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.
487
questions with no upvoted or accepted answers
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Question on globally convergent formulas for the Riemann zeta function $\zeta(s)$
Consider the following two formulas for $\zeta(s)$
$$\zeta(s)=\underset{K\to\infty}{\text{lim}}\left(\frac{1}{1-2^{1-s}}\sum\limits_{n=0}^K \frac{1}{2^{n+1}}\sum\limits_{k=0}^n \binom{n}{k} \frac{(-1)^...
- 1,061
2
votes
0
answers
136
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Closed form for the limit $\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{i=1}^n[\gcd(i,\lfloor\frac ni\rfloor)=1]$
If we define $f(n)$ as the number of coprime pairs $(i,\lfloor\frac ni\rfloor)$ for $i$ an integer from $1$ to $n$, then $f(n)\sim cn$ for a constant $c\approx 0.7883$.
Because $f(n)=\sum\limits_{i=1}^...
- 41
2
votes
0
answers
61
views
Odious twin locations related to the sequence based on $d(n) = n-d(d(n-1))-d(d(n-2))$
Let $\operatorname{wt}(n)$ be A000120, i.e. the number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $a(n)$ be the sequence of numbers $k$ such that $\operatorname{wt}(k)\...
- 3,442
2
votes
0
answers
144
views
The Laplace transform and the Lagrange compositional inversion formula
I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
- 9,931
2
votes
0
answers
231
views
Conjecture on a sieve of Flavius Josephus
Flavius Josephus's sieve: Start with the natural numbers; at the $k$-th sieving step, remove every $(k+1)$-st term of the sequence remaining after the $(k-1)$-st sieving step; iterate.
Some examples:
...
- 3,442
2
votes
0
answers
90
views
Equality of bivariate formal series
Is it possible to prove algebraically that the two series uniquely defined by the following equations are equal:
$L_1=uz+zL_1^2+z \partial_uL_1$ and
$L_2=uz+z^2+z L_2^2+2z^4 \partial_zL_2$
2
votes
1
answer
348
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Simplify multiple sum involving rising factorials
(Previously asked in MSE, no answer even with bounty offer)
In the course of a calculation, I arrived at the quantity
$$ 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)^{...
- 2,540
2
votes
0
answers
154
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The recurrence relation $x_{n+1}=x_{n}+x_{n-1}+c\,(x_{n},\,x_{n-1})$
I'm trying to study the behaviour of the recurrence relation
$$x_{n+1}=x_{n}+x_{n-1}+c(x_{n},x_{n-1})\;\;\;\;\;\;\;c,x_{n} \in \mathbb Z$$
where $(x_{n},x_{n-1})$ is the gcd of $x_{n}$ and $x_{n-1}$.
...
- 1,000
2
votes
0
answers
214
views
Two conjectures about generalised A329369
Let $m \geqslant 2$ be a fixed integer.
Let
$$\operatorname{wt}(n,m)=\operatorname{wt}\left(\left\lfloor\frac{n}{m}\right\rfloor,m\right)+n\bmod m, \operatorname{wt}(0,m)=0$$
Then we have an integer ...
- 3,442
2
votes
0
answers
90
views
Subsequence of Laguerre polynomials
Let $m\geqslant1$ be a fixed integer.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $...
- 3,442
2
votes
0
answers
51
views
Reduction of the general Lauricella hypergeometric function $F_B$ for identical parameters and variables
The Lauricella function $F_B^{n}$ of $n$ variables is defined as $$F_B^{(n)}(a_1, \ldots, a_n, b_1, \ldots, b_n, c; x_1, \ldots x_n) = \sum_{k_1, \ldots, k_n = 0}^\infty \frac{1}{(c)_{k_1 + \ldots + ...
- 121
2
votes
0
answers
89
views
A subsequence expressed in terms of a sum with a triangle
We have a sequence which generalize A329369:
$$a(2n+1, p, q) = a(n, p,q), a(2n, p , q) = pa(n, p,q) + qa(n - 2^{f(n)}, p,q) + a(2n - 2^{f(n)}, p,q), a(0, p, q) = 1$$
where $f(n)$ is A007814, exponent ...
