Convergence of sequences and different modes of convergence.

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185 views

Convergence of Ornstein-Uhlenbeck process as a scaled Brownian Motion

Let $W$ be a standard Brownian motion. Let $\alpha,\sigma^2 >0$, and let $X_0$ be a $\mathbb{R}$-valued random variable with distibution $\nu$ that is independent of $\sigma(W_t,t\geq 0)$. Now ...
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3answers
61 views

Does this integral converge or diverge?

I have the $$\int_{16}^{500} \frac{1}{x^{0.25} - 2} dx,$$ and am trying to find whether it converges or diverges. I have sketched the graph and noticed that their is an asymptote at $x=16$ (hence ...
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0answers
30 views

Help with when the integral is convergent

quick question, how can you tell for which a the integral $$\int_1^{\pi/2} \dfrac{\cos^2(2x) - e^{-4x^2}}{x^a\tan{x}}dx$$ is convergent? Thanks in advance.
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2answers
38 views

Pointwise convergence and the convergence of a distribution sequence.

The sequence of functions $f_n(x)=\{tanh(nx)\}_{n=0}^{\infty}$ does not converge uniformly to $f(x)=sgn(x)$ but only pointwise. Is it then, still possible that the sequence of distributions ...
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0answers
36 views

A special limit-sum interversion

Let $l\in\mathbb{R}$ and $(u_k)_{k\in\mathbb{N}}$ a sequence converging to $l$. Let $a_{n,k\in\mathbb{N}}$ be such that $\displaystyle\forall n\in\mathbb{N},\sum_{k=1}^na_{n,k}=1$. $\forall ...
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30 views

prove $\{z_n\}$ converges where $z_n=f(z_{n-1})$

Let $f : \mathbb C \rightarrow \mathbb C$, $f(z) = \frac{1}{2}z^2 + 1$, and $c = 1 + i$. Let $\{z_n\}$ be the sequence defined by iterating $f$ on some initial value $z_0 ∈ C$ (that is, $z_n = ...
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2answers
21 views

for which alpha is the Integral convergence

Let $\alpha>0$ and $$ f(x)=\frac{\ln x}{(x-1)^{\alpha}} $$ for $x>1$ i found that for $\int_2^{\infty}f(x) dx$ the integral is convergence for $\alpha > 2$ but for which $\alpha$ is ...
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3answers
245 views

Prove if the following sequence is convergent

$$ f(n) =\frac{6n^3 + 2n+(−4)^n}{4^n-1} $$ The sequence is dominated by $4^n$, so we divide by the dominant term, and we get $(-1)^n$ which is not a null sequence. So it is not monotonic and not ...
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1answer
26 views

Proving that the mean of a random variable is continuous, where is dominated convergence being used?

I am looking at the proof of the first part of this lemma. Previously in the text another theorem was stated: Convergence in distribution, $Y_n \implies Y$, holds iff $Ef(Y_n) \rightarrow Ef(Y)$ ...
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0answers
21 views

Show by comparison that the series $\sum_{n=2}^{\infty} \dfrac{1}{n^p (\ln(n))^q}$ is convergent if p>1 for all $q \in \mathbb{R}$ [duplicate]

So, I have to show (as the title says), that the series $\sum_{n=2}^{\infty} \dfrac{1}{n^p (\ln(n))^q}$ is convergent if p>1 for all $q \in \mathbb{R}$ by comparison. I've managed to show it for ...
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3answers
78 views

For which $\alpha$ does $\int_2^{\infty} \dfrac{\ln(x)}{(x-1)^{\alpha}} dx$ converge?

