Convergence of sequences and different modes of convergence.

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Divergence in probability

The sequence $(X_n)$ is said to diverge to $+\infty$ in probability if $\mathbb{P}\{X_n>b\}\to 1$ as $n\to\infty$ for every $b\in\mathbb{R}_+$. If $(X_n)$ diverges to $+\infty$ in probability and ...
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1answer
23 views

Uniform convergence of a complex power series on a compact set

I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ I ...
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0answers
30 views

Convergence of derivative in L1

Let $F(x)$ be a distribution function over $\mathbb{R}$ with positive derivative at origin $f(0)$. Let $Q$ be a measure on $\mathcal{B}(\mathbb{R})$. Can we have following results under some ...
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1answer
19 views

Integral of square of Brownian motion with respect to Brownian Motion

While trying to compute $\int_0^TB_t^2\ dB_t$, $B$ being the standard Brownian motion, I got stuck at showing the following. $$\sum_{i=0}^{n-1}B_{t_i}(B_{t_{i+1}}-B_{t_i})^2 \rightarrow \int_0^TB_t\ ...
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1answer
25 views

$\int_0^\pi g(x)\sin(nx) dx$ coverges to $0$ for any function $g$ is bounded variation

I have a homework as follows : Prove that $\int_0^\pi g(x)\sin(nx) dx$ coverges to $0$ for any function $g$ is bounded variation on $[0,1]$. my attempt: for any bounded variation function $g$, ...
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1answer
129 views

If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?

The random variables take values in $\mathbb{R}^d$. I have tried to prove this using characteristic functions. Let $\hat{\mu}_{X_n},\hat{\mu}_{Y_n},\hat{\mu}_{Z_n}$ be the characteristic functions of ...
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23 views

Product over a factorial convergences

I'm working on a problem and need this to converge to any value: $\displaystyle \frac{(1/2)((1/2)-1)((1/2)-2)\cdots ((1/2)-n+1)}{(n+1)!} = \Pi_{j=1}^{n} \frac{\frac{1}{2} - j+1}{j+1}$ The ...
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1answer
49 views

Prove $\sum E((X_n-X_{n-1})^2)$ is finite iff $X_n$ converges to $X$ in $L^2$

Let $(X_n)_{n \in \mathbb{N}}$ be a martingale. Prove $\sum_n E((X_n-X_{n-1})^2)$ is finite iff $X_n$ converge to $X$ in $L^2$. It is not hard at first glance, but I cannot figure it out after many ...
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46 views

Exchanging $\lim$ and $\inf$?

Suppose we have a sequence of functions $f_n(x)$ that converge to a limiting function $f(x) = \lim_{n \to \infty} f_n(x)$ for $\forall x \in [a,b]$. I was wondering under what conditions the following ...
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15 views

Relation between monotone and dominated convergence theorem and related questions

In the lecture, we often use monotone and dominated convergence. Since I have not studied maths, I have some problems understanding it, so it would be very helpful if you could try to explain it to me ...
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1answer
38 views

How to prove this Brownian motion convergence?

Let $W_t$ be a Brownian motion. How do I show the following? $$ \alpha > \frac{1}{2} \Rightarrow \lim_{t\rightarrow\infty} \frac{W_t}{t^{\alpha}} = 0 \text{ a.s.} $$ Showing convergence of this ...
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28 views

Product of power series as a product of their coefficients

Suppose that $f(x)=\sum_{j=0}^\infty a_j x^j$ and $g(x)=\sum_{k=0}^\infty b_kx^k$ have positive radii of convergence $R_1$ and $R_2$ respectively. Let $c_n=\sum_{j=0}^n a_jb_{n-j}$ for $n\ge0$; and ...
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1answer
16 views

How to derive the discrete fourier transform of $(n+2)a^nu[n]$ where $|a| < 1$?

This is a rather simple question, but I'm stuck on one step. Here's what I've done: 1) $x[n] = (n+2)(\frac{1}{2})^nu[n] = n(\frac{1}{2})^nu[n]+2(\frac{1}{2})^nu[n]$ The discrete fourier transform is ...
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40 views

Taylor serie of exponential function is not uniformly convergent on R+

Let $E_n(x)=1+x+(x^2/2!)+...+(x^n/n!)$ I'm asked to prove that this function is not uniformly convergent on $\mathbb{R^+}$ I really don't know how to start here, if anyone could give me a hint it ...
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1answer
29 views

Stuck with understanding transformation step in calculating limit of $n(\sqrt[n]{a}-1)$

Although this question has already been asked in general ( $\lim\limits_{n\to\infty} n·(\sqrt[n]{a}-1)$) , my question is different, because I am stuck with a specific transformation step: ...
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3answers
93 views

Convergence of $\sum_n \frac{n!}{n^n}$

I'm working on a problem sheet and it ask to discuss the convergence of $$\sum \frac{n!}{{n}^{n}}$$ By D'Lembert's ratio test, $$\lim_{n->\infty}\frac{{a}_{n+1}}{{a}_{n}} = 1$$ and so, is ...
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20 views

