Convergence of sequences and different modes of convergence.

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help for convergence of an integral

Does this integral converges: $$\int_{-\infty}^{\infty}\frac{\mathrm e^{ -\mathrm{i}\omega t}}{\sqrt{(-1)^\frac{1}{3}(\mathrm{i}\omega)^\frac{4}{3}}}\,\mathrm{d}\omega$$ and why ?
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2answers
235 views

Limit function of pointwise convergence is always bounded?

If $\{f_n \colon [a,b] \rightarrow \mathbb R\}$ is a sequence of bounded functions converging pointwise to $f \colon [a,b] \rightarrow \mathbb R$, then $f$ is bounded. Is the statement above ...
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2answers
56 views

Convergence of $\max_{0\le i\le n}|f(i/n)|$

Suppose that $f\colon [0,1]\to\mathbb R$ is a continuous function. How can I prove that $$\max_{0\le i\le n}\biggl|f\Bigl(\frac in\Bigr)\biggr|\to\sup_{0\le x\le1}|f(x)|$$ as $n\to\infty$? Any help ...
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1answer
41 views

Problem about Convergence in Probability (3) [closed]

Let $X_1,X_2,\dots$ be a sequence of random variables that converge to $0$ in probability. That is, for any $\varepsilon > 0$, $\lim\limits_{n\rightarrow +\infty} Pr(|X_n-0|>\varepsilon) = 0$ ...
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2answers
39 views

Problem about convergence in Probability (2) [duplicate]

Let $X_1,X_2,\dots$ be a sequence of random variables with $$ \lim_{n\rightarrow+\infty}E\left[\left|X_n\right|\right]=0 $$ Is it true or false that the sequence $X_n$ must converge to $0$ in ...
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1answer
120 views

Convergence to a constant in probability but not almost surely

Please give a example that a sequence of random variables that converge to a constant $c$ in probability but fail to converge to $c$ with probability $1$. Thanks very much.
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1answer
74 views

Convergence of $x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$

$x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$ with $x_0=a\ge0$ I calculated some values for $x_n$ with $a=0,1,2$ and recognized its monotonically decreasing (It converges to $0$). Now the standard approach ...
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2answers
68 views

Are these converging or diverging

I am having trouble working out the convergence of these series and was wondering if I could please have some assistance a) $\displaystyle\sum_{n=0}^\infty\sin(e^n)\frac{n}{n^3+1}$ and b) ...
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1answer
178 views

Convergence in normed spaces

I consider a sequence of elements $\{f_n\}_n$ in a normed space $X$ such as $\Vert f_n-f\Vert_X\to 0, n\to\infty$. Let $\{Q_n\}_n$ a complex sequence with $Q_n\to Q$ as $n\to\infty$. Does $Q_nf_n$ ...
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1answer
112 views

Is Cesaro convergence still weaker in measure?

I've encountered a question I couldn't answer, and I would appreciate any help: Is it true that $f_n \xrightarrow{m}0$ $\Rightarrow$ $ \frac{1}{n} \sum_{k=1}^{n}f_k \xrightarrow{m}0$? Where ...
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2answers
68 views

Must the sequence $X_n$ converge to $0$ in probability?

Let $X_1, X_2,\dots$ be a sequence of random variables with $\lim_{n\to +\infty} E[|X_n|] = 0$. Is it correct or wrong that the sequence $X_n$ must converge to $0$ in probability?
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4answers
136 views

check the convergence of the integral $\int_{0}^{\infty}\frac{1}{x\log x}\,dx$

Help me on checking the convergence of the integral $$\int_{0}^{\infty}\frac{1}{x\log x}\,dx$$ I have tried it in this way $$\int_{0}^{\infty}\frac{1}{x\log x}\,dx=\int_{0}^{\frac{1}{2}}\frac{1}{x\log ...
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4answers
119 views

How to prove that $\displaystyle \lim_{n \to \infty} \sqrt[n]{n}=1$? [duplicate]

I have seen this fact thrown around a lot and never really stopped to prove it; plugging in a few values convinced me of its truth. But I would like to see the result proved. To be clear, this is not ...
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2answers
192 views

Weak convergence-exercice

Let $\Omega$ be an open set in $\mathbb{R}^n$ and let $(u_n)$ be a bounded sequence in $H^1_0(\Omega).$ Who's the theorem say that we can extract a subsequence denoted $u_{n}$ as $u_n$ weakly ...
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1answer
30 views

Condition for Law of Large Numbers, Monte Carlo

In some lecture notes I am reading, there is the following; Consider $X_{1},...,X_{n}$, each with pdf $g$ (the instrumental distribution). Our aim is to estimate $E_{f}[h(X)]$ where $h(X)$ is some ...
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2answers
100 views

Does the series always diverge?

