Convergence of sequences and different modes of convergence.

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58
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3k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\...
15
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229 views

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
11
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188 views

Nested Radicals Involving Primes

How do you evaluate $\sqrt { 2+\sqrt { 3+\sqrt { 5+\sqrt { 7+\sqrt { 11+ \dots } } } } } $ ? This question appears to be rather difficult as there is no way to perfectly know what $p_{ n }$ is , ...
10
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90 views

Are the fractional parts of powers of $\pi$ divergent?

Let us define $a_n$ as the fractional part of $\pi^n$. In other words, define $a_n=\pi^n-\lfloor \pi^n \rfloor$. Then, does the following limit exist? $$\lim_{n \to \infty}a_n$$Intuitively, it ...
10
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119 views

Does $\sum_n |\sin n|^{cn^2}$ converge?

So I recently asked a question about convergence of $\sum_n |\sin n|^{cn}$ for arbitrary $c > 0$ and it turns out that the terms of the series don't even converge, for any $c > 0$, so the series ...
8
votes
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87 views

Which functions are irrelevant?

Call a bijection $f : \mathbb{N} \rightarrow \mathbb{N}$ irrelevant over $\mathbb{R}$ iff for all sequences $a : \mathbb{N} \rightarrow \mathbb{R}$, if $$\sum_{i=0}^\infty a_n$$ exists, call its value ...
8
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180 views

Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
7
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222 views

How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$?

How do I evaluate this sum :$$\displaystyle\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$$ Note : I used many criterions of convergence to show if it converges but i didn't up. Thank you for any ...
7
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170 views

Consider the problem minimize $f(x)= x^4 −1. $

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
6
votes
0answers
97 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
6
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202 views

Uniform convergence of Empirical Moment Generating Function

In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of $n$ variables $X_1,X_2, \dots, X_n$ as: $$ \begin{...
6
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160 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. $\...
6
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234 views

I need help about some compactness arguments

I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
6
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0answers
171 views

Does $\displaystyle \lim_{m \to +\infty}f_{2,m}(x)$ converge?

This is related to a previous question where, as stated there, $f_{2}(n)$ gives the greatest power of $2$ that divides $n$. Specifically the sequence $\lbrace 0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,\cdots\...
5
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62 views

Absolute convergence in Banach space

Let $X$ be a Banach space. Show that, if $\sum_{n\ge 1}x_{n}$ is absolutely convergent then $\sum_{n\ge 1}x_{n}$ is convergent and $$\left\|\sum_{n\ge 1}x_{n}\right\|\le \sum_{n\ge 1}\|x_{n}\|$$ I ...
5
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79 views

Intuition for almost sure convergence = fast enough convergence in probability

I know the meaning of convergence in probability and almost convergence. From zero-one law, we can derive that if a sequence of random variables converges in probability fast enough, then it converges ...
5
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88 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
5
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207 views

Infinite Double Exponential Sum, with Functional Equation $g(x) = g(\sqrt{x})$

What is a closed form for $$ \lim_{n\to-\infty}\sum_{i=n}^{\infty}\frac{x^{2^i}(x^{2^i}-1)}{(x^{2^{i+2}}+1)} $$ The series has the form: $$... \frac{x^{\frac{1}{4}}(x^{\frac{1}{4}}-1)}{x+1} + \frac{...
5
votes
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92 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq L^{2}\...
5
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277 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ d\mu_{\frac{1}{\sqrt{n}}M_n}\stackrel{P}\rightarrow\int_{\...
5
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154 views

Can one use $e^n$ instead of $2^n$ in Cauchy condensation test?

Cauchy condensation test is useful for testing the convergence of infinite series. The test is stated here as follows: for a positive non-increasing sequence $f(n)$, the sum $\sum_{n=1}^\infty f(n)$ ...
5
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310 views

For infinite series convergence/divergence: Why doesnt meeting the conditions of the Divergence test imply the Cauchy Convergence Critierion

Assume that the limit of the sequence is zero, $\lim_{n\to\infty}a_n=0$. So its not plainly obvious if the series $\sum a_n$ converges or diverges. I have wondered for some time. If $\lim_{n\to\...
4
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38 views

What's the implication of l'Hospital's rule on rate of convergence?

