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1answer
182 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
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2answers
237 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
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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
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4answers
809 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
666 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
398 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
295 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
387 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
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1answer
291 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
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1answer
191 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
516 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
217 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
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2answers
486 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
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1answer
386 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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votes
2answers
326 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
443 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
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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
246 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
345 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 ...
7
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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
124 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
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0answers
488 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
312 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
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0answers
197 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
415 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
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3answers
766 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
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1answer
187 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
509 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
526 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
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0answers
222 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
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0answers
219 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 $a_n$ ...
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2answers
422 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 ...
37
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 ...
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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
334 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 ...
4
votes
1answer
421 views

Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418

I am not familiar with newforms, so this may not make any sense. OEIS sequence A116418 is Expansion of a newform level 18 weight 3 and character [3] Numerical evidence suggest that up to $10^5$ $$ ...
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0answers
839 views

Regularizing the divergent sum $1^k + 2^k + \cdots$

EDIT: Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$ I was looking at ...
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2answers
822 views

Divergence of Dirichlet series

Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge? I asked ...
4
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0answers
770 views

Has n^2*|sin(n)| limit? [closed]

Is there any sub-sequence $n_k$ of natural numbers such as $\lim(n_k^2|\sin(n_k)|) = 0$ ? when $k$ tends to infinity. In other words, does $\lim(n^2|\sin(n)|)$ exist (equal to infinity)? or there is ...
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2answers
2k views

Defining the slowest divergent series

This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is: I know that a method of slowing a divergent series of positive reals is to replace the $n$-th ...
2
votes
2answers
1k views

Ratio of Sequences Sum Inequality

I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
1
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2answers
333 views

Asymptotics of Fourier coefficients of power-type functions

I would like to understand the asymptotic behaviour of the Fourier coefficients of power type functions $f(t) = |t|^{-\alpha} 1_{[-\pi, \pi]} \qquad 0 < \alpha<1.$ I suppose this is a classic ...
1
vote
1answer
215 views

exponential sum - general cases [closed]

hello community. I need some help given $k = \sum A_q e^{ia_q s}$ where $k$, and $s$ is known. Can $A_q$ s now be expressed in terms of $a_q$ . any help in that direction will be appreciated.
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0answers
154 views

Are there infinite sequences of rational cubes whose first differences are positive squares?

This is related to How many sequences of rational squares are there, all of whose differences are also rational squares? Are there infinite sequences $a_n$ of rational cubes whose first ...
1
vote
1answer
319 views

Limit of an infinite series, related to (generalised Newton's) binomial expansion

How to calculate the infinite sum of the following series, related to binomial expansion for rational number, $r$: ...