Questions tagged [sequences-and-series]

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A discrete version of Poincaré's inequality

Given a (bounded) sequence $\{q_n\}_{n\geq 0}$ such that $\lvert q_n\rvert \leq 1$ for all $n \geq 0$ and $\sum_{n\geq 0} q_n = 0$. We can impose the condition that $\sum_{n\geq 0} \lvert q_n\rvert \...
Fei Cao's user avatar
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2 votes
1 answer
988 views

Switching order of limits in double sequences

By trying to extend certain limit properties of sequences from compact subsets to the entire set, I cam up with something that can be formed in the following question. Let $a_{mn}$ be a double ...
Marko Rajkovic's user avatar
2 votes
1 answer
322 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)^{...
Marcel's user avatar
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1 answer
107 views

Can we write $e^{-\alpha x}$ as $\sum_{n=0}^\infty c_n\left(\alpha\right)\gamma\left(x\right)^n$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$

Do there exist continuous functions $c_n\colon\mathbb R^+\to\mathbb R$ and $\gamma\colon\mathbb R^+\to\mathbb R$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$ and the following equation is true ...
Chetan Vuppulury's user avatar
3 votes
0 answers
221 views

Evaluating $\sum_{n=0}^\infty n^k n!$ in p-adics, and its connection to the summation of divergent series

Often, in the discussion of the regularization of the geometric series it is mentioned that $\sum_{n=0}^\infty p^n$ converges in the p-adics, and indeed, that it converges to $\frac{1}{1-p}$. I had ...
Caleb Briggs's user avatar
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3 votes
1 answer
202 views

Is normalcy preserved under the swapping operation?

Let $\mathbb{N}$ denote the set of non-negative integers. We say that a sequence $f:\mathbb{N}\to \{0,1\}$ is normal if every finite $\{0,1\}$-sequence appears in $f$. Let the swapping operation $\...
Dominic van der Zypen's user avatar
2 votes
1 answer
211 views

Identity involving double sum with binomials

(asked previously in MSE here) In the course of a calculation, I have met the following complicated identity. Let $A$ and $a$ be positive integers. Then I believe that $$ \sum_{B\ge A,b\ge a} (-1)^{a+...
Marcel's user avatar
  • 2,510
2 votes
1 answer
374 views

A conjecture relating an integral and a sum, the floor function and squares

I've found through evidence and have conjectured on a math publication that: $$\Big\lfloor\int_1^\infty (k^{1/(k^{1+1/\sqrt{x}})} - 1)dk\Big\rfloor = \Big\lfloor\sum_{k=1}^{\infty}k^{1/(k^{1+1/\sqrt{x}...
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2 votes
1 answer
286 views

Convergence of $\sum(n^p\sin^qn)^{-1}$

I've been recently interested in the problem of convergence of the function in such form: $\displaystyle \sum_{n=1}^\infty\frac1{n^p\sin^q n}$. I saw there's been discussion here when $p=3, q=2$ and $...
Samual's user avatar
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1 answer
72 views

Proof of yet another extension of deterministic variant of "(Almost) Supermartingale" convergence theorem

In this question, there is a proof for deterministic version of "Almost Supermartingale" Question: Can we extend [1] as following? If yes, can we prove it? Let the non-negative sequences be ...
user550103's user avatar
1 vote
0 answers
44 views

Permutation of nonnegative integers applied to the numbers $n$ whose binary expansion does not begin with $11$

Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and $$n=2^{b_1}(1+2^{b_2+1}(1+2^{b_3+1}(1+\cdots(1+2^{b_{\operatorname{wt}(n)-1}+1}(1+2^{...
Notamathematician's user avatar
6 votes
1 answer
216 views

Sequence A76132 eventually periodic modulo $2,3$ and $5$

Sequence A76132 starting as $1,1,2,4,10,36,218,\ldots$ of the OEIS is recursively defined by $a(1)=1$ and $a(n)=\sum_{k=1}^{n-1}a(n-k)^k$ for $n\geq 2$. It is eventually periodic of period 1,1 and 34 ...
Roland Bacher's user avatar
1 vote
0 answers
143 views

Pairs of functions with $\sum_{n} (f \circ g)(n) = \sum_{n} (g \circ f)(n) $

I was wondering there there are any pairs of functions $(f,g)$ such that $$\sum_{n=1}^{\infty} (f \circ g)(n) = \sum_{n=1}^{\infty} (g \circ f)(n) $$ on condition that they're not commutative with ...
Max Muller's user avatar
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-1 votes
1 answer
187 views

Is the unordered sum of measurable functions measurable?

