Convergence of sequences and different modes of convergence.

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2
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1answer
69 views

Show that $\sum_{n=0}^\infty a_n z^n$ converges $\forall z\in\mathbb{C}.$

Assume that $\sum_{n=0}^\infty b_n z^n$ converges $\forall z\in\mathbb{C}.$ Let $x=\lim_{ n\rightarrow\infty}|\frac{a_n}{b_n}|$ exists. Show that $\sum_{n=0}^\infty a_n z^n$ converges $\forall ...
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3answers
42 views

I'm asked to prove the following problem on Convergence. Though I'm not sure how to apply the correct definitions.

Verify that the sequence $\left\{x_{n}=\frac{n^{2}-1}{3n^{2}-4}\mid1\leq n\leq\infty\right\}$ converges to $\frac{1}{3}$ by using the definition of limit. I've tried all the available definitions ...
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0answers
16 views

Convergence of integrals in an arbitary Rieszspace

Hello i have a question about integration theory: The situation: Let $X$ be a set, $F$ a Rieszspace of $X$ and $\varphi$ an integral on $F$. Now we can extend this to the space of all integrable ...
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1answer
32 views

how to prove $E(Y_n|B_n)\to E(Y_\infty|B_\infty)$

Suppose $B_n\uparrow B_\infty$ and ${Y_n, n \in \bar{N}}$ is a sequence of random variables such that $Y_n \to Y_\infty$. (a) If $|Y_n|\le Z \in L_1$, then show a.s. that $$E(Y_n|B_n)\to ...
5
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2answers
51 views

Convergence to half Euler's constant

Euler's constant is defined by $$\gamma=\lim_{N\rightarrow\infty}\sum_{n=1}^N\dfrac1n-\log N.$$ So we can write it as $$1+\dfrac12+\dfrac13+\ldots+\dfrac1n-\log n\rightarrow \gamma.$$ How can we ...
1
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4answers
141 views

If $ \lim_{n \to \infty} (2 x_{n + 1} - x_{n}) = x $, then is it true that $ \lim_{n \to \infty} x_{n} = x $?

If $ (x_{n})_{n \in \mathbb{N}} $ is a sequence in $ \mathbb{R} $ and $ \displaystyle \lim_{n \to \infty} (2 x_{n + 1} - x_{n}) = x $, then is it necessarily true that $$ \lim_{n \to \infty} x_{n} = ...
1
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2answers
108 views

If a sequence $s_n$ converges, does $f(s_n)$ converge?

Let $f:X \to Y$ be a function. Let $s_n$ be a convergent sequence in $X$. Then does the sequence $f(s_n)$ converge as well? Does $f$ have to be continuous? My thoughts: I believe $f$ does converge. ...
0
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2answers
42 views

If each $f_n$ is continuous on a set $S$, does $f_n$ converge pointwise to a function $f$ on $S$?

If each $f_n$ is continuous on a set $S$, does $f_n$ converge pointwise to a function $f$ on $S$?I feel I am seriously misunderstanding something. Am I asking a vacuous question?
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1answer
118 views

Does a power series with radius of convergence $R=+\infty$ converge uniformly to a continuous function everywhere?

I know that a power series with radius of convergence $R>0$ converges uniformly on $[-R_1,R_1]$ to a continuous function (where $0<R_1<R$). Would that imply that if $R=+\infty$, the power ...
2
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1answer
62 views

Fourier inversion formula with truncation

Let $f\in L^2(\mathbb{R})$, and denote $$s_N(x)=\dfrac{1}{2\pi}\int_{-N}^N\hat{f}(t)e^{ixt}dt.$$ Show that $$\lim_{N\rightarrow\infty}\int_\mathbb{R}|s_N(x)-f(x)|^2dx=0$$ So, $s_N(x)$ is the ...
0
votes
1answer
120 views

Fourier series convergence for sum of Schwartz class functions

Let $f$ be a Schwartz class function. Let $F(x)=\sum_{n\in\mathbb{Z}}f(x-2\pi n)$. Then $F$ is periodic of period $2\pi$. How can we show that the Fourier series of $F$ converges to $F$ pointwise ...
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2answers
44 views

A sequence converges uniformly

If a sequence $\{f_n\}$ converges to $f$ uniformly in $\mathbb{R}$, does it follow that $(f_n)^2$ converges to $f^2$ uniformly?
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1answer
105 views

Convergence to a uniformly distributed r.v.