- 3,442
2
votes
0
answers
196
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Uniform distribution of log(log((n!)!)) mod 1
Can it be shown that the sequence $\log(\log((n!)!))$ is uniformly distributed mod 1? I believe it should be, but I'm not certain if Stirling's approximation repeated twice combined with the ...
2
votes
0
answers
403
views
Conjecture: The sequence {$π(2n+1)!$} is equidistributed in the interval (0,1)
Let $n\in\mathbb{N}$.
From the book "Uniform Distribution of Sequences" (available here) by L. Kuipers and H. Niederreiter, (from pg. 8) I found that for any irrational $\theta$, the ...
2
votes
0
answers
226
views
Did anyone ever propose the distinction between "divergent to infinity" as opposed to "divergent but with finite average"?
There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences.
Still, in my view there is fundamental difference between divergent ...
- 9,312
2
votes
0
answers
58
views
Tail asymptotics of Durfee square identity
This post is related to the problem Asymptotics of a combinatorial series
According to the Durfee square identity:
$$\sum_{k \ge 0} \frac{q^{k^2}}{(q;q)_k^2} (q;q)_{\infty} = 1,$$
where $(q;q)_k$ is ...
- 51
2
votes
0
answers
121
views
Banach limit with added properties
Let $c=\{a:\mathbb{N}\rightarrow \mathbb{C}: \exists \alpha\in \mathbb{C} \textrm{ so that }
\lim\limits_{n\rightarrow \infty} a(n)=\alpha=:L_c(a)\}\subset \ell^\infty$, where $\ell^\infty$ is the ...
- 21
2
votes
0
answers
95
views
Log-convexity of Lassalle's sequence
Lassalle's sequence is defined by the recurrence $A_1:=1$ and for $n\geq2$,
$$A_n=(-1)^{n-1}C_n + (-1)^{n- 1}\sum_{j=1}^{n-1}(-1)^j\binom{2n - 1}{2j - 1}A_jC_{n - j}$$
where $C_k=\frac1{k+1}\binom{2k}...
- 41.8k
2
votes
0
answers
152
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How can collections of rational zeta series that are equal to $\sum_{n=2}^{\infty} (\zeta(n) - 1)^{p} $ be obtained?
It has been discovered long ago that
$$\sum_{n=2}^{\infty} \big(\zeta(n) - 1\big) = 1. \label{1} \tag{1} $$ More recently, a generalization of this result with binomial coefficients has been obtained: ...
- 4,575
2
votes
0
answers
161
views
Limit of infinite power tower $\lim_{n \rightarrow +\infty}\frac{a_0^{a_1^{a_2^{^{.^{.^{a_{n}}}}}}}}{b_0^{b_1^{b_2^{^{.^{.^{b_{n}}}}}}}}$
Let $\{a_n\}_{n \in \mathbb{N}}$ be a sequence of natural numbers. Let us define a function which roughly "make a tower of powers out of $\{a_n\}_{n \in \mathbb{N}}$", i.e.
$$F:\mathbb{N}^{\...
- 1,333
2
votes
0
answers
87
views
Rational zeta series and differential-difference equations
In an earlier question, I mentioned I was looking for generalizations of $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1. \qquad \qquad (1) $$
A variation of the above identity arises by ...
- 4,575
2
votes
0
answers
403
views
Sequences with high densities of primes: how to boost them to get even more and larger primes
I propose a methodology to help find large prime numbers with a much higher probability than picking up random numbers and testing them for primality. This would help speed up prime number generators ...
- 3,088
2
votes
0
answers
230
views
Could analytically deriving the next non-trivial zero of $\zeta(s)$ be made rigorous up to a fixed accuracy?
In this question., a very inefficient, yet rigorous analytic approach for finding the next prime was established. I wondered whether a similar approach could exist to find the next non-trivial zero ($\...
- 4,169
2
votes
0
answers
84
views
(Dis)continuity of periodic functions with non-summable Fourier series
Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$.
We assume moreover that the square-summable Fourier coefficients of $f$, ...
- 2,174
2
votes
0
answers
110
views
If $u_n \to u$ in $H^1_0(\Omega)$, does $\chi_{\{u_n = 0\}} \to g$ for some $g$ in some space, for a subsequence?
Let $\Omega$ be a bounded and smooth domain.