Let $E$ be a normed $\mathbb R$-vector space and $I$ be a nonempty set. Remember that $(x_i)_{i\in I}\subseteq E$ is called summable if there is a $x\in E$ such that for all $\varepsilon>0$, there ...
0xbadf00d's user avatar
  • 161
5 votes
1 answer
177 views

A common combinatorial description for a certain type of recurrences

For integer-valued sequences $(x_n)_{n=0}^\infty$, consider recurrences of the form $$x_n=ax_{n-1}+(bn+c)x_{n-2} \tag{$*$}\label{star}$$ for $n\ge2$, where $a,b,c$ are integers. There seem to be many ...
Iosif Pinelis's user avatar
10 votes
1 answer
366 views

Looking for a "clever" argument for a $q$-series identity

Consider the below $q$-series identity. One of the things I like about this expansion is how nicely the difference on the left hand side factors to the right hand side of the equation. $$\prod_{k\geq1}...
T. Amdeberhan's user avatar
2 votes
0 answers
154 views

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 $x_{n-1}$. ...
Augusto Santi's user avatar
1 vote
0 answers
73 views

Sum of changing termes serie [closed]

I am looking towards which value tends the sum of the inverses of the odd numbers going from 1 / (2k + 1) to 1 / (4k-1) when k tends towards + infinity : for k = 1 there is just 1/3 for k = 2 it is ...
Fabien's user avatar
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7 votes
3 answers
975 views

Question on OEIS A000085

The OEIS sequence A000085 is defined by $$ a_n \!=\! (n-1)a_{n-2} + a_{n-1} \;\text{with }\; a_0\!=\!1, a_1\!=\!1.$$ If $n$ of the form $b^2-b+1, b \in \mathbb{N}, b > 2, \;\text{then: }\;$ $$ \...
Mike's user avatar
  • 359
7 votes
2 answers
465 views

Asymptotic behavior of $\sum_{k=1}^{\infty} \sqrt{\max\{1 - k^2/x^2,0\}}$ as $x\to\infty$

Let $f:\mathbb{R}\to\mathbb{R}$ be the function $$f(x) = \sum_{k=1}^{\infty} \sqrt{\max\left\{1 -\frac{k^2}{x^2},0\right\}}.$$ Numerical experiments suggests that there exists $n\in\mathbb{N}$ and a $...
Onur Oktay's user avatar
  • 2,263
1 vote
1 answer
171 views

Relation between $\sum_{k\ge 0}\binom {n+k}{k}a_k $ and $\sum_{k\ge 0}\binom {n+k}{k}\frac{a_k}{k}$

Let $\{a_k\}(k\ge 0)$ be a sequence of nonzero real numbers which changes signs infinitely often. Suppose $|a_k|\to 0 $ and $|a_k|$ decreases fast. Let $n$ be a positive integer. What's the relation ...
Beta's user avatar
  • 365
5 votes
3 answers
296 views

Closed formula for $(-1)$-Baxter sequences

The number of the so-called Baxter permutations of length $n$ is computed by $$a_n=\frac1{\binom{n+1}1\binom{n+1}2}\sum_{k=0}^{n-1}\binom{n+1}k\binom{n+1}{k+1}\binom{n+1}{k+2}.$$ There has also been a ...
T. Amdeberhan's user avatar
2 votes
1 answer
291 views

Difference between $n$-th and $(n-1)$-th composite numbers

Let $f(n)$ = 1 if $n$ belongs to A014689, $\operatorname{prime}(n)-n$, the number of nonprimes less than $\operatorname{prime}(n)$. Here $\operatorname{prime}(n)$ is the $n$-th prime number, $\...
Notamathematician's user avatar
1 vote
0 answers
58 views

