Convergence of sequences and different modes of convergence.

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Limit with Cauchy density and arctan

Suppose $\{X_i\}$ is an iid sequence of standard Cauchy variables, i.e each has pdf $\dfrac{1}{\pi(1+x^2)}$ for $x\in \Bbb{R}$. Now I have to show that $$\lim_{n\to \infty}\Pr(X_{(n)}\le nx)$$ exists ...
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45 views

How to show that $f_n(x_n) \to f(x)$?

Define $f_n(x) = \int_0^x |T_n\left((|s|-\frac 1n)^+ + \frac 1n\right)|^{-\frac 12}\;\mathrm{d}s$. Here $T_n(x) := x$ if $|x| \leq n$ and $T_n(x) := n$ otherwise. We have that $f_n(x) \to f(x):= ...
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1answer
22 views

measure convergence

Let $(X,\mathcal{F},\mu)$ be a finite measurable space. Define $$d(f,g) = \int_X \frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}\mu(dx) $$ Proof that $d(f_n,f)\to0 \Leftrightarrow\ f_n$ converge in measure to $f$ ...
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2answers
24 views

Evaluating integral convergence

I have an the integral $$\int_{-\infty}^{6} xe^{\frac{x}{2}}\; dx$$ I know that this integral is convergent but I can not find how to evaluate its' convergence other than finding the limit of the ...
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1answer
42 views

Prove that $X_n/\lambda_n \to$ 1 in probability for $X_n \sim \text{Pois}(\lambda_n)$

Let $X_n \sim \text{Pois}(\lambda_n)$, where $\lambda_n \to \infty$ as $n \to \infty$. Prove that $X_n/\lambda_n \to 1$ in probability Should I solve this problem using chebychev's inequality? ...
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1answer
25 views

Proving convergence theorems

Suppose that ${a_n}$ converges to a. prove the sequence ${ca_n}$ converges to ca, where c is any constant. Here is the start of my proof: By definition of convergence of a sequence, $|ca_n - ca| ...
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1answer
119 views

Find the limit of a piecewise defined sequence

Given the sequence $$ d_n =\begin{cases} \frac{n}{1 + n^2}& \text{if $1 ≤ n ≤ 900$}\\ 7 & \text{if $n ≥ 901$}\end{cases}.$$ what is the limit? I have been thinking that 7 is the ...
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1answer
88 views

Absolute Convergence (Improper integral)

I need to prove that the following improper integral converges absolutely, and I don't know how: $$\int_{-\infty}^{-4-\epsilon}\frac{\cos(x)}{(x^2+3x-4)x}dx$$ where $\epsilon$ is a small number ...
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1answer
76 views

Prove that the following series uniformly converges on $[\delta, 1]$ where $0<\delta <1$, but not on $(0,1]$

Prove that with $0<\delta <1$ $\sum_{n=0}^{\infty} \frac{n^2x}{1+n^4x^2}$ converges uniformly on $[\delta, 1]$ Prove that the series does not converge on $(o,1]$. The issue is that for ...
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1answer
25 views

Can we deduce that there are infinitely many integers $n$ such that $detA_{n}=0$?

Let $δ_{n},θ_{n},ω_{n}$ be three real sequences converging to $δ,θ,ω$ respectively. Define the following matrix $$A_{n} = \begin{bmatrix} δ_{n-1} & θ_{n-1} & ω_{n-1} \\ δ_{n} & θ_{n} ...
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2answers
42 views

How can we prove that these series are convegent?

Let $r>4$ be a positive integer. How we can prove that these series are convegente: 1) $$S=\sum_{m=1}^{∞}\frac{1}{r^{m^2}}$$ 2) $$D=\sum_{m=1}^{∞}\frac{(p_{m}-2(m-1))}{r^{m^2}}$$ where $p_{m}$ ...
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1answer
50 views

The limit of convergent series

We have the following series $$\sum_{j=1}^\infty\frac{1}{7^j(5j+1)}$$ If we use the ratio test, we see that $\lim_{j\to \infty}\frac{a_{j+1}}{a_j}=\frac{1}{7}<1$. So this series is convergent. ...
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1answer
26 views

convergence of sequences $|a_{n}-L|<1/n$

$a_{n}=\frac{5n}{4n-3}$ , $L=5/4$ show that for all $n>12$ , $|a_{n}-L|<1/n$. right now, I have: $a_{n}-L=\frac{15}{4(4n-3)}<\frac{4}{4n-3}$ since $15/4=3.75<4$ then I know ...
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2answers
36 views

How do I show that this series converges pointwise?