Convergence of an integral of a function

Looking at the last line, we have $$\sum_{n=1}^{N} f(n)\leq f(1) + \int_1^N f \leq f(1) + \int_1^\infty f $$ I'm extremely disturbed by the fact the fact that the two integrals lies on the ...
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1answer
14 views

Unique solution and iteration convergent to the solution

The task is to: Prove that $x=\sin x+\frac{1}{4}$ has got a unique solution on $[\pi/4,\pi/2]$. Show that the iteration $x_0\in[\pi/4,\pi/2], x_{n+1}=\sin x_n+\frac{1}{4}$ is convergent to that ...
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1answer
60 views

need radius of convergence of $e^{-x^{2}}$

I am having difficulty finding the radius of convergence of $e^{-x^{2}}$ this is for introductory analysis course. Have looked at even and odd subsequences of powerseries, but so far unable to put ...
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2answers
22 views

Proof of convergence for a sequence composed by another sequence

How can I prove that $ y_n = \frac{x_1 + x_2 + x_n}{n}$ is convergent, knowing the fact that also $x_n$ is convergent? Thanks
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6answers
380 views

Proof of convergence of a recursive sequence

How do I prove that $x_{n+2}=\frac{1}{2} \cdot (x_n + x_{n+1})$ $x_1=1$ $x_2=2$ is convergent?
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3answers
64 views

Comparison test to prove $\frac{1}{p!}<\frac{1}{2^{p-1}}$

I'm trying to show $\frac{1}{p!}<\frac{1}{2^{p-1}}$ for $p>2$ I notice that the right hand side of the inequality represents a geometric sequence. I can compute its infinite sum but how do I ...
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4answers
45 views

Find an example of two convergent sequences $x_n$ and $y_n$, where $x_n < y_n$, for all $n \in \mathbb{N}$, but $\lim x_n \not < \lim y_n$

Ok. I am trying to solve an exercise in my last calculus assignment, which is the following: Find an example of two convergent sequences $x_n$ and $y_n$, where $x_n < y_n$, for all $n \in ...
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1answer
33 views

When converges of function imply converges of derivative?

Let $F_n$ be a sequence of differentiable real valued functions. Suppose that $$\lim_{n \to \infty} F_n(x) = F(x)$$ and that $F(x)$ is differentiable. Under which conditions does that imply ...
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3answers
71 views

Calculus: Converge of a recursive series?

I'm going crazy to solve this problem: I've a sequence defined by: $$x_1 = 1$$ $$x_2 = 2$$ $$x_{n+2} = \frac{1}{2}(x_{n}+x_{n+1})$$ And I have to prove that this sequence converges and what is its ...
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39 views

Fourier series convergence question from big Rudin.

I am working on some problems from the 3rd edition of Rudin's "Real and Complex Analysis" and I'm stumped on proving the following part from question #19 of chapter 5. Suppose $\lambda_n/\log n \to ...
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2answers
40 views

I'm unsure which test to use for this series, and how to prove?

I want to determine if this series converges or diverges: $$\sum\limits_{n=1}^\infty{\frac{3^\frac{1}{n} \sqrt{n}}{2n^2-5}}$$ I tried the Ratio Test at first, and didn't get anywhere with that. I'm ...
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23 views

Is Helly's First Theorem another Bolzano-Weierstrass Theorem working in BV(X) where X is compact?

I'm reading Helly's First Theorem of Carothers' Real Analysis: I can understand the proof. When I compare it with Bolzano-Weierstrass Theorem in $\mathbb R$, however, it seems that Helly's First ...
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4answers
64 views

Convergence of $\sum\limits_{n=2}^\infty ln(\frac{n^2}{n^2-1})$

I made sure it passed the nth term test. Next I thought the easiest way, given that it's wrapped in ln, would be to use log rules to make it $ln(n^2)-ln(n^2-1)$ and then compare it to $\frac {1}{n^2}$ ...
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39 views

Can someone help me with this Comparison Test problem involving cos?

I need to determine if this series converges: $$\sum{\frac{cos^2n}{n^2}}$$ So far, I know I'm supposed to use the Comparison Test. And I compare it with the series $\sum{\frac{1}{n^2}}$, which we ...
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4answers
214 views

Evaluate the sum $\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}$

$$\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}$$ I am having difficulty finding the function that represents this series. I have only found radius of convergence which is $(-\infty,+\infty)$ from the ...
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1answer
31 views

how to check the convergence of the following series

I was having a hard time grasping the concept of convergence and how to check convergence for the following series: $$ a_n= \frac{1}{\sqrt{n^2+n}} ...
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26 views

Wondering about limit comparison test and its application

I have a question about the limit comparison test. In particular if it is valid to use in the question I will post. I am interested in determining wether the series $$\sum_{n=2}^\infty ...
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3answers
78 views

prove: $\lim_{n \rightarrow \infty} x^{1/n} = 1$

I have to prove that $\lim_{n \rightarrow \infty}$ $x^{1/n} = 1$ for $x > 0$. I splitted it up in 3 cases: $x = 1:$ $1^{1/n} = 1$ $\forall$ $n$, so $\lim_{n \rightarrow \infty}$ $x^{1/n} = 1$ if ...
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48 views