For $n \geq 1$, let $X_{n}$ have the uniform distribution on the interval $(\alpha_{n},2+\alpha_{n})$, where $\alpha_{n}=(-1)^{n}$. The sequence $(X_{n})$ is said to be $\mathbf{stochastic}$ ...
3
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2answers
37 views

Convergence of $\sum_{n=2}^{\infty}n^2\left(\frac{1-i}{2+i}\right)^n$

Does this sequence converge/diverge and if so, does it in a (not)absolute way? $$\sum_{n=2}^{\infty}n^2\left(\frac{1-i}{2+i}\right)^n$$
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2answers
51 views

Example of an integral not converging

Consider a measurable space $(\Omega,\mathcal{A},\mu)$ with $\mu(\Omega)<\infty$. Let $f_1,f_2,\ldots$ be bounded measurable functions so, that $f_n\to f$ uniformly. Then $f$ is measurable ...
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1answer
55 views

A function of $u(0,1)$ random variables converging weakly to an exponential

This is a review problem for my final exam: Let $(X_{n})_{n\geq 1}$ be an i.i.d. sequence of random variables with $X_{i} \sim U(0,1)$. Let $M_{n}=\max_{1\leq i \leq n}X_{i}$. Show that $n(1-M_{n})$ ...
0
votes
1answer
17 views

Convergent Sequences involving functions

Let $f: (M,d) -> (N,p)$ be a function and suppose that {${f(Xn)}$} is a convergent sequence, whenever $Xn -> a$. Prove that $f$ is continuous at $a$ My thinking is that since Xn converges to ...
3
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1answer
83 views

$L^2$ norm of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$f_r(\theta)=\sum_{n=1}^\infty r^na_ne^{in\theta}$$ is a harmonic ...
0
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2answers
55 views

$\lim_{n\to \infty} \int^1_0 f_n(x)dx=0$ , is $f_n$ pointwise convergent?

Let $f_n(x)$ be a sequence of continuous non negative functions on $[0,1]$ such that $\lim_{n\to \infty} \int^1_0 f_n(x)dx=0$ . Then does $f_n$ necessarily converge pointwise? I think it is not ...
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1answer
119 views

Why is it that convergence almost surely and convergence in the r-th mean norm do not necessarily imply each other?

Why is it that convergence almost surely and convergence in the r-th mean norm $L_r$ do not necessarily imply each other?
3
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1answer
67 views

Convergence of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ If $z=re^{i\theta}=x+iy$, $$F(z)=\sum_{n=1}^\infty r^na_ne^{in\theta}$$ is a harmonic ...
1
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1answer
79 views

Convergence of Fourier series for piecewise constant function

Let $f(x)=1$ for $x\in (0,\pi)$ and $f(x)=0$ for $x\in (-\pi,0)$. Furthermore, extend $f$ to be periodic of period $2\pi$. I calculated the Fourier series of $f$ to be ...
5
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2answers
128 views

How to determine the convergence of $\sum\limits_{n=1}^{\infty}\frac{n^{n^2}}{(n+x)^{n^2}}$

I've been studying for my analysis exams, and I've come across a series I haven't been able to solve. The question is just to determine for which real values of $x$ does the series converge. I've ...
1
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1answer
46 views

Convergence of $\int_{0}^{2}{k^{p}x^{k} \over 1 + x^{2k}}\,{\rm d}x\quad\mbox{with}\quad k = 1,2,3,\ldots\quad\mbox{and constant}\quad p$

Consider the integrals $$ \int_{0}^{2}{k^{p}x^{k} \over 1 + x^{2k}}\,{\rm d}x\quad\mbox{with}\quad k = 1,2,3,\ldots\quad\mbox{and constant}\quad p. $$ For what values of $p$ do the integrands have an ...
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3answers
105 views

Radius of Convergence - $\sum_{n=1}^{\infty}2^n x^{n^2}$

What is the radius of convergence of this power series here? $$\sum_{n=1}^{\infty}2^n x^{n^2}$$
2
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1answer
132 views

Laurent series, radii of convergence.