I have the series $\sum\limits_{n=1}^\infty \dfrac{1-\cos\left(\frac{x}{\sqrt{n}}\right)}{\sqrt{n}}$ and I need to show that it converges pointwise for every $x\in\mathbb{R}$. I'm having a hard time ...
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1answer
20 views

If $u_n \to u$ in $L^2(0,T;L^2(\Omega))$ and $f_n \to f$ uniformly, does $f_n(u_n) \to f(u)$ in $L^2(0,T;L^2(\Omega))$?

Let $\Omega$ be an unbounded domain. Suppose we have $u_n \to u$ in $L^2(0,T;L^2(\Omega))$. Let $f_n\colon \mathbb{R} \to \mathbb{R}$ be a sequence such that $f_n \to f$ uniformly. We know that ...
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1answer
26 views

Proving the convergence of a recurrent sequence

Define the sequence $\{U_n\}_{n \ge 0}$ by: $$U_0 = U_1 = 1,\ and \ U_n = \sum_{i=1}^{n-1}\frac{U_i}{(i - 1)!},\ \forall \ n \ge 2$$ I calculated the first several values of the sequence, and it ...
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1answer
82 views

Convergence in norm and in mean

Here is my problem: If $\{f_k\}$ is a sequence in $L^2$ and $f_k\to f$ in mean, show that $\{||f_k||_2\}$ is a bounded sequence of real numbers. Before I start doing the problem, I would like to know ...
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1answer
46 views

The radius of convergence of a power series about a point interior to the domain of an analytic function

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a real analytic function with domain an open, non-empty set $(a, b) \subseteq \mathbb{R}$, $-\infty \leq a < b \leq \infty$ and let $c \in (a, b)$. ...
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2answers
33 views

need help with this exercise about convergence

$g_n:I \rightarrow \mathbb R$ for $I\in [0,1]$, $g_n(a)=a^n(1-a)$ show $g_n$ converges pointwise. does it converge uniformly (proof) My attempt: I am pretty confuse. I am try to understand the ...
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1answer
129 views

How to evaluate $\sum\limits_{m\in Z}(\sum\limits_{n\in Z} \frac{1}{(m-1+nz)(m+nz)})$ with $Im(z)>0$

How to prove $\phi(z)=\sum\limits_{m\in Z}(\sum\limits_{n\in Z} \frac{1}{(m-1+nz)(m+nz)})=2-\frac{2\pi i}{z}$ with $Im(z)>0$ and $(m,n)\neq(0,0),(1,0)$? For $m$ fixed, $a_m=\sum\limits_{n\in Z} ...
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1answer
36 views

Point-wise/Uniform Convergence Proof Help. $f_n : [0,1] \rightarrow \mathbb{R}$ for each $n \in \mathbb{N}$

$f_n : [0,1] \rightarrow \mathbb{R}$ for each $n \in \mathbb{N}$ $ f_n(x) = \left\{ \begin{array}{lr} 1 - nx & : 0 \le x \le \frac{1}{n}\\ 0 & : \frac{1}{n} \le x \le 1 ...
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4answers
77 views

Convergence of $\int^{+\infty}_1\frac{e^{-x}}{\sin^2x}$

I'm studying the convergence of $$\int^{+\infty}_1\frac{e^{-x}}{\sin^2x}dx$$ The solution is "It is divergent" But I can't figure it out. For $+\infty$ it should converge because $\sin^2x$ is $1$ and ...
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0answers
29 views

Check my answer - radius and domain of convergence

We want to determine the radius and domain of convergence of the sum $\sum_{n=1}^{\infty}(\frac{2n-1}{3n+2})^n(x+2)^n$ I just want to verify my solution and method. We know that the sum definitely ...
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48 views

check for convergence of finite series

Let S$_k$ = $\sum\limits_{n=2}^k $ (-1)$^a$ $\frac{1}{\ln n}$ where a:=floor($n^{0.5} $) Is S$_k$ convergent? Is S$_{2k^2}$ convergent? I have just some vague idea how to show that both of them are ...
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1answer
39 views

How to prove that if $\lim_{n\to\infty} a_n \prod_{k=1}^n b_k = l>0$ then, for any $0<c_k<b_k$, $\lim_{n\to\infty} a_n \prod_{k=1}^n (b_k-c_k) < l$?

