The sequences-and-series tag has no wiki summary.

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### Equally subspacing the support of a monotone function, maintaining its mean

SETUP:
Assume $f(\cdot)$ is continuous and strictly monotone decreasing on $[0,\infty]$, with $f(0)>0$ and $f(\infty)<0$.
Let $x_m$ be the solution of $\frac{1}{m}\sum_{i=1}^{m}f(ix)=0$, where ...

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**1**answer

256 views

### Optimality of p-Lebesgue Differentiation Theorem for Sobolev Functions

This is the third question in a series whose purpose has been to flesh out an example of the optimality of the p-Lebesgue differentiation theorem for Sobolev functions. This theorem says that for $f ...

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180 views

### sequence, such that sum of any combinations in the sequence does not equal another [closed]

Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.
...

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**1**answer

183 views

### Giving a general term of a recursive function, and upper bound for it

Let a constant $B \ge 1$, and let $l_1 = 0$, $b_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$.
Let $l_{t+1} = l_t + 1$ if $b_i < B$, and $l_{t+1} = l_t$ otherwise
Let ...

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**1**answer

343 views

### Efficient (divergent) summation for sum of zetas at negative arguments?

In a question in MSE (see bottom of my own answer) I'm considering the following series, depending on a parameter m:
$$ L(m) = -\zeta(1m)/1 - \zeta(2m)/2 - \zeta(3m)/3 - \ldots $$
where I want to make ...

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**1**answer

137 views

### Double series solution of wave equation

Let $u(x,y,t)$ be the solution of wave equation $u_{tt}=u_{xx}+u_{yy}, 0\lt x\lt 1, 0\lt y\lt 1, t\ge 0,$ $u(x,y,0)=(x-x^2)(y-y^2), u_t(x,y,0)=0$ and $ u(x,y,t)=0$ on the boundary of the square. Then
...

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**1**answer

390 views

### Extension of the Jacobi triple product identity

The Jacobi triple product and the mathematical identity of it is:
$$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$
I would ...

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**1**answer

122 views

### A series question related to solution of Laplace equation

Let $u(x,y)$ be the solution of the Laplace equation $\Delta u=0$ on the unit square $(0,1)\times (0,1)$ with boundary condition:
$$ u(x,1)=1, u(x,0)=0, u(0,y)=0, u(1,y)=0$$
The series solution is ...

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2k views

### $\sum _{k=0}^{\infty } \frac{1}{(k+m) k!} \equiv 1$ for $m=2$

I changed the title and added revisions and left the original untouched
For this post, $k$ is defined to be the square root of some $n\geq k^{2}$. Out of curiousity, I took the sum of one of the ...

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**1**answer

135 views

### Distorted Newtion binomial

This is a cross-posting of a MSE question (which did not receive any feedback there so far).
Let $\varepsilon >0$, with $\varepsilon \neq 1$. Consider the sequence $u_n$ defined by
$$
...

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**1**answer

250 views

### Does there exist a sequence of complex numbers such that…

The following question came up while I was working through an example:
Does there exist an $\ell^1$ sequence of complex numbers $a_n$, not all zero, such that $\sum_n a_n n^{-p} = 0$ for all $p ...

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**1**answer

1k views

### The unreasonable effectiveness of Pade approximation

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ...

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160 views

### Calculating $n$ for $\sigma(\sigma(n)-n) = n$ [redefined]

As in A072868 described by OEIS;
Defined by $\sigma(\sigma(n)-n) = n$.
Since these numbers are important in regard to many things, specially mersenne primes, since ${n-1 \over 2}\times ...

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**1**answer

335 views

### Limit of functions and asymptotic behaviour

Let us consider a sequence $(p_l)_l$ of polynomials on $[0,1]$ that converge uniformely, as $l\to \infty$, to a function $f$ defined on $[0,1]$.
I denote the polynomials $p_l(t) = \sum_{k=0}^{m(l)} ...

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**1**answer

366 views

### “Harmonacci” recurrence and identities for $\pi$

While playing with something totally irrelevant I stumbled upon the recurrence:
$$a_{n+1} = \frac{1}{a_n} + a_{n-1}$$
It turns out that given $a_0 = 1, a_1 = 1$,
$$lim \frac{a_{2n}}{a_{2n-1}} = ...

