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7
votes
4answers
1k views

Unique limits of sequences plus what implies Hausdorff?

It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see here) and it is also known that unique limits for nets implies Hausdorff. What I am ...
18
votes
3answers
1k views

Zeroes of the random Fibonacci sequence

Let X_n be the "random Fibonacci sequence," defined as follows: $X_0 = 0, X_1 = 1$; $X_n = \pm X_{n-1} \pm X_{n-2}$, where the signs are chosen by independent 50/50 coinflips. It is known that ...
0
votes
1answer
185 views

Distribution wanted

I have a centered random variables $Y_1,Y_2$ which first 10 moments are given respectively by $$ 0, 1, 0, 6, 0, 90, 0, 2520, 0, 113400 $$ $$0, 1, 0, 32/3, 0, 36847/100, 0, 436879364/15435, 0, ...
0
votes
0answers
140 views

Asymptotic behaviour of a recursively defined sequence

I encounter a problem in which I need to characterize the asymptotic behaviour of a sequence. $\{s_{n,k}\}$ is a stationary probability distribution, i.e., ...
2
votes
4answers
914 views

Closed-form for modified formal power series

This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series. We start with a formal power ...
3
votes
0answers
657 views

Method for variable substitution in multiple summation

I want to ask: is there any general method for variable substitution in multiple summation? For example in the following equation a new variable $\lambda=n+m-2\mu$ is introduced to transform the LHS ...
0
votes
0answers
274 views

Infinite Product

Hi there, It has been long time since I had my math classes. However, I am trying to find a close form expression (if exists) of the following infinite product $f(n) =\prod\limits_{i=n}^{\infty} ...
4
votes
2answers
587 views

Asymptotics for the number of ways to sum primes such that the sum is <= n

Hello! Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq ...
2
votes
0answers
420 views

recursive sum of products of bessel functions

I have found a way to sum products of bessel functions in the form $$S_\ell(x,y)=\sum_{n=-\infty}^\infty (-1)^{n+\ell} I_{\ell-2n}(x)I_n(y)$$ recursively, i.e. once $S_0(x,y)$ is found, via the ...
0
votes
0answers
202 views

Square variation norm and non-negative, non-increasing sequences

I am trying to understand the properties of square variation, namely, the possibility of preserving it under certain operations. I am following Albiac & Kalton's book: Let $J$ stand for the usual ...
2
votes
1answer
314 views

Can averaged limits of sequences be realized as limits of sequences?

Let a summation method take a sequence $(x_n)$ to a net $(y_\alpha)$, where $\alpha$ runs over a partially ordered set, $y_\alpha=\sum c_{\alpha,n}x_n$ ($c_{\alpha,n}\geq 0$, $\sum_n c_{\alpha,n}=1$ ...
17
votes
2answers
2k views

Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ and $\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty$?

Is there any sequence $a_n$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 <\infty$ and $$\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty\quad?$$ See ...
9
votes
3answers
2k views

Ramanujan's eccentric Integral formula

The wikipedia page on Srinivasa Ramanujan gives a very strange formula: Ramanujan: If $0 < a < b + \frac{1}{2}$ then, $$\int\limits_{0}^{\infty} \frac{ 1 + x^{2}/(b+1)^{2}}{ 1 + ...
11
votes
1answer
2k views

Ramanujan's Incorrect formula

I actually looked at one of my Questions (posted at MATH.SE) again and found a formula which actually Ramanujan had discovered. Ramanujan: If $\alpha$ and $\beta$ are positive numbers such that ...
1
vote
1answer
282 views

Maximum of a series of integrals of Hermite functions

Given the function $$f(A) := \sum_{n=1}^{\infty}\left( \int_A \varphi_0\varphi_n \right)^2,$$ where $A$ is any measurable subset of $\mathbb{R}$, and $\varphi_n$ is the $n$th Hermite function, I want ...
28
votes
8answers
3k views

Series whose convergence is not known

For most of the mathematical concepts I learn, it has more or less always been possible to find (at least google and find) unsolved problems pertaining to that specific concept. Keeping a bag of ...
0
votes
0answers
445 views

sum of the series $\sum_{n=1}^\infty \frac{(-1)^{(n-1)}}{\sqrt{n}}$?

Hi, We know that the series $\sum_{n=1}^\infty \frac{(-1)^{(n-1)}}{\sqrt{n}}$ is convergent and it is oscillating. and numerically it is almost 0.6048986434. I want to know what is the exact limit ...
0
votes
2answers
731 views

Functions defined as infinite products

Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too! The original motivation for this is the ...
0
votes
0answers
240 views

“Unbalanced” maximum length sequences?

