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.
1,735
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46
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Doubt on Series with two indices [closed]
Can someone explain why:
$$\sum_{k,l=1}^{\infty}\frac{p^2}{k+l-1}(1-p)^{k+l-2} =$$$$ \sum_{j=2}^{\infty}\sum_{l=1}^{j-1}\frac{1}{j-1}{p^2(1-p)^{j-2}}$$
And why
$$\sum_{k,l=1}^{\infty} kl = \sum_{k=1}^{...
- 11.4k
1
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0
answers
46
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Find alternative ways to do summations
Consider
$A(x)=\sum_{n,m =2}^{\infty}\frac{ (n-1)(m-1) x^{n+m}}{n (n+1) (n+m) (n+m+1)}$
where $0<x<1$.
If one uses Mathematica to sum over $n$ (or $m$), one gets the Hurwitz–Lerch transcendent, ...
- 81
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0
answers
42
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Sequence that sums up to A224071
Let $a(n)$ be A224071 (i.e., number of Schroeder paths of semilength $n$ in which there are no $(2,0)$-steps at level $1$). Here
$$
a(n) = \frac{1}{2(n+1)}\sum\limits_{k=0}^{n}(k+1)((-1)^{\left\...
- 3,464
-3
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0
answers
39
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find delta of each numbers in a number list such that each consecutive number differs at most by delta and first and last number also differ by delta [closed]
Given a sequences of numbers a0, a1, a2, ..., an, a0 and epsilon each number is randomly initialized (i don't know if this is relevant), find a sequences of delta d0, d1, ..., dN, such that for 0 <=...
3
votes
1
answer
115
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Eisenstein $E_2$ at imaginary quadratic arguments
In the paper On Epstein's zeta-function, Chowla and Selberg give a formula for evaluating the Dedekind eta function
$$\eta (\tau)=e^{\pi i\tau/12}\prod_{n=1}^\infty (1-e^{2\pi i n\tau}),\quad \Im\tau\...
- 11.8k
2
votes
2
answers
207
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Negated Fibonacci and the floor function
Let $F_n$ be A000045 (i.e., Fibonacci numbers). Here
$$
F_n = F_{n-1} + F_{n-2}, \\
F_0 = 0, F_1 = 1, \\
F_{-n} = (-1)^{n-1}F_n
$$
I conjecture that
$$
F_{-n} = \left\lfloor\frac{n+1}{2}\right\rfloor ...
- 31
1
vote
1
answer
154
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Refinement of face vectors of the simplicial noncrossing hypertree complexes of McCammond
Einziger on page 65 of "Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions" presents the antipode of a noncrossing partition Hopf algebra as a graded sequence of partition ...
CommunityBot
- 1
2
votes
2
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245
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How did Ramanujan find $\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}?$
The formula
$$\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}$$
(in older notation) appears as eq. 38 in Ramanujan's paper Modular equations ...
- 178k
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0
answers
173
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On a A057985 without recursion
Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0\to01, 1\to12, 2\to0$).
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here
$$
\...
- 3,464
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votes
0
answers
97
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Does the following series converge? To which value?
Given $f(x)=A \cdot x^5+B \cdot x^8$
with:
$A \in \mathbb{R}^-$
$B \in \mathbb{R}^-$
$h(z) = w_0 + \sum_{n=1}^\infty h_n \cdot \frac{\left(z-f(w_0)\right)^n}{n!}$
$w_0=1$
and:
$$h_n=\lim_{w \...
- 11
2
votes
1
answer
280
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Are these conditions regarding products of consecutive terms in a sequence of positive numbers equivalent?
Assume $w_n$ is a bounded (weight) sequence of positive numbers. We want to consider products of consecutive terms in this sequence. For $i,j\in \mathbb{N}$, define $M_i^j = w_i w_{i+1}\cdots w_{i+j-1}...
- 103k
2
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5
answers
453
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Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$
Can we find $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$, assuming $\{x_i\}_{i\in\mathbb{N}}$ is a set of positive real numbers?
Perhaps an easier question is, can we find $\sum_i x_i$ ...
- 56.6k
3
votes
2
answers
3k
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Is this formula expressing $\operatorname{erf} \left(\tfrac 1 {\sqrt 2}\right)$ as an infinite series already known?