Suppose we have $u_n \to u$ in $H^1_0(\Omega)$. We know that for a subsequence, $\chi_{\{u_n = 0\}} \rightharpoonup f$ to some $f$, weak-* in $L^\infty(\...
2
votes
0
answers
156
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Questions about a certain sequence of naturals generated by primorials
I'm working on the following sequence of naturals (which is NOT listed in OEIS)
$$3,5,11,17,23,29,59,89,119,149,179,209,419,629,839,1049,1259,1469,1679,...$$
whose elements are generated this way
$$3=(...
- 1,000
2
votes
0
answers
209
views
show that sequence $\{(-1)^n\Upsilon_n\}$ is convergent and strictly decreasing
Edit: Few years ago, I have posted my claim on $\Upsilon$ function regarding prime number but recently I have observed, last observation turns false that's way, (by putting $\Upsilon$ value in ...
- 597
2
votes
0
answers
163
views
Product of sines to sum
I hope this is a research level question; it is to me at least, I'm a beginning researcher in the field of the Bethe Ansatz. In the expressions I'm considering, I stumbled across the following ...
- 129
2
votes
0
answers
85
views
Closed form for unusual recurrence
We have for $k>0$, $n>0$, $m\geqslant0$
$$p_k(n,m)=k(n-1)!\sum\limits_{s=0}^{n-1}\frac{p_{k-1}(s+1,m+1)+p_{k-1}(m+1,s)}{s!}$$
also
$$p_0(n,m)=\begin{cases}
(n-1)!,&\text{$n>0, m=0$}\\
0,&...
- 303
2
votes
0
answers
131
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How many divisors of $\phi(m)$ do not divide $m-1$?
Lehmer's totient problem asks if there exists a composite number $m$ such that $\phi(m)$ divides $m-1$. Lower bounds on $m$ has been established but we do not know if a solution exists. Clearly, if we ...
- 4,638
2
votes
0
answers
177
views
From Firoozbakht's conjecture to set interesting conjectures for sequences or series of primes
In this post we denote the $k-th$ prime number as $p_k$. I present two conjectures, the first about the asymptotic behaviour of a product and the other about an alternating series. I don't know if ...
2
votes
0
answers
493
views
Bound of Coefficients of Fourier Series of Composition
Let $f(x) = \sum_{n=0}^\infty f_ne^{inx} + \bar{f_n}e^{-inx}$ and $g(x) = \sum_{n=0}^\infty g_ne^{inx} + \bar{g_n}e^{-inx}$ where $f:\mathbb{R} \to \mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$. Both ...
- 1,129
2
votes
0
answers
165
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What about series involving strong primes?
I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' ...
2
votes
0
answers
185
views
Infinite products from the fake Laver tables-Now with no set theory
We say that a sequence of algebras $(\{1,\dots,2^{n}\},*_{n})_{n\in\omega}$ is an inverse system of fake Laver tables if for $x\in\{1,\dots,2^{n}\}$, we have
$2^{n}*_{n}x=x$,
$x*_{n}1=x+1\mod 2^{n}$,...
- 27.8k
2
votes
0
answers
105
views
Understanding proof of q-theta function expression
In arXiv:math/0309252v4 at the bottom of page 11, the following result is proposed
$$ b_1 \theta(b_0 b_1^{\pm};p) \frac{z^{-1}\theta(z^2;p)}{\theta(b_0 z^{\pm};p)\theta(b_1 z^{\pm};p)} = (p;p)^{-2}\...
- 181
2
votes
0
answers
300
views
Conjectured initial values of Inkeri's primality test for Fermat numbers
This is a repost of this question .
Can you provide a proof or a counterexample to the claim given below ?
First , we shall give a definition of the Inkeri's primality test for Fermat numbers :
...
- 2,673
2
votes
0
answers
333
views
Inverse of partial sums of general harmonic series
I would like to understand better the scaling of the following summation as a function of $r$ and $p > 1$:
$$
S_r(p) := \sum_{m=1}^{r} \left( \sum_{k=r-m+1}^{r} \left( \frac{k^q}{\sum_{k'=1}^{r}...
- 199
2
votes
0
answers
377
views
Is this double integral of Fourier series always real?