Maximum length sequences (MLS) are a type of pseudorandom binary sequences with specific properties (see Wikipedia: Maximum length sequence, or m linear feedback shift register. Properties that hold ...
1
vote
3answers
1k views

Sums of uncountably many real numbers [closed]

Suppose $S$ is an uncountable set, and $f$ is a function from $S$ to the positive real numbers. Define the sum of $f$ over $S$ to be the supremum of $\sum_{x \in N} f(x)$ as $N$ ranges over all ...
5
votes
1answer
938 views

Is there a theory about these kinds of recurrence equations? Is this a known formula?

(New information at bottom) Hi. For a while, I've been toying around with solving recurrence equations of the form $$a_1 = r_{1,1}$$ $$a_n = \sum_{m=1}^{n-1} r_{n,m} a_m$$ What are these kind of ...
11
votes
5answers
2k views

Upper bounds for the sum of primes <= n

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...
8
votes
1answer
536 views

Generalized Vieta-product

It's known that $$S_2={2\over\pi} = {\sqrt{2}\over 2}{\sqrt{2+\sqrt{2}}\over 2}{\sqrt{2+\sqrt{2+\sqrt{2}}}\over 2}\dots$$ The terms in the product approaches 1, the same holds for the following ...
2
votes
1answer
367 views

Series approximation(s) of a difficult recursive equation

New user here. I'm working on trying to get asymptotic solutions to the following recursive function: $f(r)=\frac{1}{r-k}\lgroup\sqrt{\frac{2}{k^2-1}}+\sqrt{\frac{1}{2k^2-1}}\rgroup$ (Eqn. 1) ...
3
votes
1answer
662 views

Asymptotic behaviour of a sequence

Hello, I am interested in some kind of sequence that are "not finitely recurrent". Let $a_i$ be a sequence taking values in $\{0,1\}$. Consider the sequence $(u_i)$ such that $u_0=1$, and for any ...
7
votes
4answers
1k views

Determining the asymptotic behavior of a series

I am trying to determine the behavior of the following series as $n\to\infty$. Let $0<\mu<1$ be fixed and for every positive integer $n\geq 1$, consider the function $f_n(t)$ of a real variable ...
2
votes
2answers
683 views

Find the sum of a series.

It's easy to check that the sum $$ \sum_{n = 1}^{\infty}\sin{\frac{1}{2^n}} $$ is convergent. Can this sum be calculate precisely?
1
vote
1answer
389 views

Coefficient bounds of an inequality

Hello, Given positive integers $k$ and $n$. Are there upper bounds on coefficients $A$ and $B$ such that they depends only on $k$ (eg., $2 k^k$) and for all non-negative integer sequences ...
28
votes
3answers
2k views

Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?

Hi. Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway... Consider this problem. I've been trying to find a formula to expand the "regular iteration" ...
-1
votes
1answer
281 views

Derivative of Sum over Variable of derivative

I feel stupid for having to ask this, but does anybody have any idea how to handle $$\frac{d}{x}\sum_{n=k}^{g(x)}f(n,x)?$$ Example: $$\frac{d}{dx}\sum_{n=6}^{i^2+2i} \frac{1}{\ln{(i^2)}-\ln{\ln n}}.$$ ...
6
votes
3answers
858 views

Approximating e with 2s and 3s

How can I generate a series of 2s and 3s such that the average of the generated values (so far) is as close to e as possible? For example: ...
5
votes
2answers
447 views

Pseudo-alternate series

Suppose $(a_n)$ is a non-increasing sequence of positive real numbers and $\varepsilon_i = \{\pm 1\},\ \forall i \in \mathbb{N}$ such that $\sum\limits_{i=1}^\infty \varepsilon_i a_i$ is convergent. ...
1
vote
2answers
660 views

low-discrepancy sequences for sampling of distributions

Hi everybody, I am writing of physics simulation that traces charged particles. I need to set up these particles in phase space ( 3 space dimension + 3 momentum dimensions) following distributions. ...
5
votes
5answers
596 views

Can an integer or rational sequence satisfy some bounded order recurrence $\mod \ $ almost all primes but doesn't satisfy such in $\mathbb{Q}$?

Can an integer or rational sequence satisfy some bounded order recurrence $\mod \ $ almost all primes but doesn't satisfy such in $\mathbb{Q}$? The recurrences $\mod p$ can be different, possibly ...
0
votes
1answer
428 views

Sequences, semigroups, addition formulae.