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**1**answer

72 views

### Series of quotients with perturbed denominator

Let $a_n>0$ and $b_n>0$ be two strictly declining sequences such that the series
$$\sum_{n=1}^\infty \frac{a_n}{b_n}$$ is convergent. For $\sigma>0$ define $$f^N(\sigma) = \sum_{n=1}^N ...

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352 views

### What was Lambert's solution to $x^m+x=q$?

I've been thinking about Lambert's Trinomial Equation quite a bit, and I want to see his solution. The only solution I could find was in Euler's form, and I still don't quite understand how he got ...

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**2**answers

277 views

### Given a sequence of real numbers,do the following conditions suffice to guarantee convergence to 0?

If $x_{a+1}$-$x_{a}$ converges to $0$ and $x_{2a}$-$2x_{a}$ converges to $0$ , does that imply $x_a$ converges to $0$?

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95 views

### Solution of certain forms of equations

I ask about a possible method to find the solution of algebraic equations of the form
$axⁿ+byⁿ+c=0$
where $a,b,c,x,y$ are real constants and $n$ is an integer. Maybe there is a simple method, but I ...

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128 views

### Limit of Sequence of unusual Prime Product

Let $p_n$ be the nth prime and $p_L$ be closest to its square root:
\begin{equation}
p_L^2 \approx p_n \approx x
\end{equation}
Let $\sigma \in Z^+$ be a positive integer constant. Define the ...

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**1**answer

825 views

### Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...

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**2**answers

572 views

### series expansion of the q-Pochhammer symbol

The following identity arose while I was working on a recent MO question:
$-\sum_{n=1}^{\infty}\frac{1}{n}\frac{(-x)^n}{1-x^n}=\sum_{n=1}^{\infty}\frac{1}{n}\frac{x^n}{1-x^{2n}}.$
I have no doubt ...

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296 views

### Series with iterated log's: does it converge?

This came up in our office today. Let
$$f(x) = \begin{cases} x & \mbox{if } x\leq 1 \cr x\cdot f(\ln(x)) & \mbox{otherwise}\end{cases}$$
Does this series converge?
$$ \sum_{n=1}^\infty ...

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271 views

### Computing the limit of a certain recursively defined sequence

The following is not exactly a research question (it was originated from manufacturing of exercises for calculus), and has no other motivation than explaining a phenomenon. I apologize if it is ...

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121 views

### Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture
It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is ...

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1k views

### The function $\sum_{0}^{\infty} x^n/n^n$

The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I ...

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185 views

### Good set theory in which to study ordinal-indexed sequences?

I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulative hierarchy. My ...

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**1**answer

153 views

### On methods for dealing with recursively defined sequences

Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$.
By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first ...

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192 views

### sequences - recurrence relation [closed]

I have to find the expression of $(y_n)$ defined by :
$$y_{n+1}=a y_n+b z_n+c$$
where $(z_n)$ is an arithmetico-geometric sequence :
$$z_{n+1}=d z_n+e$$
and $a,b,c,d,e$ real numbers.
Thank you ...

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221 views

### Convergence of a function in a metric space to its metric.

Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric:
If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ ...

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789 views

### Another Chicken or Egg: Sequence or Series

This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...

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**4**answers

632 views

### What is the theoretical interest of finding closed-form sols. of infinite series?

Hi,
I was reading this when I came across Gourevitch's conjecture.
My understanding is that solutions to these series are of practical interest. If one encounters such a series, being able to solve ...

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232 views

### An infinite set of identities using Stirling numbers 1st kind - are they all zero?

I have the following set of series involving the Stirling numbers 1'st kind and binomials, which can be understood as a set of dot-products of row- and column-vectors of two infinite matrices (where R ...

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**2**answers

272 views

### Evaluating a limit similar to the Euler constant

In the course of studying a certain complex-valued functional equation, I have had a need to evaluate the following limit:
$$\gamma_\mathcal{T}=\lim_{n\to\infty}\left(-\frac{i}{2}\sum_{k=1}^n ...

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368 views

### Composition of two formal series

There are two formal semi-infinite Laurent series
$$
f_+(z)=z+\sum_{k=2}^{\infty} a_k z^k
$$
and
$$
f_-(z)=z+\sum_{k=0}^{\infty} b_k z^{-k}
$$
Their composition (we assume that this composition ...