I'm working on the following exercise: Prove that a Laurent series \begin{align*} \sum_{n = -\infty}^\infty a_n(z-z_0)^n = \sum_{n = 0}^\infty a_n(z-z_0)^n + \sum_{n = 1}^\infty ...
0
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1answer
102 views

Convergence of Sequence of Real Numbers

Define a sequence of real numbers recursively as follows. Let $a_1 = 1$ and $a_{n+1} = 1 + \frac{1}{1+a_n}$. First, show the sequence is not monotonic. Second, show that $a_n \geq 1$ for all $n$ and ...
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2answers
59 views

Prove that $ \left(a_{n}\right)_{n=1}^{\infty} $ converges when $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ for $ 0<q<1 $

I'm stuck on a homework question, and could really use some help. Here is said question: "Assume that for every $n$ the following occurs: $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ when $ 0<q<1 $ ...
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0answers
60 views

Prove that the sequence $1+\sum_{k=1}^{n}\frac{k+1}{3^{k}+1}$ converges using Cauchy

I need some help with a homework question i'm having difficulty with. Here is the question: "Use the definition of cauchy sequence to prove that the series ...
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2answers
39 views

$a_n =(\frac{n}{n+1})^n$ find convergence by geometric series

$a_n =(\frac{n}{n+1})^n$ I calculated a few terms: $a_1 = 1/2$ $a_2 = 4/9$ $a_3 = 27/64$ But not sure how to find $r$ so i can calculate its value as $n$ approaches infinity (if it does converge)
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1answer
45 views

Proving vector spaces are banach spaces

Let $V$ be the vector space of absolutely convergent series of real numbers under pointwise operations and with norm $\|\{a_n\}\| = \sum_n |a_n|$. Is $V$ a Banach space?
2
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1answer
96 views

Do The Integrals Tend to 0?

Consider the integrals $\int_1^\infty \frac{k}{x^2+k^p\cos^2x}dm(x),$ where $m$ is the Lebesgue measure. For what $p$ do the integrands have an integrable majorant? For what $p$ do the integrals tend ...
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0answers
49 views

Convergence in mean with logarithm

The problem is to prove that $E[\log|X_1-\bar{X}_n|]$ converges towards $E[\log|X_1|]$ for i.i.d. continuous random variables $X_1,\ldots,X_n$ with $E[X_i]=0$ and $Var[X_i]=1$, for example for Laplace ...
2
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2answers
95 views

Interval of convergence of $\sum \limits_{n=1}^\infty\ln(1+\frac{x^2}{n})$

How to determine the convergence interval of the $\sum \limits_{n=1}^\infty\ln(1+\frac{x^2}{n})$ function series? The required condition is proven, it can be convergent, but where?
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2answers
159 views

Convergence of $\sum\limits_{n=1}^\infty \frac{1}{n} \tan\left(\frac{1}{n}\right)$

Is $\sum\limits_{n=1}^\infty \frac{1}{n} \tan\left(\frac{1}{n}\right)$ convergent? The required condition is ok, but how to find a minorant or majorant series for this.
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1answer
195 views

Shifted exponential limiting distribution

Assume we have random variables $X_1,..,X_n$ which have the pdf $f(\theta):=\exp(-(x-\theta))$ for $x>\theta$ and zero elsewhere. Let $X_{(1)} =\min(X_1,...,X_n)$ then I want to find the pdf of ...
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1answer
44 views