Investigating the behavior of the following series: $$\sum_{k=2}^\infty \frac{1}{\log^{p}k}$$ I broke it into 3 parts: If $p = 0$ then it's just an infinite summation of ones, which diverges If $p ...
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2answers
86 views

Does $f_n \to 0$, a.e., implies $\int_{\mathbb R} \sin(f_n(x)) dx \to 0$, when each $f_n \in L^1$

Let $\{f_n\}$ be a sequence of $L^1(\mathbb R)$ functions converging a.e. to zero. Does $$ \lim_{n\to \infty} \int_{\mathbb R} \sin(f_n(x)) dx = 0? $$ I think the answer is no, but I can't find a ...
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2answers
131 views

check the convergence of the improper integral$\int_{0}^{1}\frac{x^{p-1}+x^{-p}}{1+x}\,dx$

How to check the convergence of the improper integral$$\int_{0}^{1}\frac{x^{p-1}+x^{-p}}{1+x}\,dx$$ I can only check that the integral is divergent for $p\geq1$, help for the cases when $p<1$. ...
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2answers
42 views

Investigate monotony, bound and convergence

I'm doing an exercise in the college, and I ran into a doubt: a friend says me that I've made a mistake, but I can't find it. The exercise asks: "Investigate if the sequence $2^n\over{(n+1)!}$ is ...
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5answers
238 views

Sequence of continuous functions which converges to a continuous limit [duplicate]

Any help with this: construct a sequence of continuous functions defined on $ [0,1] $ which converges pointwise but not uniformly to a continuous limit ? Thank you.
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1answer
107 views

Series of Functions - Pointwise and Uniform Convergence.

I was hoping for some help for the following questions. Prove that the series $\sum_{n=1}^\infty x^n(1-x)$ converges pointwise but not uniformly on $[0,1]$. Prove that the series $\sum_{n=1}^\infty ...
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3answers
60 views

finding values for absolute convergence

Find all values of real number p or which the series converges: $$\sum \limits_{k=2}^{\infty} \frac{1}{\sqrt{k} (k^{p} - 1)}$$ I tried using the root test and the ratio test, but I got stuck on ...
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3answers
170 views

Show that the sequence $(x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.

Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition. I'm not familiar with proving divergent sequence. Do anyone have any des? ...
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2answers
43 views

Radius of Convergence for $\sum \frac{[1\cdot 3 \cdots (2n-1)]^2}{2^{2n}(2n)!}x^n$

I'm trying to find the radius of convergence for this series: $$\sum \frac{[1\cdot 3 \cdots (2n-1)]^2}{2^{2n}(2n)!}x^n$$ so I have, $$R=\lim_{n\to\infty} \frac{[1\cdot 3 \cdots ...
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2answers
266 views

Composition of a continuous function with functions that converge uniformly

Here is problem that appeared in one of the past final exams for my introductory real analysis course, that I am having hard time to solve. It is Question 5 in 8 of the following file: ...
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2answers
90 views

Absolute Convergence of a Series

Find all values of real number p for which the series converges absolutely $$\sum_{k=2}^{\infty} \frac{1}{k\, (\log{k})^p}$$
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1answer
46 views

$L^p$ convergence proof check

I don't have much experience with measure theory, so I want to make sure that I'm not making any bad mistakes. I also want to be sure that the theorem is true so I can use it. Theorem: Let $\{u_i\}$ ...
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1answer
114 views

What's the lowest real $x$ such that $\zeta(x)$ converges?

It's easy to prove that$$\zeta(1)=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...$$ diverges, and $$\zeta(2)=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...$$ converges to $\frac{\pi^2}{6}$. Intuiting the ...
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1answer
43 views

Convergence and monotonicity of roots of $\frac{1}{1 + x - \exp(-x)} = n$

We suppose that $\forall x\in \mathbb{R}\setminus{0} \quad f(x)=\dfrac{1}{1+x-\exp(-x)}$ , and $x_n \in ]0;+\infty[$ as $x_n$ is an unique solution of the equation $f(x)=n$ on $]0;+\infty[$. How can ...
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2answers
84 views

Show $\sum \frac{3^{2k+1}}{k^{2k}}$ converges

I'm pretty certain it can be shown using the ratio test; I simplified $a_k+1/a_k$ to $[ 9(k)^{2k} )/( (k+1)^{2k+2} ]$ then let lim k->inf a_k+1/a_k = x thus lnx = lim k->inf ln[ 9(k)^(2k) )/( ...
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2answers
67 views

Find a convergent function in metric space

Let $C[−1, 1]$ be the space of continuous functions equipped with the metric $p(f,g) = \max\{|f(x)−g(x)| \mid x \in [−1, 1]\}$. Then the sequence of functions $(f_n):[−1,1]\rightarrow \mathbb{R}$ ...
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convergence in metric space

Let $C[-1, 1]$ be the space of continuous functions equipped with the metric $(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$. Consider the sequence $(f_n)$ of functions $f_n : [-1, 1] \to ...
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1answer
143 views

Prove convergence of improper integral using change of variable.