Consider $h(x)=f(x)/g(x)$, if l'Hospital's rule is applicable, then $$\lim h(x)=\lim\frac{f'(x)}{g'(x)}.$$ Does this fact implies $h(x)$ and $f'(x)/g'(x)$ converge at the same speed? E.g. if $f'(x)/g'(...
4
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39 views

When is a topology is defined for this special kind of uniform convergence?

Question: Let $\phi:X\to X$ be bijective and continuous. Does a topology $\tau$ exist such that $$f_n\overset\tau\to f\Leftrightarrow \phi\circ f_n\overset\infty\to\phi\circ f$$ where $\infty$ ...
4
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41 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, $f(...
4
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61 views

Elemenatry topological proof of Erdos conjecture on Arithmetic Sequences

Define $C_n(A) = \{a \in A : \forall d \in \Bbb{N}$ one of $a + d, a +2d, \dots a + (n-1)d$ is not in $A\}$ For $n \leq m$, we have $C_n(A) \subset C_m(A)$. Proof. Let $x$ be in the LHS. Then ...
4
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56 views

Prove that the Weierstrass-type function is nowhere differentiable

Given $0<\alpha\leq1$. Show that the function $$f(x)=\sum_{j=1}^\infty 2^{-j\alpha}\sin(2^jx)$$ is nowhere differentiable. I have solved the case $x=0$. Taking $t_l=2^{-l-1}\pi$, then $f(t_l)-f(0)=...
4
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73 views

My proof of: Every convergent real sequence is a Cauchy sequence.

Is my proof correct? Let $(x_n)_{ n \in \mathbb{N} }$ be a real sequence. $\textbf{Definition 1.}$ $(x_n)$ is $\textit{convergent}$ iff there is an $x \in \mathbb{R}$ such that, for every $\...
4
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0answers
54 views

A conditional normal rv sequence, does the mean converges in probability

$X_1, X_2, \dots, X_n$, are $n$ mutually independent r.v.s. $Y_1,\dots,Y_n$ are another set of mutually independent r.v.s. $X_k\mid Y_k=y_k\sim N(y_k,y_k^2)$ and $Y_k\sim\text{uniform}(-k,k)$ for $k=1,...
4
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90 views

Methods of constructing rapidly convergent series

It's fairly easy to see that the series $$1-\tfrac{1}{3}+\tfrac{1}{5}-\cdots=\tfrac{1}{4}\pi$$ is : 1. Convergent to the value given, and - 2. Very slowly converging, which can be seen just by ...
4
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153 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
4
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125 views

Riemann Zeta Function Analytic Continuation

I am struggling to understand how the analytic continuation of the Riemann Zeta function is derived to extend it to all complex values $z$ not equal to $1$, starting with the series which converges ...
4
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67 views

How to integrate scalar field over quarter torus? Infinite series does not converge.

This seems to be physics question, but the problem just concerns math. Preface If one wants to calculate the permeance $P$ of a rectangular bar: it is an easy task: $$P = \frac{\mu a b}{L} ~~~~...
4
votes
0answers
57 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e $\...
4
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79 views

A basic problem on weak convergence

Let $f_n$ be $2^n$ times the indicator of the set of $x$ in the unit interval for which the digit from $n+1$ to $2n$ is zero in the dyadic expansion of $x$ (lets call it $A_n$). I have to show that ...
4
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72 views

Prove that $||\sum_{n=0}^{+\infty}{x_n}||\le\sum_{n=0}^{+\infty}||{x_n}||$ when series $\sum_{n=0}^{+\infty}{x_n}$ are absolutely converge?