Is the following formula already known?
$$\sum_{k=0}^{\infty} \frac{2^k k!}{(2k + 1)!} = \sqrt{\frac{e\pi}{2}} \operatorname{erf}\left(\frac{1}{\sqrt{2}}\right)$$
The right-hand side is known since ...
- 192
2
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0
answers
205
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Limits related to the floor function
Here I am still interested in the function $f(n,k)=\frac{2^{k}+1}{2^{n}+1}\left\lfloor \frac{2^{n}+1 }{2^{k}+1}\right\rfloor$ and a Tauberian property that I would like to check.
Let $\lambda>1$ be ...
- 275
-2
votes
1
answer
186
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An equality between $\pi$ and $\Gamma$ function [closed]
Consider the following equality:
$$\sum_{n=1}^{+\infty} (-1)^{n+1} \frac{(\frac{(2n-3)!!}{(2n-2)!!})^2*\frac{\pi}{2}}{n}=\frac{\Gamma(\frac{1}{4})^2}{2\sqrt{2\pi}}-\frac{2\sqrt{2}*\pi^{\frac{3}{2}}}{\...
- 1,693
2
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1
answer
101
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Recursion for the Chebyshev transform of $m^n$
Let
$$
R(n, q, m) = R(n-1, q+1, m) + \sum\limits_{j=0}^{q} (-1)^{q-j}R(n-1, j, m), \\
R(0, q, m) = (m-1)^q
$$
I conjecture that $R(n, 0, m)$ is a Chebyshev transform of $m^n$.
Examples of Chebyshev ...
- 30.7k
7
votes
1
answer
492
views
Suitable closed form for the A079501
Let $a(n)$ be A079501 (i.e., number of compositions of the integer $n$ with strictly smallest part in the first position).
The sequence begins with
$$
1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, ...
- 726
6
votes
2
answers
417
views
Asymptotic behavior of the "Cauchy square" series
$\renewcommand{\ge}{\geqslant}\renewcommand{\le}{\leqslant}$
$\newcommand{\pa}[1]{\left( #1 \right)}$
Let us take $\alpha > 0$, $x_1 := \alpha$ and for any $n \ge \mathbb{N}$,
\begin{align*}
\boxed{...
- 61
2
votes
1
answer
350
views
Simplify multiple sum involving rising factorials
(Previously asked in MSE, no answer even with bounty offer)
In the course of a calculation, I arrived at the quantity
$$ f(x,y,a,b)= \sum_{n,m,i,r,q,l\ge 0}\sum_{k=0}^{n+m} K_{n,m,i,r,l,q,k}\frac{(x)^{...
CommunityBot
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1
vote
1
answer
210
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Solving (or approximating) a certain delay differential equation
I'm interested in finding the (unique?) solution to the set of delay differential equations
$$f_w(w,x) = xf(w,w^2x)+w^3x^2f(w,w^4x), $$
$$f_x(w,x) = wf(w,w^2x)$$
With the initial condition $f(1,x) = e^...
CommunityBot
- 1
0
votes
0
answers
35
views
When does the Hermite series converge pointwise and when is it uniformly bounded?
Let $\gamma$ denote the standard Gaussian measure on the real line, and consider $f \in L^2(\gamma)$. Since the Hermite polynomials $\{H_n\}_{n \geq 0}$ are a complete orthonormal system, we may ...
- 342
4
votes
1
answer
241
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Approximating a finite sum with an integral
Consider the following sum (with $a$ being a real number and $N$ an even integer)
$$S(a, N) = \sum_{m=1}^{N/2} \frac{4}{N+1} \sin^2\left( \frac{2\pi m}{N+1} \right)\sin \left( 2 a \cos \left( \frac{2\...
- 5,559
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0
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154
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Interesting conjecture by Sequence Machine
Let $a(n)$ be A344960 (i.e., position of binary complement of $n$-th word in A341258). By definition, in order to calculate $a(n)$, we need to know A341258. Below we will correspond this sequence with ...
- 3,464
2
votes
1
answer
104
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Probability that at least one of the first $k$ right-to-left maxima of $\{1, ..., n+m\}$ contains a number from $\{n+1, ..., n+m\}$
Given a permutation $\sigma$ on $[n]=\{1, ..., n\}$, we say an element $i$ is a right-to-left maximum if $\sigma(i)=\max(\sigma(1), ..., \sigma(i))$.