A lower bound for a sum related to the $j$-invariant function

There are some days that I am thinking in the following problem. For any positive integer $x$, let $t(x)$ be a real number which a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \...
Jean's user avatar
  • 515
3 votes
3 answers
442 views

Growth of the coefficients of the inversion of the $j$-invariant function

We have the $j$-invariant defined as I have that $$ j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k, $$ where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$. The inversion ...
Jean's user avatar
  • 515
6 votes
0 answers
196 views

Where to cut off a double sum?

I have to compute a double infinite sum to within a given accuracy $\epsilon$. Let us say the sum is of the form $$\sum_{m\geq 1} \sum_{n\geq 1} \frac{a_{m,n}}{m^2 n^2 \max(m,n)},$$ where $|a_{m,n}|\...
H A Helfgott's user avatar
  • 19.3k
6 votes
2 answers
363 views

Sequence of $k^2$ and $2k^2$ ordered in ascending order

Let $\eta(n)$ be A006337, an "eta-sequence" defined as follows: $$\eta(n)=\left\lfloor(n+1)\sqrt{2}\right\rfloor-\left\lfloor n\sqrt{2}\right\rfloor$$ Sequence begins $$1, 2, 1, 2, 1, 1, 2, ...
Notamathematician's user avatar
1 vote
1 answer
101 views

Simplification of $\sum_{m=0}^\infty \text B_z(m+a,b-m)x^m,\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}$ in terms of Kampé de Fériet function

Here is the goal sum where the Pochhammer Symbol with the Incomplete Beta function series $$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
Tyma Gaidash's user avatar
2 votes
1 answer
283 views

Are there any published studies on cases of infinite sums for which the Euler–Maclaurin summation method yields the exact evaluation?

The Euler–Maclaurin summation formula is as follows: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}\big{(}f^{(2k-1)}(n)-f^{(...
Max Muller's user avatar
  • 4,435
1 vote
1 answer
433 views

Closed form series for reciprocal cubic function

consider a cubic of the form f(x)=$x^3-2x+z$ Is it possible to derive a power series of coefficients for the function $x^y/f(x)$, for some $y=0,1,2...$ that does not require the use of Faà di Bruno'...
CarP24's user avatar
  • 327
7 votes
1 answer
1k views

$\sum a_n = 0$ but $\sum \frac{a_n}{n} = \infty$

I'm hoping to find an explicit construction for a sequence such that $\sum a_n = 0$ but $\sum \frac{a_n}{n} = \infty$, or a proof that one cannot exist. So far, I have a good idea of how we can ...
Caleb Briggs's user avatar
  • 1,662
13 votes
0 answers
299 views

Convergence of the series $\sum_{n=1}^\infty \frac{(2+\sin n)^n}{3^n n^a}$ for $a\in(0,1)$

This is inspired by this Math.SE question, for $a=1$. Borwein, Bailey, and Girgensohn pose in their book ([1,Problem 35]) as an open problem the convergence of the series $$\sum_{n=1}^\infty \frac{(2+\...
Clement C.'s user avatar
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2 votes
0 answers
213 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 ...
Notamathematician's user avatar
5 votes
1 answer
655 views

Is this infinite product entire?

Let $(z_i)$ be a square-summable sequence which is even summable but not absolute summable, i.e. $\sum_{i=1}^{\infty} \vert z_i \vert = \infty$,$\sum_{i=1}^{\infty} \vert z_i \vert^2 < \infty$ and $...
Guido Li's user avatar
4 votes
1 answer
298 views

Taylor coefficients of Hadamard product

I imagine this to be a very classical question in complex analysis: Consider the Hadamard product $$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$ where $E_1(z):=(1-z)e^z$ is the first elementary ...
Guido Li's user avatar
3 votes
2 answers
278 views

Does my construction always result in a stationary Poisson point process of intensity $1$? How so?

