Convergence of sequences and different modes of convergence.

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3
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1answer
56 views

Does $\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0$ imply $u\in L^2(\Omega)$?

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $u\in C^0(\Omega)$ and $(u_n)_{n\in\mathbb{N}}\subseteq C_0^0(\Omega)$ with $$\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0\tag{1}$$ Can ...
0
votes
2answers
49 views

product converging to one

I have a question concerning an infinite product. Suppose $x_n$ is a sequence of positive real numbers. My intuition says that $$\lim_n(1-\exp(-x_n))^n=1$$ for any sequence $x_n=n^\alpha$ with $\alpha ...
2
votes
1answer
37 views

Show that $|X_n|\leq Y$ a.s. implies $\sup_n |X_n|\leq Y$ a.s.

Given a sequence $(X_n)_{n\geq 1}$, show that $|X_n|\leq Y$ a.s. implies $\sup_n |X_n|\leq Y$ a.s. Here is my attempt: $|X_n|\leq Y$ a.s. means that $P(|X_n|>Y)=0$, $\forall n\geq 1$ $P(\sup_n ...
2
votes
1answer
36 views

Does $\Vert f-s_n \Vert_\infty \to 0$ still hold for $f\in C^0[a,b]$?

If $f\in C^2[a,b]$ and $s_n$ its piecewise linear interpolation at points $x_0, \ldots, x_n$ with $h_n = \max_{j=0,\ldots,n-1} (x_{j+1}-x_j)$ then one can show that $$\Vert f-s_n \Vert_\infty \leq ...
0
votes
2answers
33 views

How can I use the sequential criterion to prove/disprove the existence of a limit?

In my book (Abbott), it is written that the sequential criterion for the limit of a sequence is as much of a tool to prove as it is to disprove the existence of a limit. My question is: how? For ...
3
votes
1answer
61 views

Examples and counter-examples in Real analysis - check my answers please

I've been given some practice examples, without solutions in preparation for an upcoming exam, and was hoping I could get them double checked here. For each of the following, either give an example ...
0
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0answers
30 views

How to interpret this statement about two sequences?

Say, $(a_n)_n, (b_n)_n$ are two sequences of non-negative real numbers and both converge to zero. Moreover, we know that $$\forall \varepsilon,M > 0 \ \ \ \exists n_0 \in \mathbb N \ \ \ \forall n ...
0
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0answers
20 views

Verifying Integrals with convergence theorems

I would like to check with you guys, whether my solutions are correct so far. Problem: Verify following statements $\lim\limits_{n\rightarrow\infty} \int_0^\infty e^{-nx} \sin(e^x) dx = 0$ ...
2
votes
1answer
80 views

Verify $\lim\limits_{n\rightarrow\infty}\int^\infty_0 \! e^{-nx} \sin(e^x) \, \mathrm{d}x = 0.$

How to verify $\lim\limits_{n\rightarrow\infty}\int^\infty_0 \! e^{-nx} \sin(e^x) \, \mathrm{d}x = 0$ My idea is to use the dominant convergence theorem with $f_n(x):= e^{-nx} \sin(e^x)$ and ...
0
votes
1answer
21 views

Convergence radius

I know the Cauchy Hadamard equation to calculate the convergence radius of a power series $$\sum_{n=0}^{\infty} a_n x^n$$ Is there a way to generalize this for series of the form ...
0
votes
1answer
11 views

The majorant/minorant criterion

The majorant criterion says if a series in a Banach space has a convergent majorant, then it converges absolutely. My question is, what if a series in a Banach space has a convergent minorant, does it ...
0
votes
0answers
18 views

Analysing the convergence of improper integral with parameter

Can you, please, check if it's right what I did: Here's the exercise: Test the convergence of the following improper integral which is defined using parameter $p\in R$: ...
2
votes
1answer
59 views

$\sup_{t \in [0, \infty)} \left|[(H^{(n)} - H) \cdot X, Y]_t \right| \overset{P}{\rightarrow} 0$