How to prove that if $$\lim_{n\to\infty} a_n \prod_{k=1}^n b_k = l>0$$ then for any $0<c_k<b_k$ $$ \ \ \ \ \ \ \ \ \ \ \lim_{n\to\infty} a_n \prod_{k=1}^n (b_k-c_k) < l \ \ \ \ \ ?$$ ...
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1answer
29 views

Limit of a sequence involving partial sums

I've encountered the following curious sequence. I think the limit is correct from simulations, but I'm getting stuck on how to prove it. Any hints? Fix $\gamma \in (0,1)$. Then: $$\sum_{i=n ...
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1answer
38 views

Has a $L^1$ bounded sequence a weak converging subsequence in $L^2$?

Let $f_n \in L^2(0,1)$ with the property that $\sup_n || f_n ||_{L^1}<A< \infty$, i.e. $f_n$ is a sequence in $L^2$ that is uniformly bounded in the $L^1$-Norm. Does $f_n$ then have a weak ...
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69 views

If $\sum |a_k|$ converges, then $\sum |a_k|/{k^p}$ converges for all $p\ge 0$

Problem: If $\sum_{k=1}^{\infty} |a_k|$ converges, prove that $\sum_{k=1}^{\infty} \frac{|a_k|}{k^p}$ converges for all $p\geq0$. What happens if $p<0?$ Attempt: Suppose ...
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1answer
216 views

Limit of Gaussian random variables is Gaussian?

Consider a sequence $X_n$ of Gaussian random variables with mean $\mu_n$ and variance $\sigma_n^2$, which converges in distribution (to some limiting distribution). Can I then conclude that $\mu_n$ ...
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3answers
107 views

For what values of $p$ does $\sum_{k = 1}^{\infty} \frac{1}{k\log^p(k+1)}$ converge?

Find all $p\geq 0$ such that the following series converges $\sum_{k = 1}^{\infty} \frac{1}{k\log^p(k+1)}$. Proof: the general term for the series is $\frac{k^p}{k^p\log^p(k+1)^n} = ...
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1answer
73 views

How to determine the radius of convergence if the Taylor series cannot be written in a neat way?

I am trying to evaluate the radius of convergence of Taylor series centered at zero of function $$f(z)=\frac{\sin(3z)}{\sin(z+\pi/6)}$$ I guess the answer should be $\pi/6$ because the function will ...
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1answer
50 views

Uniform convergence on a measurable set implies convergence a.e.?

Suppose for each $\epsilon$ there exists a measurable set $F$ such that $\mu(F^c) < \epsilon$ and $f_n$ converges to $f$ uniformly on $F$. Prove that $f_n$ converges to $f$ a.e. I have been ...
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0answers
39 views

Stochastic order of Max

Consider an array $\{\{X_{ni}\}_{i=1}^n\}_{n=1}^\infty$ s.t. for each $i$, $X_{ni}=o_p(n^\alpha)$. What is the order of $M_n=\max_{1\le i\le n}X_{ni}$? I got the following ...
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1answer
36 views

Help with pointwise convergence

for a function $g_n:[0,5] \rightarrow \mathbb{R}$ where $g_n=-nx^2+5\; if\; 0\le x \le 1/n$ $g_n=0\; if\; 1/n < x \le 5$ show pointwise convergence My attempt: I am very new to this subject but ...
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64 views

A sequence of functions that converges pointwise on $(0, 1)$ but not uniformly

I am having trouble understanding why the sequence of functions defined by $$ f_n(x) = \min(1-\frac{1}{n}, x)$$ on $(0, 1)$ does satisfy the condition of pointwise but not uniform convergence on $(0, ...
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60 views

convergence or absolute convergence of $\int_0^\infty \frac{\sin(ax)} {x^p}\,dx$

$$ \int_0^\infty \frac{\sin(ax)} {x^p}\,dx $$ The given integral is convergent if $0<p<2$ and absolutely convergent if $1<p<2$ How is the convergence of the integral different by the $p$ ...
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1answer
38 views

Converging sequences in a given topological space.

let $A=[0,1]$, and $\tau=[0,1]\mathbin{\vcenter{\hbox{$\scriptscriptstyle\setminus$}}}{a_n}$ where $a_n$ is a finite or infinite sequence in A. Now let $T=(A,\tau)$, the question is what type of ...
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1answer
154 views

Does convergence in probability imply a.s. convergence in a countable space?

