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Does every integer appear in the modular sum sequence?

$\newcommand{\N}{\mathbb{N}}$Let $\N$ denote the set of non-negative integers. We inductively define a sequence $a:\N\to\N$ by: $a(0) = 0, a(1) = 1$ and $a(n) = \big(\sum_{k=0}^{n-1}a(k)\big)\text{ ...
Dominic van der Zypen's user avatar
-8 votes
2 answers
381 views

Infinite set intersection with arithmetic progressions

Let $\mathcal{A}$ be the set of all arithmetic progressions in $\mathbb{N}$ i.e \begin{align*} \mathcal{A} = \{a + b\mathbb{N} : a,b\in\mathbb{N}, b\neq 0\}. \end{align*} Does there exist a set $X \...
Family Values's user avatar
1 vote
0 answers
228 views

Is $\sum\limits_{k=1}^nk^m=S_3(n)\cdot\dfrac{P_{m-3}(n)}{N_m}$ for odd $m>1,\sum\limits_{k=1}^nk^m=S_2(n)\cdot\dfrac{P_{m-2}'(n)}{N_m}$ for even $m$?

I asked this question here When I was in high school, I was fascinated by $$ \sum\limits_{k=1}^n k= \frac{n(n+1)}{2} $$ so I tried to find the general value of the sum $$ \sum\limits_{k=1}^n k^m\;\...
pie's user avatar
  • 411
0 votes
1 answer
100 views

The sequence has a stationary accumulation point

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth (continuously differentiable), convex function with a non-empty set of minimizers and $\{x^k\}$ be a sequence such that (a) $\{x^k\}$ has an ...
Dat Ba Tran's user avatar
3 votes
1 answer
272 views

When is an upper bound on the longest irreducible program outputting something computable?

Given some way to to encode programs to strings with a finite alphabet, which we assume has a computable translation to/from Turing machines, a program is irreducible if no subsequence of it has the ...
Daniel Weber's user avatar
  • 3,029
2 votes
1 answer
118 views

A second attempt at a two-dimensional Higman's Lemma

Let $A$ be a fixed finite alphabet. If $s$ and $t$ are finite strings over $A$, define $s\leq t$ if $s$ can be obtained by deleting zero or more characters from $t$. Higman's Lemma states that if $s_1,...
Charles Rackoff's user avatar
0 votes
0 answers
47 views

Summation of the following form with non-integer n

I have the following function: $$ G(z) = \sum_{j = 0}^{\infty} \frac{\Gamma(1 + n) z^j}{j ! (\Lambda + jC)^{n+1}} $$ If $n \quad \epsilon \quad \mathbb{Z}^{+} $, the above function can be ...
CfourPiO's user avatar
  • 159
-3 votes
1 answer
161 views

Is there a summation method where the divergent series $S = U_0+U_1+U_2+\dots$ converges to a finite value(V2)?

I have a question regarding this question here. is-there-a-summation-method-where-the-divergent-series if I set $ x+2=c/c-v$ , will I have $U_n = M\left(c-\frac{c}{n+2}\right)-M\left(c-\frac{c}{n+1}\...
z.10.46's user avatar
  • 33
0 votes
0 answers
137 views

Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?

This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist) If $a\in \mathbb R\...
Sayan Dutta's user avatar
3 votes
1 answer
462 views

Is there a summation method where the divergent series $S = U_0+U_1+U_2+\dots$ converges to a finite value?

Consider the function $$ M(v) = \frac{m_0}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}, $$ where $v \in \left]-c;c\right[$, $m_0\in\mathbb{R}^{*+}$, and $c=3\cdot10^8$. Let $(U_n)$ be a sequence with ...
z.10.46's user avatar
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4 votes
0 answers
72 views

Draw an arbitrary line on a Penrose tiling. Determine a sequence of tiles can it intersect

Let us consider a Penrose tiling of $\mathbb R^2$. Starting with an arbitrary point on the tiling, draw an arbitrary straight line. Assume that this straight line never overlaps perfectly with a ...
Darren Ong's user avatar
6 votes
2 answers
607 views

Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped

I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...
user avatar
2 votes
0 answers
59 views

Factor group of all the sequences by the subgroup of bounded sequences

Consider the group G of the sequences of real numbers (the group operation is addition). It contains a subgroup H of bounded sequences. Is there any nice description of the factor group G/H ? It is ...
Nikita Kalinin's user avatar
-1 votes
2 answers
115 views

How to solve the following infinite ladder fraction? ( Through pen & paper ) [closed]

The fraction continues till infinity as shown in the image :
Chandra Prakash Bairagi's user avatar
1 vote
1 answer
117 views

Given a real $x>1$, construct an aperiodic substitution sequence whose complexity functions grow like $xn$

The Fibonacci word is a binary sequence defined as follows. We use a substitution rule $0\to 01$, $1\to 0$. Then, starting with the binary string $0$, apply the substitution rules successively. So we ...
Darren Ong's user avatar
0 votes
1 answer
92 views

Positive linear recurrent sequence

Suppose there is a linearly recurrent sequence $a_k$ satisfies $a_k\geq 0$ and $\sum_{k=1}^{\infty}a_k=1$. Can we always find a $x$ and $r<1$ such that $a_k\leq x r^k$?
gondolf's user avatar
  • 1,503
-3 votes
1 answer
202 views

All group structures on a set with cardinality $\aleph_0$

Assume we consider the additive group $(\mathbb{Z}, 0, +)$. I am wondering what other group structures are there with neutral element 0 fixed? Is there a way to classify them or find them all?
tobias's user avatar
  • 739
3 votes
0 answers
187 views

Shift Operators and the Weyl Algebra

I have a question about the action of a shift operator $E$ on polynomials $Ep(x) = p(x+1)$ in the context of linear differential operators in one variable with polynomial coefficients, i.e. ...
the_sandcastler's user avatar
2 votes
1 answer
139 views

Mellin transform (of sequences)

Is it possible to define the Mellin transform for sequences of real numbers or even for tuples? Is there any book treating this argument? Any idea or suggestion will be greatly appreciated Since the ...
MathG's user avatar
  • 131
0 votes
0 answers
203 views

What are the hidden assumptions behind Harvey Friedman's claim, CSR?

I'm doing some archeology and trying to understand a claim. As summed up by David Roberts, on the FOM list in 2011: Let the statement "every infinite sequence of rationals in [0,1] has an ...
Corbin's user avatar
  • 424
1 vote
0 answers
153 views

Infinite matrices from $\ell^p$ to $\ell^{p/(p-1)}$ that are compact operators

I wanted to ask if my proof (sketch) of the following statement is correct. Namely, let $p>1$ and define $q= \frac{p}{p-1}$ we are given an operator $K : \ell^{p} \rightarrow \ell^{q}$ defined as $...
jrranalyst's user avatar
0 votes
1 answer
211 views

Where is the source of the formula $\sum_{j=0}^\infty \bigl(j+\frac{1}{2}\bigr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}$ for an integer sequence?

The infinite series representation \begin{equation} \frac1\pi\sum_{j=0}^\infty \biggl(j+\frac{1}{2}\biggr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}, \quad n\ge0 \end{equation} for the positive ...
qifeng618's user avatar
  • 964
2 votes
0 answers
200 views

Uniform distribution of log(log((n!)!)) mod 1

Can it be shown that the sequence $\log(\log((n!)!))$ is uniformly distributed mod 1? I believe it should be, but I'm not certain if Stirling's approximation repeated twice combined with the ...
Christopher D. Long's user avatar
9 votes
1 answer
1k views

The justifiable universe

Over the years of my study of set theory, I have encountered several sentences of the form V = X: V = L, V = HOD, V = WF (the exclusive assertion of the cumulative hierarchy), and (if I understand ...
Kristian Berry's user avatar
3 votes
0 answers
71 views