So, as the title says, I have to show which $\alpha$ makes $\int_2^{\infty} \dfrac{\ln(x)}{(x-1)^{\alpha}} dx$ converge? I have really have no idea how to do this. I've managed to show that it is ...
0
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1answer
12 views

Prove that the sequence is Cauchy

This is problem 17, page 55, from section 1.2: Cauchy Sequences in the textbook Introduction to Analysis, Fifth Edition, by Edward D. Gaughan. Prove that the sequence ...
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3answers
29 views

Analysis: Basic Sequence Proof

Prove that, if $\left\{a_n\right\}_{n=1}^{\infty}$ converges to A, then $\left\{|a_n|\right\}_{n=1}^{\infty}$ converges to |A|. Is the converse true?
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46 views

Almost sure convergence and L1 convergence

I am preparing myself for the mid-term exam of my probability theory exam, and am solving questions from previous years exams. One of these questions I couldn't answer, and so far I haven't found ...
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1answer
75 views

The convergence of $\sum_{n\geq 2}\frac{1}{n^{p}(\ln(n))^{q}}$ with two different tests.

Let $p,q\in\mathbb{R}$ and consider the series $\sum_{n\geq 2}\frac{1}{n^{p}(\ln(n))^{q}}$. i) Show by the comparison test, that the series is convergent if $p>1$ and divergent if ...
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1answer
42 views

Limits and Convergence

Prove: If $\lim \limits_{n \to \infty} a_{n+1}-a_n = 0$ then $a_n$ has to converge. I understand that the distance between adjacent $a_n$ elements approaches $0$. Since $a_{n+1}-a_n$ converges it has ...
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0answers
39 views

Check whether function series is convergent

I have the following task: Check whether given function series is pointwise convergent, uniformly convergent or "almost" uniformly convergent (id est $f, f_{n} : I \rightarrow \mathbb{R} $ and $ ...
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5answers
37 views

Calculus 2 Series Convergence - Can I Use Comparison Test?

Can I use the comparison test for the following problem? $$\sum _{n=1}^{\infty }\:\frac{1}{n^2-6n+10}$$ The denominator has a negative coefficient so i'm not sure if its valid to compare it to a ...
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1answer
47 views

Prove that $f_n(x)=\frac{1-(x/b)^n}{1+(a/x)^n}$ is uniformly convergent on the interval $[a+\epsilon ; b - \epsilon]$

Title says it all; I have to prove that the function sequence $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ is uniformly convergent on the interval $[a+\epsilon ; b - \epsilon]$, with ...
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0answers
22 views

Operator norm on the space on linear functions between Euclidean spaces.

*I'm reading a text which has a preliminary section on Linear maps. I have come across a conclusion that I can't seem prove by myself. * Let $Lin(\mathbb{R}^m,\mathbb{R}^n)$ be the space of linear ...
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1answer
56 views

Convergence of $\sum\limits_{n=1}^{\infty}\left (1-n\sin \frac{1}{n}\right)^\alpha$ for parameter $\alpha$

For exactly which real values of $\alpha$ is the series $$\sum_{n=1}^{\infty}\left(1-n\sin \frac{1}{n}\right)^\alpha$$ convergent? Please give some hints.
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1answer
40 views

How is the convergent sequence $\frac{1}{n-1}$ bounded?

In a metric space all convergent sequences are bounded. This example in the real numbers should then be bounded but, it is infinite at n=1 so I do not understand how this can be true. In the proof ...
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1answer
30 views

Show that $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ with $0<a<b,$ and $x \in [a,b]$ is not uniformly convergent

So, I have to show that $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ with $0<a<b$ and $x \in [a,b]$ is pointwise convergent, but not uniformly convergent. The pointwise convergence is pretty straight ...
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4answers
23 views

need some help with a power series convergence test problem

Find the interval of convergence for the given power series: $$\sum\limits_{n=1}^\infty \frac{(x - 1)^n }{n(-4)^n}$$ First I applied the generalized ratio test, came out with $\frac{(1-x)}{4}$ ...
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1answer
38 views

Calculus 2 Series convergence - For which positive integers k is the series convergent?

