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.
1,731
questions
6
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1
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347
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Looking for infinite series resembling an exponential
I'm looking for some $f(x)$ that has the following property:
$\sum_{x=1}^\infty f(kx) = r^k$
for some real $0 < r < 1$, and at least for strictly positive integer $k$.
Does such an $f(x)$ ...
- 4,525
2
votes
1
answer
849
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Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way [closed]
I need to emulate this sequence for a program: http://oeis.org/A025302
Stuff that I've taken into account:
After finding the prime divisors of a number. I take any divisor as p and apply the ...
- 123
5
votes
2
answers
231
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Extension of Valdivia-Vogt isomorphism from $\mathscr{D}(K)$ to $\mathscr{E}'(K)$
Let $M$ be a $d$-dimensional (say, Hausdorff, paracompact, connected and oriented) smooth manifold, and $K\subset M$ compact with $\mathring{K}\neq\varnothing$. M. Valdivia has shown (based on ...
- 7,099
0
votes
1
answer
377
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A possible surprise involving Euler's constant $e$ [closed]
Let
\begin{align*}
c_n &= n!\left(e-\sum_{k=0}^n \frac{1}{k!}\right) \\
\\
u_n &= \bigg\lfloor{\frac{1}{c_n} \bigg\rfloor} \\
\\
v_n &= \bigg\lfloor{\frac{1}{1/c_n-\lfloor{u_n} \rfloor}} ...
- 3,443
-1
votes
1
answer
57
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square weighted l^2
I am looking the sequence spaces $l^1$ and $$\{(x_k)_k: \|x\|_{sq}^2 := \sum_{k=1}^\infty k^2\cdot x_k^2 < \infty\}. $$
I am particularly interested in relations between their respective norms: It ...
- 375
2
votes
1
answer
426
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Integrating nasty gaussian over square root
TLDR: trying to solve,
$$\int_1^\infty \exp\left(-\frac{x^2}{2\omega^2}\right) \frac{1}{\sqrt{ax^2+bx-1}}dx$$
After doing some reading and looking at some other questions 1, 2 (and even going through ...
- 123
0
votes
4
answers
547
views
How to compute this series: $\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$ [closed]
How to compute this series: $$\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$$ where $C_k$ is the catalan number: $C_k=\frac{1}{k+1}{2k \choose k}$. (Further, is there any general method to treat this ...
- 317
3
votes
0
answers
91
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Flow of zeros in the shifted exponential generating function?
Given a sequence $a_n$ (of real numbers, described more fully below), one may define the exponential generating function (on the complex plane) as $E(z)=\sum_{n=0}^\infty a_n z^n/n!$. The derivatives $...
- 364
7
votes
0
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205
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Has this self-similar sequence the ratio $(\sqrt2+1)^2$?
This is inspired by a math.SE question, where an infinite sequence of pairwise distinct natural numbers $a_1=1, a_2, a_3, ...$ has been defined as follows:
$a_n$ is the smallest number such that $s_n:=...
- 13.2k
5
votes
2
answers
401
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Formula for a sum of product of binomials
We know that equation $$s_1+s_2+s_3=n-1 \quad \mbox{$s_1,s_2,s_3$}\geq 1$$
has $\binom{n-2}{2}$ solution.
I want to find any good formulae for the following form :
$$\sum_{(s_1,s_2,s_3)}\prod_{i=1}^...
- 355
4
votes
0
answers
165
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Counting "motifs" with the same "energy"
This question is motivated by physics --- trying to understanding the so-called 'accidental' (i.e. non representation-theoretic) degeneracies that occur in the spectrum of the Haldane--Shastry spin ...
- 1,813
11
votes
2
answers
660
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Value of $c$ such that $\lim_{n\rightarrow\infty}\sum_{k=1}^{n-1}\frac{1}{(n-k)c+\log(n!)-\log(k!)}=1$
What is the value of $c$ such that
$$\lim_{n\rightarrow\infty}\sum_{k=1}^{n-1}\frac{1}{(n-k)c+\log(n!)-\log(k!)}=1?$$
Numerically, it seems that the answer is $c=\log 2$. But I'd like to see a reason ...