Some exercise on the regularity of a summability method

I was reading the book of Johann Boos "Classical and modern method in summability theory" and I came across an exercise from the Chapter 2 (page 50, exercise 2.3.15). Here is the statement ...
popa13's user avatar
  • 31
7 votes
0 answers
138 views

Characterization of tempered distributions from tempered sequences

Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported and infinitely smooth functions for its usual topology. Let $\mathcal{D}'(\mathbb{R})$ be the topological dual of $\mathcal{D}(\...
Goulifet's user avatar
  • 2,226
1 vote
0 answers
99 views

Generalizion of Euler identity with infinite sum of inverse squares

For $x,y\in \mathbb{R}\backslash \mathbb{Z}$ let $$ f(x,y)=\sum_{n\in\mathbb{Z}} \frac{1}{(n-x)(n-y)} $$ Is there a closed formula for $f(x,y)$? What is known: We have $$ f(x,x)=\left(\frac{\pi}{\sin(\...
user35593's user avatar
  • 2,286
3 votes
1 answer
188 views

"Approximating" linear recursion with homogenous polynomial coefficients by linear recursion with constant coefficients

In a lecture I once attended, I remember the speaker using a result of the following nature: $``$Let $\{A_n\}_{n=1}^\infty \subset \mathbb R$ be a sequence satisfying a recursion of the form $$P(n) ...
asrxiiviii's user avatar
2 votes
1 answer
110 views

Decreasing sequences in a finitely generated closure algebra

I am interested in finitely generated closure algebras (as a special case of Heyting algebras), and in decreasing sequences of elements within such an algebra that have no lower bound. Call two ...
Andrew Apps's user avatar
1 vote
1 answer
280 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 \...
ABIM's user avatar
  • 5,079
2 votes
0 answers
235 views

Cardinal numbers and the Bolzano-Weierstrass theorem

Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows: $\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...
Gabriel Medina's user avatar
3 votes
1 answer
89 views

Meetability of $\pm 1$-functions on $\omega$

If ${\cal S}$ is a collection of functions $f:\omega\to\omega$ we say that ${\cal S}$ is meetable if there is a "global function" $g:\omega\to \omega$ such that for every $f\in {\cal S}$ there is $n\...
Dominic van der Zypen's user avatar
1 vote
0 answers
96 views

Existence of limit computable map

Is there a limit computable function $\Phi$ with the following properties? Let $T\subset \mathbb{N}^{<\mathbb{N}}$ be a tree (coded as its characteristic function) and $f\in\mathbb{N}^\mathbb{N}$ ...
Manlio's user avatar
  • 332
9 votes
0 answers
402 views

Can this infinite sum for the Riemann zeta function be generalised?

I recently derived the following identity (which is probably a rediscovery of something well-known to experts). $$\sum_{k=1}^\infty{\frac{(-1)^{k+1}k^{4n+1}}{e^{k \pi}-(-1)^k}}=\frac{1}{2}\zeta\left(-...
Greg Egan's user avatar
  • 2,902
3 votes
1 answer
217 views

Is there a two-dimensional Higman's lemma?

A string over a finite alphabet $A$ can be thought of as a function $f:\{1,2,...,m\} \rightarrow A$ for some natural number $m$. A 2-Dim string over $A$ is a function $f$ where $f:\{1,2,\ldots,m\}\...
Charles Rackoff's user avatar
12 votes
1 answer
626 views

Integrals of power towers

Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...
Vladimir Reshetnikov's user avatar
1 vote
0 answers
36 views

Positive density of certain arithmetic sequence

Let $\{a_i\}_{i=0}^{\infty}$ be a sequence of positive real numbers bounded above by some uniform constant. For each $k,n\geq 1$ let $$ A^{(k)}_n:=\prod_{i=k}^{n+k-1}a_i $$ and suppose that there ...
Stefano Luzzatto's user avatar
4 votes
1 answer
1k views

The range of the Euler totient function and multiplication by 28

If $n$ is in the range of the Euler totient function, certain multiples of $n$ are likewise guaranteed to be totient values. The simplest nontrivial example of this is that, if $n$ is in the range of ...
DavidLHarden's user avatar
  • 3,625
1 vote
1 answer
164 views

