Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

20
votes
0answers
723 views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
9
votes
0answers
158 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. ...
8
votes
0answers
498 views

Difference of differentiation under integral sign between Lebesgue and Riemann

Here is a consequence of Lebesgue dominated convergence theorem on differentiation under integral sign. Function $f(x, t)$ is differentiable at $x_0$ for almost all $t \in A$, and $t \to f(x, t)$ ...
5
votes
0answers
93 views

Convergence of Euler-transformed zeta series

I am trying to prove that the expression $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^\infty\sum_{m=0}^n\frac{(-1)^m}{2^{n+1}}{n\choose m}(m+1)^{-s}$$ converges to an analytic continuation of the alternating ...
5
votes
0answers
110 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.. ...
5
votes
0answers
75 views

Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a ...
5
votes
0answers
109 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
votes
0answers
126 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?
5
votes
0answers
194 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 ...
4
votes
0answers
49 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
votes
0answers
40 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
224 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 \ ...
4
votes
0answers
61 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
votes
0answers
39 views

General and basic question about convergence of a series

Let $(a_{i,j})_{i,j=1}^n$ be a sequence of real numbers such that the following series converges $$ S = \lim_{n\to\infty}\sum_{i=1}^n\sum_{j=1}^na_{i,j} $$ It is known that for each $i$th the ...
4
votes
0answers
88 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
votes
0answers
91 views

For every space $X$, $C_p(X)$ is a topological group.

I try to show that for every space $X$, $(C_p(X), +)$ where $$+:C_p(X)\times C_p(X)\to C_p(X):(f,g)\mapsto f+g$$ and for every $x\in X$, $(f+g)(x)=f(x)+g(x)$ is a topological group. The family ...
4
votes
0answers
190 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 ...
4
votes
0answers
160 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
votes
0answers
338 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
votes
0answers
274 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 ...
3
votes
0answers
26 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
3
votes
0answers
46 views

extending the convergence of measures

Let $F$ the real vector space of all applications $\phi: X \times Y \rightarrow \mathbb{R}$ where $(X,\mathcal{B}_1, \mu)$, $(Y,\mathcal{B}_2 )$ measurable spaces with $X$ and $Y$ are compact metric ...
3
votes
0answers
27 views

Asymptotic Bounds for the distribution of $f_n(X_n)$.

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of $\mathbb{R}^{k}$-valued random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ converging almost surely to $X$. ...
3
votes
0answers
70 views

Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
3
votes
0answers
35 views

Help proving(well, disproving) the convergence of $\sin^3(x)$

So I'm stuck on a question where it's asking for the power series and radius of convergence of $\sin^3(x)$ I've done the power series ok, but my problem is that when I apply the ratio test it's ...
3
votes
0answers
67 views

snails on a tetrahedron

suppose you have $4$ snails arranged on the surface of an equilateral tetrahedron(all sides equal, each snail is numbered from $1$ to $4$, and that the snail numbered $1$ moves toward snail numbered ...
3
votes
0answers
51 views

Proving $\sum^\infty_{n=1}a_n$ converges absolutely iff each each sub series converges

We have a series $\displaystyle\sum^\infty_{n=1}a_n$ and a sub series $\displaystyle\sum^\infty_{k=1}a_{n_k}$ where $n_k\in\mathbb N$. Prove that $\displaystyle\sum^\infty_{n=1}a_n$ converges ...
3
votes
0answers
43 views

Problem of convergence of characteristic functions

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
3
votes
0answers
47 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 ...
3
votes
0answers
32 views

Convergence of a matrix product

Let $A=o_{a.s.}(1)$; $A:k\times k$ matrix and $Vu=O_p(1)$; $V:k\times k$; $u: k\times 1$. Specifically, $Vu$ converges in distribution to $\mathcal N(0,I_k)$. Can we show that $VAu=o_p(1)$ or ...
3
votes
0answers
50 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 ...
3
votes
0answers
32 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
3
votes
0answers
80 views

Convergence of sequences of sets

Assume we have the indicator function $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and the sequence of its approximation functions $\{ I_{\nu}(x) ...
3
votes
0answers
77 views

Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
3
votes
0answers
59 views

convergence of series implies convergence of coefficients

Is it true that $$\sum_{i=0}^\infty a_{i_n} y^i \rightarrow \sum_{i=0}^\infty a_{i} y^i \quad \forall y \in [0,1]$$ implies $$a_{i_n} \to_{n \to \infty} a_{i} \quad \forall i$$ where $0 \leq ...
3
votes
0answers
165 views

derivative of limit function vs limit of derivatives

Suppose that we have a sequence of differentiable functions $f_n:\mathbb{R} \rightarrow \mathbb{R}$ such that $f_n$ converges to some function $f$. Then it is not necessary that the sequence of ...
3
votes
0answers
54 views