I am interested in the efficient computability of sequences. Is it possible some ``interesting sequences'' be computed via addition formulae/semigroup operation? Here is an artificial example. ...
4
votes
2answers
2k views

power series of the reciprocal… does a recursive formula exist for the coefficients. [closed]

Hello If $f(x)=\sum _{n=0}^{\infty } b_nx^n$, and $\frac{1}{f(x)}=\sum _{n=0}^{\infty } d_nx^n$. Then the coefficients of the reprical of f can be written down. The first few terms are: $d_0 = ...
4
votes
0answers
665 views

Computability of OEIS A034891 …partitions of n into prime parts (1 included)

On the seqfan mailing list RGWv gave short algorithm for computing A000041 number of partitions of n the partition numbers: f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == ...
0
votes
2answers
400 views

“Exotic” Banach spaces of sequences

Does there exist a linear subspace of $\mathbb C ^{\mathbb N}$ that can be endowed a Banach space topology that is not finer than the locally convex topology of pointwise convergence? Best, Martin
10
votes
11answers
2k views

Longest coinciding pair of integer sequences known

There are arbitrarily many pairs of integer sequences (of arbitrary origins) that coincide upto an $N$ but differ for an $n > N$. I assume, the coincidence will be considered accidentally then by ...
0
votes
3answers
1k views

Inverting a power series? … Cornish Fisher

Hello In the derivation of the cornish fisher expansion, the following equation is obtained: $$ \sum_{n=2}^{\infty} b_n H_{n-1}(x_\alpha) = \sum_{j=1}^{\infty}\frac{(x_\alpha - ...
9
votes
3answers
561 views

Convergence of alternating harmonic sums

I owe the idea of asking this question to Max Muller and his curiosity. What is the set of $\alpha$ in the interval $0\le\alpha < 1$ for which the alternating sum $$ ...
0
votes
2answers
712 views

Infinite sums of inverses of degree $3$ polynomials

In undergraduate courses we compute the sum $S$ of some series of the form $\frac{1}{P(n)}$ where $P(x)$ is some simple polynomial of degree $2$ with integer coefficients, by the following procedure: ...
3
votes
0answers
319 views

Finch's sequence over $\mathbb{F}_3$

In http://algo.inria.fr/csolve/seqmod3.pdf -- "Periodicity in sequences mod 3" Steven Finch (also cited in Sloane's OEIS A112683) defines the following sequences in $\mathbb{F}_3$: For each positive ...
6
votes
2answers
2k views

Sums of arctangents

$$ \begin{align} \arctan(x) & = \arctan(1) + \arctan\left(\frac{x-1}{2}\right) \\ & {} - \arctan\left(\frac{(x-1)^2}{4} \right) + \arctan\left(\frac{(x-1)^3}{8}\right) - \cdots \end{align} $$ ...
45
votes
4answers
2k views

Nonexistence of boundary between convergent and divergent series?

The following is a FAQ that I sometimes get asked, and it occurred to me that I do not have an answer that I am completely satisfied with. In Rudin's Principles of Mathematical Analysis, following ...
1
vote
1answer
398 views

Matrix version of Berlekamp Massey algorithm

What are the most obvious generalizations of Berlekamp Massey algorithm [1] to matrix sequences? [1] Massey, J. L., "Shift-register synthesis and BCH decoding", IEEE Trans. Information Theory ...
4
votes
4answers
702 views

Two Equal Series?

Hey all, I'd like to know if anyone has a link to a proof of the following? Take two infinite sequences $a_n$ and $b_n$ such that $$\sum_{n=1}^\infty a_n^s=\sum_{n=1}^\infty b_n^s=finite$$ for all ...
7
votes
1answer
585 views

How to rearrange only negative part of a conditionally convergent series to get any sum greater then initial?

Suppose that $\sum^\infty_{n=1} u_n = s,$ where the series converges conditionally, and $s'>s$. How to prove the existence of such a permutation $\sigma,$ such that 1) $u_n\geq 0 \rightarrow ...
4
votes
1answer
445 views

mertens-function in the light of divergent summation - what summation method were best adapted

Just reading about the Mertens-function in the other thread Mertens function I remember an earlier attempt to apply divergent summation to the series which is constructed of the Moebius-function at ...
0
votes
1answer
472 views

Series of squared Fourier coefficients

Hi, if the Fourier series development of $g(t)$ (periodic, $C^\infty$) is $$ g(t)=\sum_{-\infty}^{+\infty}a_n e^{in\omega t} $$ does the series $$ \sum_{-\infty}^{+\infty}\frac{a_n^2}{n^2}? $$ ...