1. Notation We start with establishing some (standard, I think) notation. Let $(\Omega, \mathcal{A}, P)$ be a given probability space. For any filtration $\mathcal{G} = (\mathcal{G}_t)_{t \in ...
1
vote
2answers
41 views

How do I prove that this sequence converges? $\sum_{0}^{\infty} \frac{n-3}{n+2}^{n^2-n}$

I've been having trouble checking whether this sequence converges or not: $$\sum_{n=0}^{\infty} \frac{n-3}{n+2}^\left({n^2-n}\right)$$ At a first glance I thought I should try the root test but that ...
1
vote
0answers
14 views

Weak convergence of one Gaussian Process to another one- determined by covariance function?

I am wondering whether I can make this statement: Assume we have centered Gaussian processes $G_n$ and $G$. Furthermore it holds that $Cov(G_n(x),G_n(y))\stackrel{\mathbb{P}}\rightarrow ...
-1
votes
3answers
58 views

Convergence of series $\sum \limits^{\infty }_{n=1}\frac{n^{(n+\frac{1}{n} )}}{(n+\frac{1}{n} )^{\frac{1}{n} }}$

i need help for find method or methods for solve this series and find the convergence. I very appreciate for any help and yours comments. $$\sum \limits^{\infty }_{n=1}\frac{n^{(n+\frac{1}{n} ...
0
votes
1answer
17 views

Definite integral convergence answer check

this problem is part of my homework assignment and I would like you guys to see if I'm right. So I need to find out for which $a$ is the integral ...
0
votes
2answers
37 views

Convergence of $\int_{2}^{+\infty} \frac1{x \ln^\alpha x}dx$

I can't analyse the convergence of this integral: $$\int_{2}^{+\infty} \frac1{x \ln^\alpha x}dx$$ with $\alpha \in R$. I have tried to find some functions and use comparison theorem, but I haven't ...
3
votes
2answers
59 views

How to prove that this sequence converges? $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$

I'm trying to prove this sequence converges: $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$ I noticed that this is continuous function which its derivative is always less than $0$ for $ x \gt 1 $, so I ...
0
votes
1answer
31 views

Super-Linear Growth after Reordering

Suppose we have a sequence/function $f: \mathbb{N} \to \mathbb{N}$. (For the scope of this question, let $\mathbb{N} = \{ 0, 1, 2, \dots \}$.) From $f$, we are going to create a function $g: ...
0
votes
2answers
34 views

Convergence of random variable times function: $nX_n$

If $X_n\xrightarrow[]{p}X$, can I prove that $n(X_n-X)\xrightarrow[]{p}0$ if $X$ is a natural number. I know that if $Y_n$ is bounded in probability $Y_nX_n\xrightarrow[]{p}0$, or that if $n$ is a ...
-1
votes
2answers
74 views

Convergence of series $\sum^{\infty }_{n=2}\frac{\log[(1\text{+}\frac{1}{n} )^{n}(n+1)]}{\log(n)^{n}\log\text{(}n+1)^{n+1}} =\log_{2}\sqrt{e}$ [duplicate]

I need help with this problem, I'm very lost with the algebraic expression. I'd appreciate your opinions. $$ \sum^{\infty }_{n=2}\frac{\log[(1\text{+}\frac{1}{n} ...
0
votes
0answers
13 views

Composition of Lp convergent function and continuous function

Let $f$ be a continuous function on $\mathbb R$ such that for $U\subset \mathbb R ^n$ bounded it holds that $\forall w\in L^p(U) ~~ f(w)\in L^q(U)$. Let $~u_k \rightarrow u$ in $L^p(U)$ . Does ...
1
vote
0answers
40 views

Generating fractional taylor series

I was considering the notion of taylor series which posit that the sum $$ \sum_{i=0}^{\infty} \frac{1}{i!} a_ix^i $$ Where: $$ a_i = \frac{d^if}{dx^i}_{x= a} $$ Converge to the function f in a ...
0
votes
3answers
67 views

Do their exist power series with non circular regions of convergence?