- 481
6
votes
1
answer
416
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On convergent series - in the spirit of Abel and Dini
Nonexistence of boundary between convergent and divergent series?
I'm hoping the following is true:
Suppose $a_i $ is a positive sequence and $\sum_i a_i < \infty.$ Then there exists a positive ...
- 239
2
votes
0
answers
105
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Understanding proof of q-theta function expression
In arXiv:math/0309252v4 at the bottom of page 11, the following result is proposed
$$ b_1 \theta(b_0 b_1^{\pm};p) \frac{z^{-1}\theta(z^2;p)}{\theta(b_0 z^{\pm};p)\theta(b_1 z^{\pm};p)} = (p;p)^{-2}\...
- 181
2
votes
1
answer
77
views
Defect of subnormality and repeated normalizer series
Let $G$ be a a finite group and $S$ a subnormal subgroup of $G$. The lenght of a fastest chain of subgroups $(U_i)_{1\le i\le n}$ such that $U_1=S$, $U_i$ normal in $U_{i+1}$ and $U_n=G$ is called the ...
- 581
1
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0
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101
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unknown sequences of rational numbers with sum of a transcendental number [closed]
Starting with a transcendental number like $e$, it is known that we can write it as a sum of infinitely many rational numbers of the form :
$e= \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{...
- 289
5
votes
0
answers
151
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A relation concerning the "sum of squares" counting function $r_2(n)$
This is a re-post from MSE as I did not get any response there.
Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here ...
- 2,019
6
votes
1
answer
372
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Asymptotic behavior of a certain trigonometric partial sum
Let $a<0$ and $b>0$ be real numbers such that $a<-2b$. Let $n>1$ be a positive integer and consider the following partial sum:
$$
f(n) = \frac{1}{(n+1)^2}\sum_{i=1}^{n}\sum_{j=1}^{n} (-1)^{...
- 2,682
7
votes
0
answers
945
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A problem of Erdős on convergence of $\sum (-1)^nn/p_n$ and equidistribution of $\pi(n)$ modulo 2
Erdős asked1 whether the series
$$\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$$ converges.
Here, $p_n$ denotes the n-th prime.
I can show that this series converges simultaneously with the series $\sum_{...
- 3,679
6
votes
3
answers
466
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Randomly picking $k$ members of $\{1,\ldots,n\}$
Every day, I randomly pick a sample consisting of $k$ members of $\{1,\ldots,n\}$ where $k\leq n$. I stop as soon as every number of $\{1,\ldots,n\}$ has been picked at least once. Let $S$ be the ...
- 45.4k
1
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1
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127
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$\sum_{n=0}^{N-1}\exp(-i\frac{\pi}{N}n^2)=\sqrt{N}\exp(-i\frac{\pi}{4})$
The question is: N is an even positive integer, then $\sum_{n=0}^{N-1}\exp(-i\frac{\pi}{N}n^2)=\sqrt{N}\exp(-i\frac{\pi}{4})$. I thought the terms on the left are the solutions set of some polynomial ...
- 11
3
votes
1
answer
385
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Constant "periodization" of a function
Let $w$ be a rapidly decaying function on $\mathbb{R}$ such that
$$ \sum_{n \in \mathbb{Z}} w(x+n) = 0$$
for all $x \in \mathbb{R}$. Does that imply that $w$ is identically zero? What if we assume ...
- 9,591
1
vote
1
answer
1k
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Finding closed form of recurrence relation in two variables [closed]
I have a recurrence,
$$F(n, m) = F(n-1, m) + F(n, m-1) + F(n-1,m-1) $$
$$F(n,1) = 0$$
$$F(1,n) = 2*(n-1)$$
I would like to compute $F(N,M)$ in terms of $N$ and $M$. The system is defined for $1 \...