How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is ...
3
votes
0answers
109 views

Monotonic convergence of powers of a stochastic matrix

Let $P$ be a stochastic matrix (nonnegative and each row summing to 1). Assuming that $P^n$ converges to $\textbf{1}\pi$ as $n \rightarrow \infty$, where $\pi$ is a row vector (stationary distribution ...
3
votes
0answers
95 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 $\mathfrak{S}$ ...
3
votes
0answers
115 views

sequence convergence

Assume all terms $a(n) >0$ , $\sqrt{a(1)}\geqslant 1 +\sqrt{a(0)}$, and $$\left|\dfrac{a(n+1)}{a(n)}-\dfrac{a(n)}{a(n-1)}\right|\leqslant \dfrac{1}{a(n)} $$ for all $n>0$. Prove that ...
3
votes
0answers
162 views

Cluster point of $a_{n}:=n+(-1)^{n}n$

I am trying to find the cluster point of the sequence $a_{n}:=n+(-1)^nn$. Can you please check my solution? The subsequence diverges for increasing even $n$ since $2n$ grows infinitely. The ...
3
votes
0answers
129 views

Infinite product convergence

Reading some notes I have taken during class, I am unable to see the connection between these lines: $g(z)=\prod_{k\geqslant 0}(2z\lambda_k + 1)$, where $\lambda_k=4/((2k+1)^2 \pi^2)$ And The zeroes ...
3
votes
0answers
152 views

Cauchy criterion

We know that the series $\sum_{n=1}^\infty \dfrac{1}{n(n+1)}$ coverge to 1. I want use Cauchy criterion to show the series converges as an exercise. Is the following proof correct? Given $\epsilon ...
3
votes
0answers
61 views

The existence of the limit $\lim_{n\to\infty} S_{i_n}\circ…\circ S_{i_1}(x)=x_0$ for some $i_1, i_2,…\leq N$

Let $(X,\rho)$ be a Polish space. We are given continuous functions $S_i: X\rightarrow X$ ($i=1,...,N$) and a constant $0<r<1$. We assume the following: For each $x,y \in X$ there exists ...
3
votes
0answers
210 views

Convergence of sequence in $C^{k, \alpha}$ composed with $C^\infty$ function

Is it true that if a sequence $u_n \to u$ in $C^{k, \alpha}$ norm, and if you have a function $f \in C^\infty$, then $f(u_n) \to f(u)$ in $C^{k, \alpha}$? I think so, since this is true for ordinary ...
3
votes
0answers
329 views

Uniform convergence of analytic functions

Let $f_{n}(z), g(z)$ be entire functions, for all $n\geq 1$. Suppose that $g(x)$ doesn't vanish on $\mathbb H\cup\mathbb R$ (so we have $\frac{f_{n}(z)}{g(z)}$ analytic on $\mathbb H\cup\mathbb R$). ...
3
votes
0answers
122 views

Another Question based on Abel's theorem of multiplication of series

I am trying to show that the $qth$ power of the series $$a_{1}\sin \theta +a_{2}\sin 2\theta +\ldots +a_{n}\sin n\theta +...$$ is convergent whenever $q(1-r)<1$, r being the greatest number ...
3
votes
0answers
175 views

How to prove that $\lim_{\lambda \rightarrow 0} f*h_\lambda (x)=f(x) $ a.e.

Assume that $h_\lambda(x)=\frac{1}{\pi} \frac{\lambda}{\lambda^2+x^2}$, for $\lambda>0$, $x \in \mathbb{R}$. I know that if $f\in L^p$ then $\lim_{\lambda \rightarrow 0} \|f*h_\lambda -f\|_p =0$, ...
3
votes
0answers
132 views

Uniform convergence of measures

Let $A\subset[0,1]^n\subset\mathbb{R}^n$ be a closed semi-algebraic set. Let $f_k: [0,1]\rightarrow\mathbb{R}$, $f_k(x_i)=\mu^{n-1}(A_{|_{x_i}}+B^{n-1}_{1/k})$ where $A_{|_{x_i}}=A\cap H^i_{x_i}$ ...
3
votes
0answers
141 views

Convergence of marginal distribution in the case of convergence in distribution

Suppose a random variable $Z_n=(X_n,Y_n)$ takes a value on $A= [0,1] \times B=[0,1]$. We can consider a conditional distribution of $Y_n$ given the realized value of $X_n$. Now suppose $Z_n$ converges ...