Convergence of sequences and different modes of convergence.

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2
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2answers
67 views

Showing $s_n = \left(\frac{1}{2}\right)(s_{n-1} + s_{n-2})$ is Cauchy.

Let $(s_n)$ be a sequence defined as $s_1 = 1, s_2 = 2$ , and $s_n = \left(\frac{1}{2}\right)(s_{n-1} + s_{n-2})$. Prove that $(s_n)$ is Cauchy. I can see how it is convergent and Cauchy but not sure ...
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1answer
31 views

Convergence radius of complex power series

If $a_n\neq 0$ for all $n \geq n_0$ and $\lim|\frac{b_n}{a_n}|=1$, then $\rho(S)=\rho(T)$. Since S=$\sum a_nz^n$ and T=$\sum b_nz^n$. I tried to use the definition of convergence radius $$\limsup ...
2
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0answers
141 views

Series convergence question

This question occured to me in the context of ARMA time series analysis: Let $\alpha_n\geq 0,\, n= 1,2,\dots$ such that $\sum_{n=1}^\infty\alpha_n = 1$ and define the sequence ...
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0answers
56 views

Limit of a sequence of Marlov processes

Let $(X_n)$ be a sequence of Markov processes on, say, $[0,1]$, that converges in finite dimensional distributions to a process $X$. Is it true that $X$ must also be a Markov process ?
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1answer
35 views

$\sum a_{n}$ is convergent, $\sum a_{n}^2$ is divergent. Prove $\sum a_{n}$ is conditionally convergent.

Suppose that the series $\sum_{n=1}^{\infty}a_{n}$ converges while $\sum_{n=1}^{\infty}a_{n}^2$ diverges. Prove that $\sum_{n=1}^{\infty}a_{n}$ converges conditionally.
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0answers
34 views

Convergence of $\int^{+\infty}_{0} \frac{\arctan x }{x^a} \sin x\, dx$

I´m trying to find out if this integral is convergent (and for what values of $a$) or not: \begin{equation*} \int^{+\infty}_0 \frac{\arctan x}{x^a} \sin x \ dx,\quad a \in \mathbb R \end{equation*} ...
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2answers
36 views

Uniform convergence of sequence of functions with infinite roots to a limit with finite roots

Consider a sequence of continuous functions $(f_n)$ defined over $[0,1]$ such that, for all $n$, the set: $$A_n = \{x\in [0,1] : f_n(x) = 0\}$$ is infinite in cardinality. Can $(f_n)$ uniformly ...
0
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1answer
17 views

Determing taylor series from other series

Consider $\cos(x)$ and $\cos(3x^2)$. How to determine the latter's Taylor series from the formers at $a = 0$? I'd write $$\cos{x} = \sum_0^\infty (-1)^n\frac{x^{2n}}{(2n)!}$$ Now, I could just ...
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1answer
14 views

On convergence in probability given a bound on the random variable.

I am dealing with the end of a proof: Could somebody please clarify for me the extra steps needed to show that $P(||\bar{W}_n - \mu||_{\infty} > 3\varepsilon) \rightarrow 0$ as $n \rightarrow ...
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1answer
25 views

Convergence in the Tychonoff topology on $\mathbb{R}^\mathbb{R}$.

This is Example 10.2(b) (p.70) in Willard's General Topology: In the product space $\mathbb{R}^{\mathbb{R}}$, a sequence $f_n$ converges to $f$ iff $f_n(x) \rightarrow f(x)$ for each $x \in ...
5
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1answer
188 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 ...
0
votes
3answers
63 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.
3
votes
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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0answers
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 = ...
0
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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 ...
3
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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 ...
0
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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
22 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 ...
0
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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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0answers
47 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 ...
0
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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 ...
1
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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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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
57 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 ...
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1answer
45 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 ...
2
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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 $n\in \mathbb{N}$. ...
2
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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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0answers
13 views

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 ...
0
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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 ...
5
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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 ...
2
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2answers
35 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 ...
5
votes
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 ...
6
votes
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 ...