Modular arithmetic and elementary symmetric functions

Denote the elementary symmetric functions in $n$ variables by $e_k(x_1, x_2,\dots, x_n)$. In the special case $x_j=j$, simply write $e_k(n)$ for $e_k(1, 2, \dots, n)$. Next, define the sequence $$a_{+}...
T. Amdeberhan's user avatar
5 votes
2 answers
374 views

Asymptotic rate for $\sum\binom{n}k^{-1}$

This MO question prompted me to ask: What is the second order asymptotic growth/decay rate for the sum $$\sum_{k=0}^n\frac1{\binom{n}k}$$ as $n\rightarrow\infty$?
T. Amdeberhan's user avatar
-1 votes
1 answer
196 views

Is this a Borel summable $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ with $ a_k$ alternating sequence?

let $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ a divergent series such that $b_k=(-1)^k (k!)a_k >0 $ for $k>1$ , and $b_k$ signed this from $k=1$ to $20$ ,The asymptotic of the titled series ...
user avatar
5 votes
2 answers
238 views

Different notions of computable binary sequence

The standard definition of computability, for a sequence $s\in\{0,1\}^\omega$, is that there is a Turing machine outputting $s[i]$ on input $i$. I'm looking for strengthenings of this notion; for ...
grok's user avatar
  • 2,489
2 votes
0 answers
1k views

Is there an infinite product like this for $\cos x$?

There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example $$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
John Finkelstein's user avatar
2 votes
2 answers
254 views

Reference request for function by which to compute coefficients of continued fraction of algebaic number

The simple continued fraction is in the form $$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance. Obviously,the coefficients $x_i$can be computed by computable function $x_i=f(i),...
XL _At_Here_There's user avatar
1 vote
0 answers
162 views

Is there any irrational algebraic number among the set? [closed]

Suppose $S$ is set of numbers such that every number in it expands in decimal digits,every digit is 0 or 1,and $\lim_{n\rightarrow\infty}\frac{C_{n}(0)}{n}=\frac{1}{2}$ where ${C_{n}(0)}$ and ${C_{n}(...
XL _At_Here_There's user avatar
4 votes
0 answers
132 views

Irreducibility of polynomials corresponding to sequences

I have no experience with this, so I dont know if this is too easy for MO. Let $(a_n)$ be a strictly monotone sequence of natural numbers, then define the set of nice numbers of $(a_n)$ as $X(a_n):=\{...
Mare's user avatar
  • 26.3k
2 votes
2 answers
1k views

Algorithm to check if vertex belong to infinite path in Graph theory

My purpose is to understand if in a graph $G = \langle V, E\rangle$ given 4 vertices in input (a, b ,c and d) they belong to an infinite path. With infinite path I mean a vertex succession that has a ...
Philip's user avatar
  • 133
2 votes
2 answers
226 views

What's "serialization" really called, and is there any theory surrounding it?

Define an operator $\mathop{\vec{\bigcup}}$ as follows: Definition. Whenever $A$ is an $I$-indexed family of sets, where $I$ is a totally-ordered set, we have $$\mathop{\vec{\bigcup}}_{i \in I} A_i ...
goblin GONE's user avatar
  • 3,693
0 votes
2 answers
170 views

Closed form of $\sum_{i=k}^\infty i h {i \choose {k-1}} h^{k-1} (1-h)^{i - (k-1)}$?

Is there a closed form solution to the expression below? Or, if there is no closed form solution but the series converges, is there some upper bound on this expression? $$\mathbb E_{i \sim Q}[i] = \...
A.M.'s user avatar
  • 3
3 votes
1 answer
454 views

Put 10 balls in the jar then randomly take 1 out. Do it infinitely many times. Find the probability of resulting in an empty jar [closed]

The original discussion (in Chinese): https://www.zhihu.com/question/58702489 The original problem was from an probability theory exam. The problem is translated as: Assume an infinitely large jar ...
Liwei Cai's user avatar