Consider $f(x)$ a function from $\mathbb{R^+}$ to $\mathbb{C}$ such that $f(x) \sim_0 x$ and $\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} x^2 f(x) dx=0$
Can we demonstrate that following integral is ...
- 1,121
2
votes
0
answers
80
views
Characterisation of Frobenius algebras via sequences
Given a commutative Frobenius algebra, finite dimensional over a field $k$.
We assume that the algebra is connected and in fact given by quiver and relations. Let $S$ be the unique simple modules of ...
- 26k
2
votes
0
answers
205
views
A sum with integer parts
Let $ \mathcal{A} $ be a set of reals such that $ \sum_{a \in \mathcal{A} } \frac{1}{a} = \infty $ and $ \sum_{a \in \mathcal{A} } \frac{1}{a^2} < \infty $. For instance, $ \mathcal{A} = \mathbb{N}^...
- 549
2
votes
0
answers
144
views
A sum involving fractional part function
I was exploring some sum when I came across this sum which I have no idea the value, here is the sum
Let $ N $ be an integer with the prime decomposition $ N = p_1^{k_1} p_2^{k_2} ... p_m^{k_m} $.
...
- 21
2
votes
0
answers
152
views
More on rearrangements of series . . .
Earlier I posted this question. First I'll quote the question before refining it and elaborating on it:
For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for ...
- 11.9k
2
votes
0
answers
187
views
Generalize upper semicontinuous regularization using Borel Hierachy
Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$.
...
- 603
2
votes
0
answers
463
views
The sum of an hydrogen atom related infinite series, continued
In continuation of this MO question: The sum of an hydrogen atom related infinite series.
Can the sum
$$\sum\limits_{n=1}^{\infty}\frac{n-\frac{1}{2}}{n}\left[\frac{\Gamma(n)}{\Gamma\left(n+\frac{1}{...
- 16.4k
2
votes
0
answers
247
views
Convergence of a tetration series
Let $0<a\neq 1$ be a fixed real number and denote by $a^{ \frac{x}{}}:=\mbox{uxp}_a(x)$ the ultra exponential function (ultra power) that is a unique extension of tetration (and also its linear ...
- 995
2
votes
0
answers
111
views
Linear topologies on the finite dual of the polynomial algebra
Let $\Bbbk[X]$ be the polynomial algebra in one indeterminate over a field $\Bbbk$, endowed with the primitive-like bialgebra structure, i.e. $\Delta(X)=X\otimes1+1\otimes X$ and $\varepsilon(X)=0$.
...
- 718
2
votes
0
answers
143
views
Numerical algorithm for extracting the coefficients of transseries
Assume a function $f(x)$ is given numerically for $x>0$, i.e. for any $x>0$ there is a numerical procedure to obtain $f(x)$ to any desired precision.
Also assume that the function $f(x)$ has a ...
- 996
2
votes
0
answers
82
views
Convergence of a sequence involving a truncated exponential
Let $n\in\mathbb{N}_{>0}$, $\gamma\in\mathbb{R}_{>0}$. Let $\{a_n\}_{n}$ and $\{b_n\}_{n}$ be two sequences defined as follows
$$
a_n := \sum_{k=0}^{n-1}{2k \choose k}\frac{1}{\gamma^{2k+1}} \...
- 2,682
2
votes
0
answers
163
views
Link of a power series by the Bernoullis for a Riccati equation to zonotopes?
On pg. 85 of The Rise and Development of Theory of Series up to the Early 1820s by Ferraro is a series soln. of
$$ d^2z/z = -x^2dx^2 $$
related to the reputed first appearance of a Riccati-type eqn.,...
- 9,931
2
votes
0
answers
143
views
Closed form for $\frac{1}{2 \pi z}\sum _{n=-\infty }^{\infty } \sum _{m=-\infty }^{\infty } e^{-z \sqrt{m^2+n^2}} \cos (m y) \cos (n x)$
I looking for a closed form of this double sum, which is not tractable with mathematica.
$$\frac{1}{2 \pi z}\sum _{n=-\infty }^{\infty } \sum _{m=-\infty }^{\infty } e^{-z \sqrt{m^2+n^2}} \cos (m y) ...
- 21