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3 votes
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series representation of bivariate functions

Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g_k(x)h_k(y)$ for all $(x,y)$ in ...
Chao's user avatar
  • 53
3 votes
1 answer
379 views

Convergence of a power series

Consider the numbers $$a_n=\frac{1}{n+1}\sum_{k=0}^{n}\frac{2^{k-1}\binom{n+1}{k}B_k}{2^{s+k-1}-1}, \ n\geq0,$$ where $s\neq1;0;-1;-2;-3;...$ is a fixed real number, and the $B_k$ are the Bernoulli ...
L.L's user avatar
  • 463
3 votes
1 answer
232 views

An Euler-Mascheroni double sum

An interesting representation of the Euler-Mascheroni constant $$ \gamma~=~ \lim \limits_{n\to \infty} \sum \limits_{k,s=1}^n \frac{s-k}{k\left( s\,n +k\right)},\label{1}\tag{$*$}$$ can be proved ...
Karl Fabian's user avatar
  • 1,676
3 votes
1 answer
147 views

Number and asymptotic for cyclic sequences

Cyclic sequence is equivalence class of cyclic shift action. If $a = (a_1, ... , a_i)_c$ is cyclic sequence then $(a_1, a_2, \ldots a_{i-1}, a_i)_c = (a_2, a_3, \ldots, a_i, a_1)_c = \ldots = (a_i, ...
G H's user avatar
  • 123
3 votes
1 answer
128 views

Weaker version of the lemma of K.L. Chung

Let $\{u_n\}_{n\in\mathbb{N}}$ be a sequence of nonnegative real numbers (i.e., $u_n\geq 0$ for all $n\in\mathbb{N}$). Assume furthermore that, for some positive constant $C$, the following holds: $$...
giorgi nguyen's user avatar
3 votes
1 answer
89 views

A density lemma for families of sequences indexed by the unit interval

Suppose for every $x \in [0, 1]$, we have a subset $S_x$ of the natural numbers with asymptotic density $1$ such that if $n \in S_x$, there is an open neighbourhood $U$ of $x$ (depending on $x$ and $n$...
Nate River's user avatar
  • 6,155
3 votes
1 answer
205 views

Understand the properties of this function

We define a function $f(t):=\sum_{n=0}^{\infty}e^{-nt}= \frac{1}{1-e^{-t}}= \frac{e^{\frac{t}{2}}}{e^{\frac{t}{2}}-e^{-\frac{t}{2}}}=\frac{2e^{\frac{t}{2}}}{\sinh\left(\frac{t}{2} \right)}$ observe ...
Kreimer's user avatar
  • 31
3 votes
1 answer
373 views

Ability to have function sequence converging to zero at some points

Consider the continuous and non negative function $c : \mathbb R \to [0,1]$ defined by $$ c(x) = \begin{cases} \cos \frac{\pi x}{2} &\text{for } x \in [-1,1]\\ 0 &\text{otherwise} \end{cases}$$...
mathcounterexamples.net's user avatar
3 votes
1 answer
449 views

Prove that when converge, the following expansions are equal

Prove $f_1(x)=f_2(x)=f_3(x)$ when converge. $$f_1(x)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=0}^m\binom mk(-1)^{m-k}f(k)$$ $$f_2(x)=\lim_{n\to\infty}\binom xn\sum_{k=0}^n\frac{x-n}{x-k}\binom nk(-1)^...
Anixx's user avatar
  • 10.1k
3 votes
0 answers
454 views

Surprisingly difficult limit of a sequence

Is there an easy way to prove that $|\operatorname{Re}(a_n)| \to \infty$ where $a_n=\left(\frac{1}{2}+i\frac{\sqrt{7}}{2}\right)^n$? Of course $|a_n| \to \infty$, but we have $$ \operatorname{Re}(a_n)=...
J.Mayol's user avatar
  • 489
3 votes
0 answers
114 views

Choose a sub series of a random series, such that its expectation can be a given real number

Suppose $a>0$, and we have an infinite series of Bernoulli random variables $B_k$ with $$\mathbb{Pr}{\large[}B_k=1{\large]} = \frac{1}{1+e^{a\cdot 2^k}}$$ Then $$\text{E}\left[\sum_{k=-\infty}^{\...
Jone Sweden's user avatar
3 votes
0 answers
79 views