My construction is as follows: Let $X_k$ be a real-valued continuous random variable centered at $k$ (an integer), having distribution $F_k(x,s)$ where $k$ is the location parameter and $s$, a ...
Vincent Granville's user avatar
2 votes
1 answer
110 views

Modulo $2$ binomial transform of A243499 applied $k$ times

Let $m \geqslant 1$ be a fixed integer. Let $f(n)$ be A007814, exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. ...
Notamathematician's user avatar
0 votes
1 answer
149 views

Modulo $2$ binomial transform of A124758

Let $f(n)$ be A153733, remove all trailing ones in binary representation of $n$. Here \begin{align} f(2n)& = 2n\\ f(2n+1)& = f(n)\\ \end{align} Then we have an integer sequence given by \begin{...
Notamathematician's user avatar
1 vote
0 answers
56 views

Inverse modulo $2$ binomial transform of generalised A284005

Let $m \geqslant 1$ be a fixed integer. Let $\operatorname{wt}(n)$ be A000120, $1$'s-counting sequence: number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Let $f(n)$ be A007814, ...
Notamathematician's user avatar
8 votes
0 answers
513 views

Concave and other bounded functions: Series representation and converging polynomials

Main Question Suppose $f:[0,1]\to[0,1]$ is continuous, polynomially bounded, and belongs to a large class of functions (for example, the $k$-th derivative, $k\ge 0$, is continuous, Lipschitz ...
Peter O.'s user avatar
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1 vote
0 answers
154 views

Open tours by a biased rook (proof verification)

Related questions: Number of open tours by a biased rook on a specific $f(n)\times 1$ board which end on a $k$-th cell from the right Sum with products turned into subsequences Combinatorial ...
Notamathematician's user avatar
2 votes
2 answers
170 views

Modulo $2$ binomial transform of $m^n$

Let $m \in \mathbb{R}$. Let $f(n)$ be A007814, exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. Let $g(n)$ be ...
Notamathematician's user avatar
5 votes
1 answer
212 views

Absolute summability of multiplication operators on $\ell_p$

A linear bounded operator $T:X\to Y$ between Banach spaces is called absolutely summing if for every unconditionally convergent series $\sum_{i\in\omega}x_i$ in $X$ the series $\sum_{i\in\omega}\|T(...
Taras Banakh's user avatar
  • 40.7k
1 vote
2 answers
146 views

Natural boundary with non-zero "thickness"

Many series (in fact, "most" Taylor series) have a natural boundary at the unit circle. This boundary is only 1 dimensional, in the sense that only along the unit circle is it true that ...
Caleb Briggs's user avatar
  • 1,662
3 votes
0 answers
145 views

Combinatorial interpretation of inverse modulo $2$ binomial transform of A284005

My question is related to the following: Sum with products turned into subsequences We have an identity $$a(n, -1) = \sum\limits_{j=0}^{2^{\operatorname{wt}(n)}-1}(-1)^{\operatorname{wt}(n)-\...
Notamathematician's user avatar
9 votes
1 answer
534 views

Infinite series with inverse trigonometric functions

Consider the infinite series $$ F(s)=\sum_{k=1}^\infty \frac{1}{k^{2s+1} \sin(k \pi \sqrt{2})} $$ Prudnikov Vol.1 , gives a result for this sum in 5.4.16.13 for $s=1.$ $$ F(1)=-\frac{13 \pi^3}{360 \...
Paul's user avatar
  • 91
3 votes
1 answer
290 views

Sum with products turned into subsequences

Let $p, q \in \mathbb{Z}$. Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and $$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$...
Notamathematician's user avatar
7 votes
1 answer
690 views

One conjecture by sequencedb.net

Let $a(n)$ be A214973, number of terms in greedy representation of $n$ using Fibonacci and Lucas numbers. Let $b(n)$ be A329320, sequence which arises from attempts to simplify computing of A329319. ...
Notamathematician's user avatar
1 vote
0 answers
58 views

Subsequences related with square table

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 $...
Notamathematician's user avatar
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 $...
Notamathematician's user avatar

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