Sequence Converging to Supremum

Show if $C$ is a subset of $\mathbb{R}$ which has a supremum $\alpha$, then there is a sequence $(c_n)$ from $C$ which converges to $\alpha$. So far I have by definition of a supremum, for each ...
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1answer
37 views

On (absolute) convergence of $f_c:= c + \sum_{n=0}^{+ \infty} \frac{a_n}{n+1}x^{n+1}$

Let $R> 0$ and let $g: (-R,R) \longrightarrow \mathbb{R}$ be given by the convergent power series $$g(x):= \sum_{n=0}^{+ \infty} a_nx^n$$ for $|x| < R$. Let $c \in \mathbb{R}$ and let $f_c: ...
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0answers
64 views

Convergence of the Integrals

Consider the integrals $\int_1^\infty\frac{x^2+kx}{x^4+k^px^2+k^2}dm(x)$ for k=1,2,3,.... For what p do the integrands have an integrable majorant? For what p do the integrals tend to $0$? I'm ...
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0answers
88 views

Proving completeness in normed vector space

Given norm $$ \| \{ a_n \} \| =\sum_{i=1}^\infty | a_n|,$$ where $E$ is a vector space of absolutely convergent series of real numbers under pointwise operations. How would one prove completeness of ...
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1answer
64 views

Can someone show me step by step how to handle this convergence problem?

I just took my final and one of the questions read: Use the integral test to determine the convergence or divergence of the series $$\sum_{n=1}^{\infty} \frac{e^n}{1+e^{2n}}.$$ I struggled ...
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1answer
71 views

Help with Homework Problem

Let $f_k(x)=|x-1/k|^{-2/3} (k=1,2,3,...).$ Do the $f_k$ have an integrable majorant, meaning a function bounding $f_k$ that satisfies the dominated convergence theorem, on the interval [0,1] with ...
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1answer
60 views

Question about convergence of a power series and when the series is not zero

Following is a past exam question I am trying to solve as a preparation for my own exam. Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers with $a_n \leq M$ for some $M\in\mathbb{R}^{+}$ and ...
0
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1answer
58 views

Another Question on Convergence

Let $f_n$ be a sequence of non-negative measurable functions on $(X,\mathscr{A},\mu)$ such that for every $x\in X$, the sequence $f_n(x)$ is convex. Show that the limit of $f_n(x)$ as n goes to ...
0
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1answer
35 views

Convergence Problem With Parametric Family

Let $f_t(x)=\left|\sin tx\right|^t$, $x\in[0,1]$, $t\in(0,+\infty)$. Does the parametric family converge on $[0,1]$ in measure with respect to the Lebesgue measure as $t$ approaches $+\infty$? Does it ...
0
votes
1answer
48 views

Convergence/Divergence of sums

I was asked to determine if the next sums converge absolutely, converge conditionaly or diverge. For the first question I tried to use Leibniz: Define $a_n=\frac{1}{n^a ln(n)}$. It's easy to show ...
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0answers
64 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
4
votes
4answers
215 views

Convergence of $a_n=(n+1)^{100}e^{-\sqrt{n}}$ for $n\geq 1$

Question is to check for Convergence of : $a_n=(n+1)^{100}e^{-\sqrt{n}}$ for $n\geq 1$ what i could do is : ...
1
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1answer
283 views

Conditions for convergence of moments given uniform convergence of distribution functions

Setup: Let $S_n = n^{-1} \sum_{i=1}^n X_n$ denote a sample mean and let $S_n^*$ denote a stationary bootstrap re-sample of $S_n$. Let $F_n(x)$ denote the cumulative distribution function of $\sqrt{n} ...
0
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0answers
50 views

Does a a.s. convergence of $X_n\to X$ and a $E|X_n|^{2+\epsilon}\to E|X|^{2+\epsilon}$ combined, imply a $L^2$ convergence?

Does a a.s. convergence of $X_n\to X$ and $E|X_n|^{2+\epsilon}\to E|X|^{2+\epsilon}$ convergence combined, imply a $L^2$ convergence $X_n\to X$ (namely, $E|X_n- X|^2 \to 0$)? Or more practically: ...