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3
votes
2answers
290 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 $0$?
0
votes
1answer
117 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 ...
1
vote
0answers
149 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 ...
7
votes
1answer
1k 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 ...
4
votes
2answers
856 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 ...
0
votes
0answers
317 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 ...
5
votes
1answer
302 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 ...
2
votes
0answers
135 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 ...
14
votes
2answers
2k 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 ...
0
votes
1answer
199 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 ...
1
vote
1answer
186 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 ...
0
votes
2answers
255 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 ...
0
votes
0answers
227 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 $A$ ...
3
votes
4answers
811 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" ...
8
votes
4answers
670 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 ...
5
votes
1answer
408 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 ...
4
votes
2answers
297 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 ...
8
votes
0answers
396 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 ...
0
votes
1answer
198 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 $$ ...
1
vote
1answer
292 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 ...
0
votes
1answer
126 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
1
vote
1answer
192 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 ...
1
vote
2answers
522 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 ...
4
votes
0answers
218 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 ...
1
vote
1answer
129 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 $gcd(p,q) = 1$ ...
1
vote
2answers
502 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 ...
1
vote
1answer
402 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 ...
6
votes
2answers
334 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 ...
9
votes
1answer
454 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 $(x_0, x_1, \dots)$ ...
7
votes
3answers
535 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 ...
4
votes
0answers
251 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 $s=1,$ ...
4
votes
1answer
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.
1
vote
1answer
353 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 ...
8
votes
2answers
1k views

Cesaro means and Banach limits

Consider the class of bounded sequences to which every Banach limit (non-negative shift-invariant continuous functional on $l^\infty$ taking convergent sequences in the usual sense to their limits) ...
0
votes
1answer
132 views

An operator realizing the Borel transform

Let $y(z) = \sum_k y_k z^k$ be a holomorphic function in a vicinity of the point $z=0$. Define its Borel transform $By$ as a function $By(z) = \sum_k \frac {y_k}{k!} z^k$. The well-know formula ...
0
votes
0answers
504 views

“closed form” finite sum

If a finite sum has a definite integral representation, for which it can be proved the underlying indefinite integral is not an elementary function, then does this imply the original finite sum can ...
2
votes
1answer
322 views

Is it true that $c_0(X)^* = \ell_1(X^*)$ ?

I'm trying to prove this that but I can't . Any help/reference ?
2
votes
0answers
200 views

Proving that an increasing iterative sequence increases at a decreasing rate

In this question Proving a sequence of integrals increases (iterated minimax distributions) Pietro Majer proved that $$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = ...
9
votes
2answers
319 views

Finding local patterns in a circular list

Consider a list $\boldsymbol{x}=x_0,x_1,\ldots,x_{n-1}$, which we consider to be circular by taking the subscripts modulo $n$. The entries in the list are distinct integers. A local pattern is a ...
0
votes
1answer
420 views

Sequence of smooth functions converging to sgn(x)

I'm looking for a sequence of smooth functions $f_i(x)$ converging to Sign$(x)$, each of which additionally have the following property: \begin{equation} f_i(x_1+x_2) = g_i(x_1, f_i(x_2)) ...
0
votes
3answers
790 views

Simplifying finite sum over 1/(ax+b)

Can I simplify: \begin{equation} \sum_{x=x_0}^{x_1} \frac{1}{ax+b} \end{equation}
1
vote
1answer
188 views

About one series. Are there some related special functions?

Hello, I have the following series: $$ \sum_{n=2}^\infty \frac{t^n}{\Gamma(a n)} = ?,\qquad t\ge 0, $$ where the parameter $a\in (0,1]$, $\Gamma$ is the Gamma function. When $a=1$, the above sum ...
8
votes
2answers
510 views

The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$

Let $\operatorname{ GPF}(n)$ be the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
5
votes
1answer
546 views

random hyperharmonic series

The Harmonic Series is defined as: $\sum_{n} \frac{1}{n}$ where $n=1,2,3,4....$. This series is known to be divergent. A generalization of this series can be made by raising each term to $p$: ...
3
votes
0answers
223 views

Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$)

I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the $\small lngamma() $ function. Precisely $$ \small f_p(x) = \sum_{k=0}^{\infty} ...
1
vote
0answers
221 views

Is there an infinite sequence of positive integers $a_n$ s.t. $\sum_{n=1}^\infty{\frac{1}{a_n}}$ converges and $a_n$ contains arbitrary long arithmetic progressions?

This is somewhat related to Erdős conjecture on arithmetic progressions Is there an infinite sequence of positive integers $a_n$ s.t. $\sum_{n=1}^\infty{\frac{1}{a_n}}$ converges and $a_n$ ...
6
votes
2answers
423 views

Alternating sums of GCDs

The exciting question on alternating sums of binomial coefficients triggered me to ask the following much simpler question (sorry if it is too simple, but I'm not a number theorist, so I must be ...
36
votes
2answers
3k views

Alternating sum of square roots of binomial coefficients

Let $$ c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}. $$ It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...
63
votes
7answers
5k views

Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?

Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$? (I think it is.) If so, how would one prove this? (To confirm: This is the power series for $e^x$, except with the ...
3
votes
1answer
337 views

Products of trigonometric functions with increasing frequencies

I am looking at weighted $L_2$ norms of a class of Littlewood polynomials, related to Walsh and Rademacher functions which made me look for pseudo-closed forms or computationally efficient expressions ...