So far just about any series of the form $$ \sum_{i=0}^{\infty} \left(a_ix^i \right)$$ Has tended to have a circular disk of convergence (of some radius, sometimes even 0). Is there a reason this ...
0
votes
2answers
37 views

Fixed points that are NOT convergent points

Are there any fixed points that are NOT converget (aka attractig fixed points) in the sequence $x_n = 5\ln x_{n-1}$? How do you determine this?
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0answers
70 views

$l_1$ space not being inner product space

Hello so I am just reading some book in functional analysis but there is some calculations that I am getting different results for, so in the book he tries to prove that indeed $l_1$ is normed space ...
2
votes
2answers
217 views

When is this integral convergent?

Let $a \in \mathbb{C}$. Consider the integral $$\int_{-\infty}^{+\infty} \frac{e^{-ax}}{1 + e^x} dx,$$ for which values of $a$ is this convergent? Is it right to say that $a$ has to be purely ...
0
votes
0answers
48 views

Almost sure convergence, product of variables

Consider the sequence of random variable given by distribution : $$\mathbb{P}(X_n=1)=1-\frac{1}{n},$$ $$\mathbb{P}(X_{n}=0)=\frac{1}{n}$$ and $Y_n=X_n \cdot Y$ for random variable Y. Does the $Y_{n}$ ...
1
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0answers
20 views

Convergence in probability of function composition.

I need to show that $G_n \stackrel{P}{\to}_n F_0$, i.e. for any $\epsilon>0$ $$ P(|| G_n - F_0||>\epsilon) \to_n 0 $$ We know the following: $G_n$ and $F_0$ are a bilinear functions from ...
0
votes
1answer
48 views

Almost sure convergence of random variables

$(X_n)$ is a sequence of random variables having the following distribution: $$P(X_n=1)=1- \frac{1}{n},\; P(X_n=0)=\frac{1}{n}$$ (we don't assume that those variables are independent). $X$ is some ...
1
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1answer
35 views

Convergence of conditional probability

Can anyone help me with this question: suppose $X$ and $Y$ are non-negative random variables. Under what condition does $\lim_{\delta\rightarrow 0}\dfrac{P(Y\leq ...
0
votes
1answer
53 views

Convergance of some series of random variables

Random variables $(X_{n})$ are independent and have the distribution $P(X_{n}=1)=p, P(X_n=-1)=1-p$, $\frac{1}{2}<p<1$. Prove that $$X_1+X_2+\dots+X_n \to \infty $$ almost sure. Let ...
0
votes
1answer
99 views

Alternate proof for Arzela-Ascoli

Im trying to finish a beautiful excercise, which consist of giving an alternate proof for the following corollary of Arzela-Ascoli´s Theorem. Given $X,Y$ metric spaces, $X$ compact, $Y$ complete, and ...
2
votes
2answers
185 views

Proof that the sequence $s_n = \frac{1}{n}$ converges to $0$

Prove that the sequence $s_n = \frac{1}{n}$ converges to $0$. I am writing this proof in order to help other people to understand better how to prove if a sequence converges and in particular why ...
3
votes
2answers
54 views

Show that $\frac{X_1+\dots+X_n}{n}$ converges to $\infty$ a.s. for $X_n \sim U([0,n])$ independent

Random variables $(X_{n})$ are independent and $X_{n}$ has an uniform distribution on $[0,n]$ for n=1,2,... Prove that: $$\frac{X_{1}+X_{2}+\dots+X_{n}}{n}\rightarrow \infty$$ almost sure. We can ...
0
votes
2answers
26 views

