Questions tagged [infinite-sequences]
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80 questions
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Sum of an infinite series
I have the following infinite series
$$S=\sum_{n=1}^\infty \frac{x^n}{n!}e^{\frac{c}{n}}, ~(x>0,c>0)$$
How could I perform this summation? Or is there any good analytical/closed-form ...
- 19
5
votes
2
answers
372
views
Weak Archimedean property instead of Archimedean property
We say that a sequence $(z_n)$ of real numbers is a modulated Cauchy sequence, whenever there exists a function $\alpha:\mathbb{N} \rightarrow \mathbb{N}$ such that:
$$
|z_i-z_j| \le \frac{1}{k} \quad ...
3
votes
1
answer
130
views
Left shift of transfinite sequences
Background: Let $(D,\, {\leq})$ be a partial order. We consider sequences $a = (a_n)_{n \in \mathbb{N}}$ of elements of $D$. For two sequences $a$ and $b$, we lift $\leq$ (point-wise) to sequences by
...
- 351
-1
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1
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122
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Divergent summation [closed]
Let $(x_i)_{i=0}^\infty$ be a sequence such that $0<x_i<1\ \forall i \in \mathbb{N} \cup {0}$.Consider the following series:
$$\sum_{i=1}^\infty \frac{x_i}{\left(\sum_{k=0}^{i-1} x_k \right)^2}.$...
- 21
5
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1
answer
200
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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{ ...
- 48.8k
-8
votes
2
answers
410
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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 \...
1
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0
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236
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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
$$
\sum\limits_{k=1}^n k^m\;\...
- 541
0
votes
1
answer
106
views
The sequence has a stationary accumulation point
Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth (continuously differentiable), convex function with a non-empty set of minimizers and $\{x^k\}$ be a sequence such that
(a) $\{x^k\}$ has an ...
- 21
3
votes
1
answer
309
views
When is an upper bound on the longest irreducible program outputting something computable?
Given some way to to encode programs to strings with a finite alphabet, which we assume has a computable translation to/from Turing machines, a program is irreducible if no subsequence of it has the ...
- 3,319
2
votes
1
answer
119
views
A second attempt at a two-dimensional Higman's Lemma
Let $A$ be a fixed finite alphabet.
If $s$ and $t$ are finite strings over $A$, define $s\leq t$ if $s$ can be obtained by deleting zero or more characters from $t$. Higman's Lemma states that if $s_1,...
-3
votes
1
answer
168
views
Is there a summation method where the divergent series $S = U_0+U_1+U_2+\dots$ converges to a finite value(V2)?
I have a question regarding this question here.
is-there-a-summation-method-where-the-divergent-series
if I set $ x+2=c/c-v$ , will I have
$U_n = M\left(c-\frac{c}{n+2}\right)-M\left(c-\frac{c}{n+1}\...
- 33
0
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0
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138
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 791
3
votes
1
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475
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Is there a summation method where the divergent series $S = U_0+U_1+U_2+\dots$ converges to a finite value?
Consider the function
$$
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 ...
- 33
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votes
0
answers
78
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 ...
- 785
6
votes
2
answers
772
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, ...
user504691
2
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0
answers
62
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 ...
- 5,043
-1
votes
2
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126
views
How to solve the following infinite ladder fraction? ( Through pen & paper ) [closed]
The fraction continues till infinity as shown in the image :
1
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1
answer
123
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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 ...
- 785
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votes
1
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93
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$?
- 1,503
-3
votes
1
answer
218
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?
- 749
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votes
0
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197
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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. ...
2
votes
1
answer
146
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 ...
- 131
0
votes
0
answers
211
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 ...
- 436
1
vote
0
answers
163
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 $...
- 11
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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 ...
- 1,091
2
votes
0
answers
209
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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 ...
9
votes
1
answer
1k
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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 ...
3
votes
0
answers
78
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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 ...
- 31
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votes
0
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146
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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}(\...
- 2,306
1
vote
0
answers
103
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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(\...
- 2,286
3
votes
1
answer
201
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"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
$$P(n) ...
- 739
2
votes
1
answer
113
views
Decreasing sequences in a finitely generated closure algebra
I am interested in finitely generated closure algebras (as a special case of Heyting algebras), and in decreasing sequences of elements within such an algebra that have no lower bound.
Call two ...
- 21
1
vote
1
answer
303
views
Comparison of product topology and colimit topology in sequence spaces
In Munkres Theorem 20.4 it is shown that the (relative) uniform topology induced by:
$$
d(x,y)\triangleq \sup_{n \in \mathbb{N}} d(x_n,y_n)
$$
is strictly finer than the product topology on $\prod_{n \...
- 5,407
3
votes
0
answers
242
views
Cardinal numbers and the Bolzano-Weierstrass theorem
Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows:
$\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...
- 561
3
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1
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Meetability of $\pm 1$-functions on $\omega$
If ${\cal S}$ is a collection of functions $f:\omega\to\omega$ we say that ${\cal S}$ is meetable if there is a "global function" $g:\omega\to \omega$ such that for every $f\in {\cal S}$ there is $n\...
- 48.8k
1
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98
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Existence of limit computable map
Is there a limit computable function $\Phi$ with the following properties?
Let $T\subset \mathbb{N}^{<\mathbb{N}}$ be a tree (coded as its characteristic function) and $f\in\mathbb{N}^\mathbb{N}$ ...
- 342
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votes
0
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407
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Can this infinite sum for the Riemann zeta function be generalised?
I recently derived the following identity (which is probably a rediscovery of something well-known to experts).
$$\sum_{k=1}^\infty{\frac{(-1)^{k+1}k^{4n+1}}{e^{k \pi}-(-1)^k}}=\frac{1}{2}\zeta\left(-...
- 2,902
3
votes
1
answer
218
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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\}\...
12
votes
1
answer
633
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 ...
- 6,819
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 ...
4
votes
1
answer
1k
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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 ...
- 3,645
1
vote
1
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167
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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
$$a_{+}...
- 43.1k
5
votes
2
answers
379
views
Asymptotic rate for $\sum\binom{n}k^{-1}$
This MO question prompted me to ask:
What is the second order asymptotic growth/decay rate for the sum
$$\sum_{k=0}^n\frac1{\binom{n}k}$$
as $n\rightarrow\infty$?
- 43.1k
-1
votes
1
answer
199
views
Is this a Borel summable $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ with $ a_k$ alternating sequence?
let $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ a divergent series such that $b_k=(-1)^k (k!)a_k >0 $ for $k>1$ , and $b_k$ signed this from $k=1$ to $20$ ,The asymptotic of the titled series ...
user119110
5
votes
2
answers
251
views
Different notions of computable binary sequence
The standard definition of computability, for a sequence $s\in\{0,1\}^\omega$, is that there is a Turing machine outputting $s[i]$ on input $i$.
I'm looking for strengthenings of this notion; for ...
- 2,519
2
votes
0
answers
1k
views
Is there an infinite product like this for $\cos x$?
There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
2
votes
2
answers
257
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),...
- 3,084
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}(...
- 3,084
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):=\{...
- 26.5k
2
votes
2
answers
1k
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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 ...
- 133