Suppose we sample a random permutation from $\{1, ....
- 30.7k
1
vote
1
answer
124
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Factorial series $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$ and hypergeometric functions
For positive integer $D$, define $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$.
For $D \le 6$, sage finds closed form in terms of hypergeometric functions
at algrebraic arguments and fails to find closed ...
- 13.7k
5
votes
1
answer
571
views
Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution
Examples of infinite dimensional involutions
Edit 2/25/23, as suggested by YCOR below: (Start)
The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
CommunityBot
- 1
2
votes
1
answer
138
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Example of a conditionally convergent series $\sum_{n=1}^\infty b_n$ such that $n^2(b_n-b_{n+1})$ is bounded
Let $(b_n)_{n \in \mathbb{N}}$ be a real sequence such that $(nb_n)$ is bounded. I know that if the series $\sum_{n=1}^\infty b_n$ is conditionally convergent, then $(n^2b_n)_n$ is not bounded. But, ...
- 118k
1
vote
1
answer
167
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Structural differences between closed forms of two related infinite products?
In this question, I went quite a bit over the top, so I now tried to rephrase it in a much simpler way.
Take $a \in \mathbb{R}, s \in \mathbb{C}$ and:
$$\displaystyle C(s,a) := \prod_{n=1}^\infty \...
CommunityBot
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11
votes
3
answers
825
views
Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$
I was working with some Dirichlet series and I realized that I have never seen any general conditions under which
\begin{equation}
\sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\...
- 99.2k
-4
votes
0
answers
51
views
Probability of sequence of estimated values belonging to a set
Suppose I have estimated residuals of an ARCH(1) process: $\hat{\varepsilon}_1, \dots, \hat{\varepsilon}_n$ from the sample of length $n$. On the other hand, I have "true" residuals: $\...
- 15
0
votes
0
answers
40
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Formalization of sample convergence
Let's say I have a sample of $X_1, \dots, X_n$, where I know that $X_i$ were generated by some ARCH(1) process. It means that
$$X_i = \sigma_i z_i,$$
where $z_i \stackrel{iid}{\sim} N(0, 1)$ and $\...
- 15
1
vote
1
answer
76
views
Sequence with two restrictions
Does there exist a positive sequence $\{a_n\}_{n=3}^\infty$ that satisfies both of the following conditions?
$\sum_{k=3}^n\frac{a_k}{k}\to\infty$ as $n\to\infty$.
$\frac{\sum_{k=3}^n a_k\sqrt{\frac{\...
- 23.1k
1
vote
1
answer
126
views
Bound on a two-dimensional recursive series
For $n,k\in\mathbb{N}$, let $f(n,k)$ be defined as follows.
If $n \geq k$ and $n > 2$, then
$$
f(n,k) = \frac{k(n-k)}{n(n-1)}f(n-2,k-1) + \frac{k(k-1)}{n(n-1)}f(n-2,k-2) + \frac{n-k}{n}f(n-1,k) + \...
- 30.7k
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1
answer
51
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Boundedness from convergence of some demeaned sequence
Let $\{a_n\}_{n=1}^\infty$ be a sequence of real numbers and define the sequential average $\bar{a}_n=\frac{1}{n}\sum_{i=1}^n a_i$ and a demeaned sequence
$$
b_n=a_n-\bar{a}_{n-1},\quad n\geq 1,
$$
...
- 5,116
2
votes
0
answers
112
views
A sequence linked to irrationality
Let $0 < c < 1$ be a real number and $ x \in \mathbb{R}$. We define the sequence $(u_n)_{n \in \mathbb{N}}$ by :
$$u_0 = x$$
$$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
- 4,575
32
votes
3
answers
2k
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Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
Updated on Feb.16.2024
Fortunately for three of these series, Eqs. (1), (3) and (4), I have found a proof using classical methods which I placed in the Answers section below. (I doubt that there is a ...
- 2,210
0
votes
1
answer
61
views
Given $F[N,M]=\sum_{m=0}^{N-1}(-1)^{N-1-m}(m+1)^M)/(m!(N-1-m)!)$, show $F[N,N-1]=1$ and $F[N,M]=0$ for $M<N-1$ [closed]
The function defined by
$$
F[N,M]=\sum_{m=0}^{N-1}\frac{(-1)^{N-1-m}(m+1)^M}{m!(N-1-m)!}
$$
where $N,M$ are positive integers. I want to show
$$
F[N,N-1]=1,\ F[N,M]=0
$$
for $N>2$ and $M<N-1$. ...
