lfloor n/3\rfloor}$

I am considering the sequence $$a_n=a_{\lfloor 2n/3\rfloor}+a_{\lfloor n/3\rfloor}$$ with $a_0=1$, and I would like to calculate the limit $$\lim_{n\to\infty} \frac{a_n}{n}$$ I have seen this famous ...
9
votes
0answers
162 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 ...
9
votes
1answer
235 views

Is anything known about $ \small{b_0+\tfrac{a_1}{\left(b_1+\tfrac{a_2}{\left(b_2+…\right)^n}\right)^n}} $?

What is known about this generalized "continued fraction" $$ b_0+\frac{a_1}{\left(b_1+\frac{a_2}{\left(b_2+\frac{a_3}{\left(b_3+\dotsb\right)^n}\right)^n}\right)^n} $$ when the integer $n\ge 2$? ...
9
votes
0answers
285 views

Evaluate $\sum_{n=0}^{\infty} \frac{n!}{(n^2)!}$

I'm interested in a method of evaluating $\sum_{n=0}^{\infty} \frac{n!}{(n^2)!}$. If there was a linear equation with leading coefficient $1$ in the denominator or a quadratic with leading ...
9
votes
0answers
135 views

Closed form for $\sum_{n=1}^\infty \frac{1}{P(n)}$, where $P(n)$ is the partition function.

Is there a closed form for the following infinite series? $$\sum_{n=1}^\infty \frac{1}{P(n)}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$$ where $P(n)$ is the partition function.
9
votes
1answer
251 views

Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\cdots = -\frac{1}{12}$? The Euler-...
9
votes
0answers
242 views

Invariant functions on the space of finite sequences of reals

Let $S$ be a space of all finite sequences of real numbers (we don't endow it with metric or topology in general). Before asking the main question, some notation. 1. For each $\mathbf s\in S$ we ...