- 13
10
votes
4
answers
1k
views
Adventure with infinite series, a curiosity
It is easily verifiable that
$$\sum_{k\geq0}\binom{2k}k\frac1{2^{3k}}=\sqrt{2}.$$
It is not that difficult to get
$$\sum_{k\geq0}\binom{4k}{2k}\frac1{2^{5k}}=\frac{\sqrt{2-\sqrt2}+\sqrt{2+\sqrt2}}2.$$
...
- 41.8k
1
vote
1
answer
171
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Approximate the following series on the euclidean grid
I had trouble in finding a closed form solution for the following series, so now I am trying to find a good approximation for it. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the ...
- 55
3
votes
2
answers
357
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Infinite sum of reciprocals of squares of lengths of tangents from origin to the curve $y=\sin x$
This question is actually from MSE. I had to post it here due to the lack of response there even after placing a bounty. Here goes the question
Let tangents be drawn to the curve $y=\sin x$ from ...
- 139
5
votes
2
answers
861
views
Searching for a proof for a series identity
The below identity I have found experimentally.
Question. Is this true? If so, may you provide a "slick" (or any) proof.
$$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
- 41.8k
2
votes
1
answer
94
views
Quotient with positive second derivative in the limit?
I am studying the quotient of
$$f(\varepsilon) = \sum_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$
and $$g(\varepsilon) = \sum_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$
for some $\...
- 119
0
votes
1
answer
118
views
May Champernowne constants $C_m$ be related to other numbers than $m$?
[This question is related to another question concerning normal numbers I asked at Math SE.]
Has it ever been found worth to ask the question if the Champernowne constants $C_m$, especially $C_2$ ...
- 9,628
3
votes
2
answers
558
views
How to estimate a recursive inequality with an upper bound
The below is a simplification of part of a proof I'm working on, in numerical analysis. It is similar to a paper that I studied some months ago, for which I got some advice here on MathOverflow.
I ...
- 167
3
votes
1
answer
188
views
Matching two sequences between each other
Given the sequence of symbols $A$ (contains ~10,000 symbols) and sequence of blocks $B$ (contains ~3,000 blocks, ~30 symbols inside each block) I need to exclude some blocks from sequence $B$ so that ...
- 153
34
votes
1
answer
3k
views
A remarkable almost-identity
OEIS sequence A210247 gives the signs of $\text{li}(-n,-1/3) = \sum_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$.
Mikhail Kurkov noticed that it ...
- 53.6k
2
votes
2
answers
135
views
Divergence rate of geometric sum of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that
$$
0<\lim_{\beta\rightarrow 1}(1-\...
- 479
2
votes
1
answer
196
views
Guess (or upper bound) the general formula for a double sequence
Let $t,s \geq 0$ be integers. We have the following recursive formula:
$$f(t+1,s) = f(t,s) + f(t,s-1) + \sum_{0\leq a,b,c \leq h(t):\\a+b+c = s-1}f(t,a)f(t,b)f(t,c),$$ where
$$h(t) = \frac{1}{2}3^t -\...
- 505
7
votes
1
answer
338
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Descartes' rule of signs for infinite series
Consider the function given by
$$f(x)=1-a_1x-a_2x^2-a_3x^3-\cdots$$
where each $a_k\geq0$ and some $a_j>0$. If $f(x)$ is a polynomial then Descartes' Rule of signs tells us there is exactly one ...
- 41.8k
10
votes
2
answers
308
views
Denominators of certain Laurent polynomials
Consider the following somos-like sequence
$$x_n=\frac{x_{n-1}^2+x_{n-2}^2}{x_{n-3}}.$$
It's known that $x_n$ is a Laurent polynomial in $x_0, x_1$ and $x_2$. I got interested in the denominators of ...
- 41.8k
1
vote
1
answer
142
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About a Dirichlet series [closed]
I would like to know if the following assertion is true:
Let consider a real decreasing sequence $(t_n)$ of positive numbers with limit zero, if the series $\sum\limits_{n=1}^\infty(t_n)^a$ is ...