For which positive integers k is the series given below convergent? $$ \sum _{n=1}^{\infty }\:\frac{\left(n!\right)^6}{\left(kn\right)!} $$ I tried using Root/Ratio tests but that didn't work out. Not ...
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2answers
58 views

Example of a Power Series Given Interval of Convergence

This was a thought question assigned to our calc II class, and I wasn't sure how to approach it. Give an example of a power series whose interval of convergence is $(0, \frac{4}{3}]$. Show ...
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1answer
56 views

Disprove that if $\sum_n a_n$ converges, then $\sum_n a_n^2$ converges

I am attempting to answer a question from some Real Analysis exercises. The question asks if the series $\sum_n a_n$ converges, then does $\sum_n a_n^2$ converge, diverge, or is it impossible to tell. ...
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0answers
21 views

Is this method for showing convergence legit?

I am working with some comparison testing for the first time, and am unsure if the method I have adopted is legit. For example, imagine I want to compare some improper integrals (or series) with the ...
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2answers
65 views

How to determine if $\sum_{n=1}^{\infty}\left (\frac{n^2-5n+1}{n^2-4n+2}\right)^{n^2}$ converges or diverges

$$\sum_{n=1}^{\infty} \left(\frac{n^2-5n+1}{n^2-4n+2}\right)^{n^2}$$ Using root test seems not a efficient way since I got stuck without knowing what to do next ...
2
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1answer
44 views

Construction of a function $u$ such that $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ and $u \not\in W_0^{2,2}(\Omega)$

I'm wondering about an example of a function $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ such that $u \not\in W_0^{2,2}(\Omega)$. Clearly $W_0^{2,2}(\Omega) \subset W^{2,2}(\Omega) \cap ...
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1answer
51 views

Convergence of the sequence, $f_{n}(x)$.

Correct me if I am missing something or show me the better way. Let $0<a<b$ and consider the sequence of functions $$f_{n}(x)=\frac{1-(x/b)^{n}}{1+(a/x)^{n}}$$ for $n\in \mathbb{N}$. ...
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3answers
58 views

How to prove that $\sum\limits_{n=1}^{\infty}\frac{n+4^n}{n+6^n}$ converge by basics test (comparison, integral, ratio, root)

I have problem to prove that this series is converge. I know that it's converge without any proof but don't know how to prove it. $$\sum_{n=1}^{\infty}\frac{n+4^n}{n+6^n}$$
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Speed of convergence of squares of RVs

My problem appears to be pretty simple. I've obtained the speed of convergence — Berry-Esseen bound — for the expression $$ \underset{C \in \mathcal{C}}{\mathrm{sup}} | Q_{X, n} (C) - \Phi (C)| ...
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1answer
43 views

Convergence of series with integral test

Given that the following series is convergent, determine the values of p. $$\sum_{n=2}^{\infty}\dfrac{1}{n(\log(n))^p}$$ So far what I have done is using the integral test, in order to use integral ...
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2answers
49 views

Convergence of $\sum_{n=1}^{\infty} \frac{n^3}{\ln(2)^n}$

would the sum from 1 to infinity of $\frac{n^3}{(\ln{2})^n}$ converge? In the limit n tends to infinity the denominator grows more quickly and so the terms go to zero. Using the ratio test I get ...
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1answer
117 views

Equivalence between conditions for convergence

Let $(X_k)$ be independant random variables such that $X_k\sim\mathcal{P}(p_k)$ (Poisson distribution with parameter $p_k$). So in particular we have $ \sum_{n=1}^NX_k \sim \mathcal{P}(\sum ...
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2answers
34 views

Cute convergence problem. Proving convergence of sequence regarding reciprocals of least common multiple converges.