- 30.7k
4
votes
0
answers
257
views
Four infinite series involving Riemann zeta function
Can you provide a proof for at least one of the claims given below?
It is known that $\pi=\displaystyle\sum_{n=1}^{\infty}\frac{3^n-1}{4^n} \cdot \zeta(n+1)$ where $\zeta$ denotes Riemann zeta ...
- 99.2k
2
votes
0
answers
84
views
Curious sequences associated to continuous fractions
Given a strictly positive initial rational number $x_0$ in $\mathbb Q_>$
we define a sequence $x_0,x_1,\ldots$ recursively by
setting $x_{n+1}=x_n+1/S(x_n)$
for $S(x)=a_0+a_1+\ldots+a_k$
where $[...
- 99.2k
29
votes
4
answers
2k
views
Advanced software for OEIS?
Is there (if not, why?) a software where I can input a sequence of integers, like into the OEIS, and then it makes some simple transformations on it to check whether the sequence can be obtained from ...
CommunityBot
- 1
4
votes
1
answer
510
views
Are the irrotational and solenoidal parts of a smooth vector field linearly independent?
Let $\textbf{F}\in \mathbb{R}^3$ be a smooth vector field for all space. It is well known using Helmholtz decomposition that we can decompose $\textbf{F}$ into two vector fields in $V$: $$\textbf{F} = ...
- 44.9k
4
votes
2
answers
319
views
Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$
I was wondering about sets that do not contain any $3$-term AP, and came to know that the official name of such a set is Salem–Spencer set. I was considering the question of counting the number of ...
- 3,413
3
votes
1
answer
213
views
How to find the coefficient of $x^k$ in the expression $\prod_{p=2}^n (1+xp) $
I got this general formula for $ n\in N$ (I showed it here)
$$\int_0^1 \left(\frac{x}{1-x} \ln x \right)^n dx=n \sum_{p=0}^{n-1}a(n,p+1) (-1)^{n-p} \zeta(p+2)+n! $$
where $a(n,k)$ is the coefficient ...
- 30.7k
1
vote
0
answers
86
views
Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$
I posted this question on Math Stack Exchange, but there were no helpful comments or answers
https://math.stackexchange.com/q/4874446/1298448
How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
- 194
2
votes
0
answers
29
views
joint rank sequences
An algebraic question I have been working on led me to a sequence that appears in OEIS as A186355: "adjusted joint rank sequence of $(f(i))$ and $(g(j))$ with $f(i)$ before $g(j)$ when $f(i)=g(j)$...
- 16.5k
0
votes
0
answers
59
views
Pairs of permutations such that $p(n)<2^k$ iff $n<2^k$
Let $p(n)$ be an arbitrary permutation of natural numbers such that $p(n)<2^k$ iff $n<2^k$.
Let $q(n)$ be an inverse permutation of $p(n)$.
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
...
- 3,464
5
votes
0
answers
80
views
Formula and smallest solution for the A260711
Let $a(n)$ be A260711 without initial $0$ (i.e., numbers of the form $x^2 - y^2$ with $x > y$ where $x$ and $y$ are odd, $x + y$ is a power of $2$).
The sequence begins with
$$
8, 16, 32, 48, 64, ...
- 3,464
8
votes
1
answer
359
views
You have $n$ rectangles of area $1$ and variable height. Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?
You have $n$ rectangles of area $1$ and variable height. Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$?
How to ...
CommunityBot
- 1
10
votes
1
answer
558
views
About certain infinite products with the property $f(a)=f(1/a)$
In the paper Transformations of infinite series, Bryden Cais gives the following transformations of infinite products
Theorem 4. If
$$
f(t) = \frac{\cosh(\pi t)-1}{\sinh(\pi t)}\frac{\cosh(2\pi t)+1}{...
- 33.9k
3
votes
0
answers
193
views
A problem about the series $\sin(n^p)$ [closed]
Prove that when $p>0,$ the series $$\sum_{n=1}^\infty \sin(n^p)$$
is divergent
- 49