Some exercise on the regularity of a summability method

I was reading the book of Johann Boos "Classical and modern method in summability theory" and I came across an exercise from the Chapter 2 (page 50, exercise 2.3.15). Here is the statement ...
popa13's user avatar
  • 31
3 votes
0 answers
204 views

Infinite partial fraction expansions to compute fractional iterations and recurrences

Let say a function $f$ is defined iteratively over the set of positive integers, for instance $f(t+1)=f(f(t))$ or $f(t+1)=f(t)+f(t-1)$. Based on the recurrence relationship and initial conditions, how ...
Vincent Granville's user avatar
3 votes
0 answers
155 views

Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: https://math.stackexchange.com/q/1780149/52694 Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the ...
Mike Battaglia's user avatar
3 votes
1 answer
105 views

How to show monotonocity and the limit? [closed]

Let me reformulate my recent question. Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density: $$\phi(x) = C\left\{ \begin{array}{lcc} \sqrt{...
smyroosh's user avatar
3 votes
0 answers
275 views

Maximizing the discrepancy in Jensen's inequality for a certain function

Let $\underline{b}=\{b_1,\dots,b_n\}$ be a fixed sequence of positive numbers, and let $a>0$ be a parameter. Define $$ D(a;\underline{b}):=\frac{1}{\frac{1}{na}+\frac{1}{\sum_{i=1}^n b_i}} -\sum_{...
dima's user avatar
  • 959
2 votes
1 answer
362 views

more on "sinc-ing" integrals and sums

This is a follow up on the MO question here. I kept being fascinated and bemused by these functions. Denote $\text{sinc}(x)=\frac{\sin x}x$. Experiments suggest that $$\sum_{n=1}^{\infty}\text{sinc}^...
T. Amdeberhan's user avatar
2 votes
2 answers
314 views

Convergence of series related to partial fraction expansion of cotangent function

I am looking at the convergence of the series $$ \cos(t\theta) = \frac{\sin(\pi t)}{\pi} \cdot \Bigg[\frac{1}{t} + 2t \sum_{k=1}^\infty (-1)^k \frac{\cos(k\theta)}{t^2 - k^2}\Bigg].$$ Here $t\in\...
Vincent Granville's user avatar
2 votes
2 answers
261 views

Prove a family of series having integer coefficients

I encountered a certain family of infinite series in some work, which is given by $$F_r(x)=\frac1{2^r}\sum_{k=0}^r\binom{r}k\frac1{1+x(2k-r)^2}.$$ I've convincing date to believe the following is true,...
T. Amdeberhan's user avatar
2 votes
1 answer
1k 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
3 answers
821 views

Riemann series theorem and uncountable number of sums which sum to every value

I asked on MSE this question which I am going to copy-paste here: "Wikipedia: "In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century ...
user avatar
2 votes
1 answer
93 views

Does this condition on $f$ imply essential boundedness on compacts?

Let $f: \mathbb R \to \mathbb R$ be a nonnegative measurable function, and $\{q_n\}$ some enumeration of the rational numbers. Suppose for every $0 < r < 1$ it holds that $$\sum_{n = 0}^\infty r^...
Nate River's user avatar
  • 6,155
2 votes
1 answer
332 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
  • 21
2 votes
1 answer
210 views

asymptotic estimate for log-tan sum

I am finding the following first order estimate. Question. As $y\rightarrow\infty$, $$\sum_{n=1}^{\infty}\frac{\log n}n\,\arctan\frac{y}n\,\, \sim\,\,\frac{\pi}4\log^2y.$$ Is it true?
T. Amdeberhan's user avatar
2 votes
1 answer
309 views

Find better than $ 4^n\prod_{k=1}^{n-1}\cos^2(k)\sim e^{o(n)}$

Let $$u_n=\prod_{k=1}^{n-1}\cos^2(k)$$ then $$\frac1n \ln(u_n) = \frac1n\sum_{k=0}^{n-1} \ln(\cos^2(k)) \underset{n\to\infty}\longrightarrow \frac1{2\pi} \int_0^{2\pi} \ln(\cos^2(x))\,{\rm d}x = -\ln(...
Paul's user avatar
  • 1,503
2 votes
2 answers
218 views

Convergence for a non-linear second order difference equation

In my work, I need to study the convergence of sequence defined by the non-linear recurrence relation $$ u_0,u_1>0, \qquad \forall n\in \mathbb N, \; u_{n+2}=a\ln(1+u_n)+b\ln(1+u_{n+1}) $$ with ...
Paul's user avatar
  • 1,503
2 votes
2 answers
268 views

If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?

Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...
YC Su's user avatar
  • 605
2 votes
1 answer
152 views

The asymptotic behavior of $F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$, $b>a>0$

Let $b>a>0$. Given $ \lambda>0$, what is the asymptotic behavior of $$F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$$ as $\lambda\to 0^{+}$ and as $\lambda \...
Medo's user avatar
  • 852
2 votes
1 answer
200 views

Proof of a discrete isoperimetric inequality

The following inequality appears in the proof of certain isoperimetric-type inequalities for analytic functions in two dimensions: $$\sum_{m=0}^{\infty}\frac{|c_m|^2}{m+1} \leq \pi \left(\sum_{m=0}^{...
MathLearner's user avatar
2 votes
1 answer
230 views

Does exist a variant of Frullani's theorem valid for $f(x)=\pi(x)/x$ or $f(x)=\psi(x)/x$, where the numerators are prime-counting functions?

Frullani's theorem is a deep theorem in real analysis with applications, see the Wikipedia Frullani integral and other uses and contexts (see [2]). I wrote two imaginative examples of what can be ...
user142929's user avatar
2 votes
1 answer
69 views

Decaying of a certain ratio of binomial sums

Consider the two sequences $$a(n)=\sum_{k=1}^n\binom{n}k\sum_{j=1}^{k/2}\binom{k}{2j}\frac{(2j)!}{j!}$$ and $$b(n)=\sum_{k=0}^n\binom{n}k^2k!$$ QUESTION. Is this true? $$\frac{a(n)}{b(n)}\...
T. Amdeberhan's user avatar
2 votes
1 answer
246 views

Does this sequence contain a nonnegative number?

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
MSMalekan's user avatar
  • 2,118
2 votes
1 answer
120 views

Difference between finite partial sums from two divergent series

Fix a sequence $(r_i)_{i\in\mathbb{N}} \subseteq (0, 1)$ such that $\lim_i r_i=0$ and $\sum_{i\in \mathbb{N}} r_i=\infty$. According to the answer in this post, for any $c>0$ there exists $N,M\in\...
Sanae Kochiya's user avatar
2 votes
1 answer
183 views

Example of a conditionally convergent series $\sum_{n=1}^\infty b_n$ such that $n^2(b_n-b_{n+1})$ is bounded

Let $(b_n)_{n \in \mathbb{N}}$ be a real sequence such that $(nb_n)$ is bounded. I know that if the series $\sum_{n=1}^\infty b_n$ is conditionally convergent, then $(n^2b_n)_n$ is not bounded. But, ...
Kanydo Mat's user avatar
2 votes
1 answer
192 views

Does every real number $r\in [0,1]$ have a rational sequence $q_n\to r$ s.t. $q_n$ has (simplified) denominator $n$? [closed]

This seems pretty trivial but I can't seem to figure it out. I think it's obviously true, given an unconstrained convergent sequence we just have to add some filler elements, but I'm having trouble ...
J.R.'s user avatar
  • 291
2 votes
1 answer
200 views

Laguerre polynomial and series

Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$. Consider the sum $$\sum^\infty_{j=0} \frac{1}{(b-j)}L^{m}_j(x)$$ where $b\not\in\Bbb N,x>0$ and $m\in \Bbb N$. I have found this series ...
zoran  Vicovic's user avatar
2 votes
2 answers
431 views

Why is the following recurrent sequence convergent?