Let $(\Omega, \mathcal F,\mathbb P)$ be such that $\Omega$ is countable. I'm trying to find a simple example of random variables $X_n$ which converge to $0$ in probability but not a.s. If $\mathcal F ...
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1answer
50 views

Convergence of Fourier series for a sum which is not uniform convergent

Given $$\sum_{n=1}^\infty\frac{\cos nt}{n}$$is it a fourier series in a. $L^2(\mathbb T)$? b. $C(\mathbb{T})$? Usually when we get a series we use Weierstrass M test in order to find ...
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1answer
28 views

show series converges, calculus

Show that each of the following series converges 1) $\displaystyle \sum_{k=1}^\infty\dfrac{\log k}{k^p}$, $p > 1$. 2) $\displaystyle \sum_{k=1}^\infty \dfrac{1}{k^{\log k}}$. Can anyone please ...
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3answers
57 views

Does a positive constant $\nu$ exist so that $\varphi(n)>\nu\cdot n$ for all $n$?

Does a positive constant $\nu$ exist so that $\varphi(n)>\nu\cdot n$ for all $n$? Clearly this problem is exactly the same as asking if $\prod\limits_{i=1}^\infty \frac{p_i-1}{p_i}=0$. This is ...
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1answer
67 views

Is there an elementary way to show $\sum_n (\sin n)^{2^n}$ converges?

So I asked earlier whether $\sum_n (\sin n)^{n^k}$ converges for some $k$, and if so then what $k$, and I got an interesting comment/answer saying that supposedly it boils down to the irrationality ...
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2answers
34 views

The convergence of series

Is it true that if $\alpha_k\to0$ and $\sum\limits_{k=0}^\infty\alpha_k=\infty$, we have $\sum\limits_{k=0}^\infty(\alpha_k)^2= c$ (some constant) or at least it does not converge to $\infty$?
2
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2answers
36 views

Convergence of the product $\prod_{i=0}^{b^\frac n2 -1} \left(\frac{b^n-i}{b^n}\right)$

Suppose we have a set of $b^n$ different numbers. Every time we randomly choose a number from this set and put it in a list of length $b^\frac n2$. So we want to fill this list with unique numbers. ...
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0answers
104 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 ...
2
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1answer
52 views

Does $\sum_{n=1}^\infty |\sin n|^{cn}$ converge? What about the terms?

Does $\sum_{n=1}^\infty |\sin n|^{cn}$ converge for some $c > 0$? If so, what $c$? And for the $c$ that give divergence, what about the terms of the series? Do they converge? I saw a question where ...
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1answer
60 views

using taylor series to prove $\lim_{{x}\to+{\infty}}e^{-x}=0$ equal to zero without using its algebraic fact.

To be more specific. We know $e^{-x}$=$\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}{x^k}$, without using the fact that $e^{-x}$=$\frac{1}{e^x}$ and using taylor expansion on $e^x$, how do we prove ...
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1answer
24 views

Show convergence of a series for $p\in (0,1), p\neq\frac{1}{2}$

Let $p\in (0,1)$ and consider $$ \sum_{n=0}^{\infty}\frac{1}{\sqrt{\pi n}}(4p(1-p))^n. $$ For $p=\frac{1}{2}$ it is $(4p(1-p))^n=1$ and thus $$ \sum_{n=0}^{\infty}\frac{1}{\sqrt{\pi ...
3
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1answer
147 views

Prove that a Lipschitz continuous function is differentiable at a point ${\bf x}_0$

Consider $f: B({\bf x}_0,r)\to \Bbb R$, that apart from being Lipschitz continuous, has directional derivatives at the point $x_0$ and $\frac{\partial f}{\partial{\bf v}}({\bf x}_0)=\sum_{i=1}^n ...
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0answers
60 views

Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...