Proof for a property of convergent series

Would someone be so kind as to demonstrate the proof for me? edit: $$\sum_{n=v+1}^\infty a_n=a_{v+1}+a_{v+2}+\cdots$$ The $N^\text{th}$ partial sum is: $$\begin{align}\sum_{v+1}^{v+N} ...
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1answer
42 views

Limit points and subsequences

I'm having trouble with proving this exercise: Let $(a_{n})_{n = 0}^\infty$ be a sequence of real numbers, and $L \in \mathbb{R}$ . Prove: $L$ is a limit point of $(a_{n})_{n = 0}^\infty ...
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17 views

Infinite series with a binomial

I'd like to know, is there any place where I can find the proof of this? in some radius of convergence?
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54 views

Question about series on knopp book

The book of K. Knopp, Theory and Application of infinite series , 1990 on page 312, exercise 135, its written "For every fixed $\rho$ in $0<\rho<1$, the expression ...
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52 views

Convergence Problem in Normed Space

Probably easy, but I'm stuck atm: A sequence converges in norm 1 if and only if it converges in norm 2, for all sequences. Are the two norms necessarily equivalent?
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Does $f\cdot \chi_{A_n}\to f\cdot \chi_A$?

Let $(\Omega,S,\mu)$ be a measure space. Suppose $f$, a non-negative function is integrable on $A_1\supset A_2 \supset A_3\dots$, a decreasing sequence of measurable sets $\{A_n\}_n\subset S$ and ...
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Convergence of sequences, General Topology

Prove: Show that if $x_{n}\rightarrow x$ and $|x_{n} - y_{n}| \rightarrow 0$ then $y_{n}\rightarrow x$. proof: Let $\epsilon > 0$, since $x_{n}$ converges to $x$, then there exists a positive ...
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1answer
27 views

Integral on set sequence is not convergent to their countable intersection

Let $(\Omega,S,\mu)$ be a measure space. Suppose $f$ is integrable on $A_1\supset A_2 \supset A_3\dots$, a decreasing sequence of measurable sets $\{A_n\}_n\subset S$ and denote ...
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2answers
18 views

Uniform continuous function convergence VS continuous funcition convergence

Suppose that $f:\Bbb{R}\to\Bbb{R}$ is uniformly continuous. Let $f_n(x)=f(x+1/n)$. a) Prove that $f_n$ converges uniformly to $f$ on $\Bbb{R}$ b) Does this remain true if $f$ id just continuous? ...
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258 views

Proof that a sequence that converges to zero has a subsequence whose series is convergent

How would I go about proving that if $a_n$ is a real sequence such that $\lim_{n\to\infty}|a_n|=0$, then there exists a subsequence of $a_n$, which we call $a_{n_k}$, such that $\sum_{k=1}^\infty ...
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19 views

Convergence of a sequence in a normed vector space [duplicate]

help with homework problem... I feel like its easy, I'm just missing something Show that $\{||x_k||\}$ converges in $\mathbb{R}$ if $\{x_k\}$ converges in a normed vector space V. merci :) its ...
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50 views

$\{a_n\}$ sequence $a_1=\sqrt{6}$ for $n \geq 1$ and $a_{n+1}=\sqrt{6+a_n}$ show it that convergence and also find $\lim_{x \to \infty} \{a_n\}$

$\{a_n\}$ sequence $a_1=\sqrt{6}$ for $n \geq 1$ and $a_{n+1}=\sqrt{6+a_n}$ show that it convergence and as well find $\lim \limits_{n \to \infty} a_n$ In order to show that that sequence convergence ...
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1answer
43 views

Pointwise convergence of a sequence of piecewise functions $\{f_n\}$

For $n \ge 1$, define functions $f_n$ on $[0,\infty)$ by $$f_n (x) = \begin{cases} e^{-x} &\quad\text{for}\quad 0 \le x \le n\\ e^{-2n} (e^n + n - x) &\quad\text{for}\quad n \le x \le n ...
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1answer
19 views

Generalization of Lebesgue Convergence theorem

A generalization of Lebesgue Dominated Convergence Theorem tells us that if $f_n \to f$ in probability and $|f_n| \le g \in L^p$, then $|f| \in L^p$ and $f_n \to f$ in $L^p$ norm. Suppose $f_n \to f$ ...
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2answers
29 views

Using Comparison test the series for convergence. Stuck with understanding the hint.

I'm currently having some trouble trying to start this question. I'm meant to test the series $\sum_{k=1}^{\infty} \frac{k^3}{3^k}$ for convergence. The question provided a hint to start off by ...
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206 views

Prove $\lim\limits_{n\to\infty}\int_0^1(\cos \frac1x)^n\mathrm dx=0$

Prove $$\lim_{n\to\infty}\int_0^1 \left(\cos{\frac{1}{x}} \right)^n\mathrm dx=0$$ I tried, but failed. Any help will be appreciated. At most points $(\cos 1/x)^n\to 0$, but how can I prove that the ...