Let $a, b , c, d$ be reals. Define the sequence $(x_n)$ as: $$x_0 = a,\,\, x_1 = b$$ $$x_n = \left(1 - \frac{b^2}{n^2}\right)x_{n-1} + \frac{1}{n-1}\sum_{k=0}^{n-2}\binom{2n+1}{2k+1}^{-1} (x_{k+1}-x_k)...
A. PI's user avatar
  • 121
2 votes
1 answer
103 views

Are there theorems dealing with the "amount of oscillatory divergence" of series?

Are there a set of theorems dealing with "amount of divergence" series? Let me explain by example. The Dirchlet $\eta$ series $\sum_n (-1)^{n-1} n^{-x}$ converges when $x > 0$. We may say ...
Shree's user avatar
  • 203
2 votes
1 answer
260 views

Squaring a semi-convergent series

Let $S=\sum_{n=1}^\infty a_n$, be a semi-convergent series with $T=\sum_{n=1}^\infty a_n^2 < \infty$ and $\sum_{n=1}^\infty |a_n|=\infty$. Under which conditions are the following formulas valid? ...
Vincent Granville's user avatar
2 votes
2 answers
782 views

Weak version of Karamata's Tauberian theorem

I first posted this on mathematics. However, I got no answer there and it seems adapted here too. Also, it seems to be harder than I first thought. Karamata's Tauberian theorem states the following. ...
M. Dus's user avatar
  • 2,090
2 votes
1 answer
95 views

Quotient with positive second derivative in the limit?

I am studying the quotient of $$f(\varepsilon) = \sum_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$ and $$g(\varepsilon) = \sum_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$ for some $\...
Xing Wang's user avatar
  • 119
2 votes
1 answer
244 views

Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm?

For any sequence $\omega\in[1, \infty)^{\mathbb{N}}$, define the weighted $\ell^p_\omega$-norm of the sequence $v$ by $$\Vert v\Vert_{\ell^p_\omega} := \left(\sum_{k=1}^\infty \omega_k^p |v|_k^p\right)...
Philipp Trunschke's user avatar
2 votes
1 answer
290 views

One series converges iff the other converges

In Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this sequence of partial sums converges $$ \begin{split} \sum_{1<n\leq N}\frac{a_{n}}{\...
tongyang2357's user avatar
2 votes
1 answer
497 views

Truncated Euler products, Dirichlet eta function, and convergence issues

Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as $$W(\sigma,...
Vincent Granville's user avatar
2 votes
1 answer
168 views

A question about series involving a Sobolev functions

Let $\Omega\subset\mathbb{R}^n$ open, bounded and smooth. Let $\lambda_j$ and $e_j$, $j\in\mathbb{N}$, be the eigenvalue and the corresponding eigenfunctions of the Laplacian operator $-\Delta$ in $\...
inoc's user avatar
  • 339
2 votes
1 answer
148 views

Lower semi-continuity of length-dependent functional

Let $f:\mathbb{R}\rightarrow [0,\infty]$ be a lower semi-continuous function and define the functional $$ \begin{aligned} F_f:&\ell^1 \rightarrow [0,\infty]\\ (x_n)_{n=0}^{\infty} &\to \sum_{n=...
ABIM's user avatar
  • 5,405
2 votes
1 answer
142 views

Limiting behaviour of elementary sequence

I am curious about the limiting behaviour of a certian sequence of functions $$f_n:=\left(\sum_{k=1}^{\infty} 2^{-k} e^{i2^{k}/n}\right)^n$$ -where $i$ is the imaginary unit-to get a conjecture about ...
Sascha's user avatar
  • 536
2 votes
1 answer
280 views

Does the following function series converge?

Let $$ f_n(x)=\frac{\frac{1}{(n-1)!}\sum_{k=0}^{\lfloor \alpha n-x\rfloor}C_{n-1}^{k}~(-1)^k(\alpha n-x-k)^{n-1}}{\frac{1}{n!}\sum_{k=0}^{\lfloor \alpha n\rfloor}C_{n}^{k}(-1)^k(\alpha n-k)^{n}}, $$ ...
RyanChan's user avatar
  • 550
2 votes
2 answers
152 views

Divergence rate of geometric sum of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that $$ 0<\lim_{\beta\rightarrow 1}(1-\...
Marc's user avatar
  • 479
2 votes
1 answer
152 views

Growth rate of elementary sequences

We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...
António Borges Santos's user avatar

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