I think it should be proved that: Since $$||\sum_{n=0}^{N}{x_n}||\le\sum_{n=0}^{N}||{x_n}||$$ so $$\lim_{N\to+\infty}||\sum_{n=0}^{N}{x_n}||\le\lim_{N\to+\infty}\sum_{n=0}^{N}||{x_n}||$$ so $$|...
4
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0answers
108 views

Using formal power series to solve nasty equations

Consider a function $f:[0,\infty)\times \mathbb R\to\mathbb R$, and suppose that given some $a>0$, I would like to solve for $x\in\mathbb R$ satisfying \begin{align} f(\delta, x) = a. \end{align} ...
4
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153 views

$\alpha <1$, series $n^{-\alpha} \sum _{k=1}^n \sin (k^2)$ is bounded

Could you help me answer the question, if there exists $\alpha <1$ such that series $n^{-\alpha} \sum _{k=1}^n \sin (k^2)$ is bounded?
4
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118 views

maximum renewal rate of a Markov chain

Consider a Markov chain $(X_t)$ on a state-space with a countably generated $\sigma$-algebra and assume this Markov chain allows for small sets of order one. This means there exist sets $\mathfrak{S}$ ...
4
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172 views

$\sum f_n(x)$ converges absolutely and uniformly in $[a,b]$, but $\sum |f_n(x)|$ doesn't converge uniformly

I'm studying uniform convergence, and am looking for some examples of series $\sum f_n(x)$ that converge absolutely and uniformly in $[a,b]$, but $\sum |f_n(x)|$ does not converge uniformly in the ...
4
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441 views

Interchanging the order of limits

Would you advise me on the references of Pringsheim Convergence about interchanging the order of limits? Where can I find the most general statement?
4
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425 views

How to prove stability of this dynamic system?

I'm trying to prove stability of the following dynamic system but I think my Mathematics knowledge is not deep enough. My dynamic system consists of a state vector $x \in \mathbb{R}^n$. The system ...
4
votes
0answers
196 views

Hölder Condition Implying Uniform Convergence

Define $g(z)=\frac{1}{2\pi i}\int_{-k}^k \frac{h(\zeta)}{\zeta-z}d\zeta$, where $h$ is continuous and defined on $[-k,k]$. Let $|h(x)-h(y)|\leq |x-y|^\alpha$ for all $x,y\in[-k,k]$ and for some $\...
3
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35 views

Pass to the limit under the sign of integral

I need to show that this integral converges to its limit, showing this only for a subsequence is also good enough for me. Consider $w_k \rightarrow w $ in $L^2$, $u_k \rightarrow u $ weakly in $H^...
3
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75 views

Revision of Borel-Cantelli, $(X_n)_{n \geq 1}$ sqn of $\geq 0$ identical RVs with $E(X) < \infty$ then $X_n/n \to 0$ a.s., are my arguments correct?

I care to understand the concept behind the Borel-Cantelli Lemma better, hence I would appreciate it if someone could take the time to check if my arguments below are clear and rigorous. Statement:...
3
votes
0answers
25 views

weak convergence and unbounded functions with bounded moment

I want to prove the following: Given a topological space (it is a Lusin space, but I think that does not matter) $\Omega$, a function $f \in C(\Omega,\mathbb{R})$ and a sequence of Radon measures $P^{...
3
votes
0answers
87 views

On the Laplace transform $\int_0^\infty e^{-sx}d \left( \ \int_2^{e^{1+x}}\frac{dt}{\log t}\right) $

I've read the basics about Laplace transform, and I know that since for $\Re s>1$, $\frac{e^x}{1+x}$ has exponential order, then $$F(s)=\int_0^\infty e^{-sx}\frac{e^{1+x}}{1+x}dx$$ is well defined, ...
3
votes
0answers
80 views

Monte Carlo with non uniform weighting

So, I just want to check if what is in my mind is in fact true. Assume, that we have are given a distribution $p_{z}(k)$ over the whole $\mathbb{Z}^+$. We are interested in approximating $p_v(v)$ over ...
3
votes
0answers
52 views

Does the series $\sum_{k=1}^{\infty} \left[\ln\left(1+\frac{x}{k}\right) - \frac{x}{k} \right]$ converge?

I tried all the theorems, that I knew in analysis, to know if the mentioned series converge but none of them is relevant except one: The Ratio Test for Series, but unfortunately this is not working as ...
3
votes
0answers
71 views

Prove $ \limsup a_n$ is a real number

($a_n$) is a real sequence bounded only from above. Let $S :=$ {$t \in \Bbb R:$ $t$ is the limit of a convergent subsequence of ($a_n$) }. Suppose the supremum of S is a real number. Prove that $ \...