This may be trivial, but I could use some help... Consider a real function $f: (0,1) \rightarrow \mathbb{R}$, continuous, positive, but not necessarily bounded. Let $g: [0,1] \rightarrow [0,1]$ be a ...
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0answers
152 views

weakly convergent sequence in $l^1$ [duplicate]

Prove that every weakly convergent sequence in $l^1$ converges. By Riesz on $L^p$ spaces, every linear functional $L\in (l^1 )^*$, is $L(x) = \langle u,x\rangle$ for some $u\in l^{\infty}$ ...
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2answers
115 views

Convergence of sum of random variables

Let $X_n$, $n\geq 0$, be i.i.d. random variables such that: $\mathbb E(X_1)=0$, and $0<\mathbb E(|X_1|^2)<\infty$. Given that $\alpha >\frac{1}{2}$, I need to show that ...
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1answer
40 views

Uniform convergence of a series through induction principle

I'm working on this paper. Can you please explain me the following passage of the proof that the series (2.9) converges uniformly? Given: $$ |\delta_{k+1}| \le ...
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1answer
58 views

Prove rearrangement of harmonic series tends to 1 or -1

Prove that a rearrangement of the series $\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}$ has a subsequence that tends to 1 or -1 or prove otherwise. I did a proof that brought me to a converging limit to ...
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1answer
46 views

Showing that a sum diverges

Suppose that $a_{j} \geq 0$ and that $\sum a_{j}$ diverges. Prove that $\sum\frac{a_{j}}{1+a{j}}$ diverges. The hint that is given is show that it if it converges $a_{j} \rightarrow 0$. I don't ...
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1answer
169 views

Convergence in distribution - Gamma distribution

If we have a random variable defined as $Y_{n}=\displaystyle\frac{X_n-n\alpha}{n\alpha^2}$, where $X_n$ is $\operatorname{Gamma}(n,\alpha)$ distribution, how can I prove that $Y_n$ converges in ...
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0answers
91 views

sequence with almost surely convergence & moment of order 2

Let's say we have iid random variables $(X_n)$ s.t. $X_1$ admits a moment of order 2. For $n \geq 1$, does the following sequence converge almost surely or not? Why? And how to see/show? $$A_n = ...
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2answers
90 views

Convergence of infinite series with p-test and constant

My question is: Does the infinite series $\sum_{n=1}^\infty \frac{1}{n^{\frac{4}{5}}+10^{10}}$ converge or diverge? I know that $\frac{1}{n^{\frac{4}{5}}}$ diverges by the $p$-test, and that adding ...
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2answers
79 views

Equality of limits with respect to different metrics.

Suppose that $X$ is a set equipped with two metrics, say $d_1$ and $d_2$. Let $\{x_n\}_{n\in\mathbb{N}}\subset X$ be a sequence of points which converges to $x\in X$ with respect to metric $d_1$. ...
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1answer
42 views

The existence of a subsequence of harmonic functions that converges pointwise

Let $u_{n}$ be a family of harmonic functions on $\mathbb{R}^n$, and there exists a point $x_{0}$ such that $\{u_{n}(x_{0})\}$ is bounded. Then does it exist a subsequence of $u_{n}$ that converges ...
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1answer
77 views

Stochastic variable equals indicator function?

An exercise in my statistics & probability theory course goes as follows: $\Omega = [0,1], \mathcal{B} = \mathcal{B}([0,1]), P$ the Lebesgue measure on $[0,1]$. We have the sequence of ...
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1answer
50 views

Convergence in norm operator topology

I have to prove that a sequence $A(\varepsilon)$ of operators between Hilbert spaces $A(\varepsilon):H_1\to H_2$ converges, when $\varepsilon\to 0^+$, to an operator $B:H_1\to H_2$ in the uniform norm ...
0
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1answer
614 views

Whether convergence in L2 norm implies convergence a.e.? [duplicate]

How to prove or disprove$$\lim_{n\to\infty}\|f_n-f\|=0\;\Rightarrow \;\lim_{n\to\infty}f_n(x)=f(x)\; a.e.?$$ Any hint is appreciated.
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2answers
67 views

Prove or disprove a result for a double sequence. [closed]

Suppose that a double sequence $\{a_{n,k}\}=\left\{\frac{1}{n^{\frac{k-1}{k}}}\right\}$. Prove or disprove $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k}a_{n,k}=0$.
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1answer
138 views

Vector space of convergent sequences, prove it's complete

In the space of convergent sequences, such that for a convergent sequence $(a_n)$ we have $\sum _{n=1} ^{+ \infty} a_n ^2 < \infty$, we define a norm $(a_n) \rightarrow \sqrt{\sum _{n=1} ^{+ ...
3
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1answer
222 views

Convergence Counterexamples

I'm trying to compile a list of counterexamples for convergence implications (or rather, the lack of). I have an incomplete list and I hope to get it all together in one piece. I'm currently working ...
3
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1answer
87 views

Series convergence $\frac{(z+2i)^n+(z-i)^n}{n^2 \cdot 2^n}$

Could you tell me what can I do to determine convergence of $\frac{(z+2i)^n+(z-i)^n}{n^2 \cdot 2^n}$ for $z \in \mathbb{C}$ ? Thank you.