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**1**answer

186 views

### A series representation

How to find the end of a series representation of the product
$$
\prod_{\substack{i=1...\infty\\\ j=0...i\\\ k=0...j}}\frac{1}{1-x^{i-j}y^{j-k}z^{k}}?
$$
For example for product
$$
...

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274 views

### Maximal difference between k randomly drawn numbers from 1 to n – Looking for formula to sequence

Hello!
I have an interesting problem that seemed simple to me, but I'm unable to solve it on my own.
Suppose I am drawing k numbers out of n numbers labeled from 1 to n.
Considering all ...

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**1**answer

119 views

### what this type of series expansion is

I obtained a series expansions as this type
$$f(x)=g(x)^{\textstyle \sum_{i=0}^{n}\alpha_{i}x^{-i}+O\left(\tfrac{1}{x^{n+1}}\right)}$$
what is the exact name of this formula

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185 views

### Growth of a particular sequence

Not sure if this is research level? While analyzing a particular algorithm, I came across the following series:
$\sum_{k=1}^{m-1}\frac{3^{k}}{\prod_{i=1}^{k-1}2^{2^{i}}}$
Is there a closed form ...

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**2**answers

439 views

### Convergence of Newton series for sin ax

Let's define half discrete-analytic function as a function whose Newton series converges to that function for each $x>0$:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k f\left ...

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210 views

### $\sum_{p,q \text{ primes } p \le q} 1/(pq\log(pq))$

The sum $$ \sum\limits_{p,q \text{ primes } p \le q} \frac{1}{pq\log(pq)}$$
is related to a conjecture of Erdős about primitive sequences.
It converges because the sequence is primitive. If my ...

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121 views

### expanding the sqare of sum

If there any way to expand the following?
$$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}}$$
and more generally, a way to expand
$$\left(\sum_{i=1}^nx_i\right)^{\frac{p}{q}}$$
where $gcd(p,q) = 1$
...

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330 views

### Converting a recursive definition to an explicit one

Is there an explicit form for $a_x$ (whole numbers x) given that $a_x = \displaystyle\sum_{i=1}^{x-1} \binom{x-1}{i} a_i$?
I've listed out the first few terms:
for $x=0,1,2,3,4,5,6, 7$
we have $a_x ...

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**1**answer

237 views

### Generalization of Cauchy product

Hello,
working on some machine learning problem I end up facing a problem which looks like generalizing the notion of Cauchy product.
I briefly go back to Cauchy products before exposing my ...

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280 views

### Equidecomposable graphs, unimodality and asymptotics

I will call two graphs $G$ and $H$, $r$-equidecomposable (in analogy with Hilbert's third problem) if they can be written as unions of disjoint subgraphs
$$G\cong \bigsqcup_{i=1}^r G_i\quad ,\quad ...

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297 views

### Does every strictly increasing, unbounded sequence of positive real numbers contain arbitrarily long, finite subsequences which are “sort of increasing” or “sort of decreasing” (as defined below)?

Is the following true?
If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $(x_0, x_1, \dots)$ ...

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519 views

### The digit sum: $s(na)=s(nb)$

Not that I was serious about the following question, but I think it is a must-to-ask as a follow-up to this MO post.
For integer $n\ge0$, let $s(n)$ denote the sum of the digits in the decimal ...

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212 views

### Useful lower bound on an infinite sum

Fix integer $s.$ I have encountered the following infinite sum.
$$\sum_{k=0}^\infty \Pi_{l=1}^k\left[1-(1-2^{-l})^s\right]$$
Is there a useful lower bound on this expression? For instance, if $s=1,$ ...

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**1**answer

2k views

### sum calculation

I would like to calculate, or bound from above, the following sum
$$
\sum_{i=0}^n(n-2i)^p{p \choose i},
$$
here $p\geq 2$.
Any references are very welcome.
Thank you.

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**1**answer

301 views

### Cesaro means for $\alpha<1$ and Banach limits

I am interested in conditions in terms of standard scales of summation methods that guarantee the existence of an averaged limit for all almost convergent sequences. For the Cesaro summation method ...