This is the first problem of the second day of the $2014$ CIIM. Let $\{a_n\}$ be a strictly increasing sequence of positive integers. Prove the sequence ...
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1answer
55 views

The convergence of a sequence with infinite products

I have a problem to determine convergence (sum over n). $$\sum_{n=0}^\infty \dfrac {a\left( a+1^{p}\right) \ldots \left( a+n^{p}\right) }{b\left( b+1^{p}\right) \ldots \left( b+n^{p}\right) }$$where ...
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1answer
39 views

Divergent series of random variables

I've been trying to prove that given a sequence of independent random variables with identical distribution $\{X_n\}_{n \in \mathbb{N}}$ such that $P(X_1 \neq 0)>0$, so also $P(X_i \neq 0) >0 \ ...
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2answers
63 views

Divergent, convergent series

Let $p$, $q \in \mathbb{R}$ and see the series $$ \sum_{n=2}^{\infty} \frac{1}{n^p(\ln n)^q} $$ View with the comparison criterion that if $p> 1$ then the series is convergent for all $q$, and ...
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1answer
14 views

proof of limit involving convergent sequence

Let (a$_n$)$_{n \in N}$ be a convergent sequence in R with the limit a $\in$ R. Furthermore, let y$_n$ = $\frac 1n$$\sum_{i=1}^n$a$_n$. Show that $\lim \limits_{n \to \infty}$ y$_n$ = a. -- I ...
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1answer
24 views

Representing functions as power series and finding $c_0,c_1,c_2$…

I have a problem with representing functions as power series: I was trying to find $c_0$, $c_1$, $c_2$, ... and the Radius of Convergence but I'm not sure how to do this, I'm a bit lost here could ...
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votes
3answers
150 views

Radius of Convergence and Interval of Convergence help

I had a bit of a problem with a question I had. $$\sum_{n=1}^{\infty}n^{n}(x-5)^{n}$$ While looking for the radius and interval of convergence through the ratio test I ended up with $|x-5| ...
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2answers
30 views

A sequence of distributions converges to a certain distribution.

Given the sequence of functions: \begin{equation} f_n(x)=tanh(nx) \end{equation} and knowing that: \begin{equation} \lim_{x \to \pm \infty}f_n(x)=f(x)=\begin{cases} -1, & x<0 \\ 1, & ...
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1answer
38 views

Convergence of $\sum_ {n=1}^\infty (1-n\sin\frac1n)^\alpha$ and $\sum_ {n=1}^\infty 2^n (\tan x)^{n^2}$

I was trying to solve a question of an entrance exam. I am having trouble in a particular type of problems. Please help me to solve. (Actually my last 2 questions are also from these exam papers. I ...
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0answers
36 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 ...
3
votes
1answer
10 views

Derivative limit is uniformly convergent

If we consider the sequence of functions: $g_{n}(x)=\frac{f(x+h_{n})-f(x)}{h_{n}}$ where $h_{n}>0$ is a sequence of real numbers converging to $0$, and $f$ is a $C^{1}$ function. How can you show ...
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6answers
44 views

Show that $\sum_{n=0}^{\infty}xe^{-n^2x}$ converges pointwise on $(0,\infty)$

$\sum\limits_{n=0}^{\infty}xe^{-n^2x} = x\sum\limits_{n=0}^{\infty}e^{-n^2x}$. Apparently the series converges pointwise on $(0,\infty)$ by a limit comparison test, but I cannot see what series I ...
0
votes
1answer
126 views

For which values of $\alpha$ is $\int_{1}^{\infty}f(x)\, \mathrm{d}x$ convergent?

Let $\alpha>0$ and define $f(x)=\ln(x)/(x-1)^{\alpha}$ for all $x>1$. Before I got this problem, I was asked to determine the values of $\alpha$ such that for each of ...
0
votes
1answer
36 views

$X_n\rightarrow X$ in probability, but $\mathbb{E}(X_n)$ does not converge to $\mathbb{E}(X)$ [duplicate]

What is an example of a sequence $X_1,X_2,...$ such that $X_n\rightarrow X$ in probability, but $\mathbb{E}(X_n)$ does not converge to $\mathbb{E}(X)$?