Find the domain of convergence of series $\sum_{n=1}^{\infty }(nx)^n$

Find the domain of convergence of series $$\sum_{n=1}^{\infty }(nx)^n$$ I tried to find it by using the Root Test as follows: $$=(nx)^{n/n}=nx$$ I know for convergency, the $(nx)$ should be less ...
-4
votes
1answer
90 views

$\sum\limits_{n=1}^{\infty}\sin ( \frac{5^n + 2^n}{n!})$ converges? [closed]

I was trying to determine weather or not $\sum\limits_{n=1}^{\infty}\sin ( \frac{5^n + 2^n}{n!})$ converges using perhaps the D'Alembert test, but it doesn't really seem to fit..are there other ways? ...
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votes
1answer
57 views

$\sum\limits_{n=1}^{\infty}\sin ( \frac{n}{2^n})$ converges? [closed]

I was trying to determine weather or not $\sum\limits_{n=1}^{\infty}\sin ( \frac{n}{2^n})$ converges using perhaps the D'Alembert test, but given the sine I cant really see it happening..are there ...
1
vote
1answer
27 views

Exponential(1) distributed random variable convergence

I am stuck with convergency in probability... I have the following exercise: Let $(X_k)_{k\ge1}$ be a sequence of independent exponential-(1) distributed random variables. Show that $n^\alpha ...
1
vote
5answers
81 views

Show if $\sum_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt{n}}$ is absolute convergent or divergent

Show if $\sum_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt{n}}$ is absolute convergent or divergent First i subbed numbers in $$\lim_{n \to \infty} \frac{(-1)^n}{1+\sqrt{n}} = \frac{-1}{1+\sqrt{1}} + ...
0
votes
1answer
23 views

Does this contravene the dominated convergence theorem?

The function $f_n$: $\mathbb{R}\rightarrow \mathbb{R}$ is defined by $f_n(x):=\chi_{[0,\infty)}(x)\frac{1}{n}\exp(-\frac{x}{n})$, where $n\in \mathbb{N}$. $f(x):=\lim\limits_{n \rightarrow ...
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votes
2answers
42 views

Analysis: limit of $x^n$ if $x\gt 1$ [closed]

could someone tell me how I can show $x^n \to \infty$ if $x \gt 1$ From limit to infinity definition $x^n \to \infty$ if for all $M \gt 0$ there exists an N in the natural numbers such that for all ...
4
votes
3answers
74 views

Is the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$ necessarily $0$?

I have the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$. I know that every successive partial product will necessarily be smaller than the last, as we are multiplying always by a number smaller than ...
3
votes
2answers
83 views

Probability of get a kidney donated

It is really a probability problem. I use the story of kidney donation because it is easier to describe. Consider the following scenario: Time is discrete. At each period, the measure of patients ...
1
vote
1answer
37 views

Convergence of a sequence with both sin and cos

I'm trying to figure out whether the following series converges absolutely or conditionally or whether it diverges. I am stuck on the following one that involved both sin and cosine: $$ ...
5
votes
3answers
100 views

An Improper integral with parameter $\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$

I have to analyse the convergence of this integral: $$\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$$ where $p\in R$. I have thought to write: ...
0
votes
5answers
81 views

Proof that $n(1-e^\frac{-\alpha}{n})$ converges to $\alpha$

I'm supposed to prove that $$\lim_{n\to\infty}n(1-e^{\frac{-\alpha}{n}})=\alpha.$$ It looks like it's supposed to be super easy but I'm stumped. Any suggestions?
0
votes
0answers
22 views

Conceptual question regarding convergence and continuity

Let $X,Y$ be metric spaces. Let $(f_n)$ be a sequence of functions from $X$ to $Y$ equicontinuous that converges pointwise to a function $f:X \to Y$. Then, $\{f,f_1,f_2,...\}$ is equicontinuous. ...
3
votes
1answer
18 views

Show convergence of sequence with infinite series as inequality

I've got a question concerning some weird kind of Lipschitz constant function, but it's an introduction course in Mathematics so Lipschitz hasn't passed the course yet. I should be able to prove this ...