- 337
9
votes
0
answers
177
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Infinite series identities in search of a proof
This comes in relation to the Fishburn numbers.
I stumbled on the following relation for which I ask a proof if true.
Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then
$$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
- 41.8k
16
votes
1
answer
641
views
Does every real function have this weak derivation property?
After this question : Does every real function have this weak continuity property?
Natrualy there are an other (more difficult) :
Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}...
- 3,795
53
votes
3
answers
4k
views
Does every real function have this weak continuity property?
In my research I came across the following question :
Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x_n)_n$, taking infinitely many values, ...
- 3,795
-1
votes
2
answers
367
views
Are there integral solutions for $(2a-1)(2^{(b+c)}-3^c )=2^b-1$?
Can anyone prove this assertion? Or at least suggest a method of attack? It has come up in my research.
There do not exist $a,b$ and $c$ such that$$
(2a-1)(2^{(b+c)}-3^c )=2^b-1
$$where $a>0,b&...
4
votes
1
answer
129
views
Convergence of a real sequence (stochastic approximation)
Let $(\gamma_n)_{n \geq 1}$ be a sequence of positive real numbers satisfying $$\sum_{n \in \mathbb{N}}\gamma_n = + \infty \text{ and }\sum_{n \in \mathbb{N}}\gamma_n^2 < + \infty$$
I would like ...
- 113
3
votes
1
answer
410
views
What is the shortest length of an Egyptian fraction expansion for a given $p/q$?
An Egyptian fraction expansion is a sum of reciprocals of integers, for example:
$$\frac{4}{17} = \frac{1}{5} + \frac{1}{29} + \frac{1}{1233} + \frac{1}{3039345}$$
Every positive rational number $p/...
- 4,114
7
votes
1
answer
214
views
Is anything known about this class of series involving the divisor function?
I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it!
Let $N\in\mathbb{N}$, let $q$ be a point in the open ...
- 661
1
vote
0
answers
94
views
Show that the norm's bound is an exponent
Let $f$ be a continuous function on $S^2$. Consider $g\in C^{\infty}(R)$, such that $g(x)=1$ for $|x|\leq 1$ and for $|x|\geq 2$. Let $h(x)=g(x)-g(2x)$. The notation $proj_k$ denotes the orthogonal ...
- 343
3
votes
0
answers
203
views
Gray Code for Combinations
Question:
are Gray Codes known for enumerating all fixed-size subsets of a given finite set?
Background of the question is trying to find the lightest k-clique in a symmetric TSP instance of size n ...
- 12.7k
1
vote
1
answer
163
views
How many points appear in the plane when the chain of n-gons is close?
Let $A_{11}A_{12}\cdots A_{1n}$ be a regular $n$ polygon, we call $A_{11}A_{12}\cdots A_{1n}$ is the $1st-n-gons$. Now we construct the $2nd-n-gon$ based two condition as follows:
$2nd-n-gons$ is ...
- 1,086
2
votes
2
answers
455
views
Is the exponential version of Catalan-Dickson conjecture true?
The aliquot sum function $s:\mathbb{N}\rightarrow \mathbb{N}$ assigns to any natural number $n$ the sum of its proper divisors. Perfect numbers are fixed points of this function. The open conjecture ...
-1
votes
1
answer
72
views
Create approximations of finite integer sequence
Given a function of real numbers f(x), I can create approximations to arbitrary precision using Taylor polynomials.
Is there something equivalent in the discrete case when I have a sequence of ...
- 103
1
vote
0
answers
40
views
linear difference inequality with error
Let $(\alpha,\gamma,\beta)$ real number such that $\alpha+\gamma+\beta=1$, let $(e_n)_{n\in\mathbb{N}}$ be a summable non-negative sequence and let $(u_n)$ be a non-negative sequence such that
$\...
- 11