Convergence of sequences and different modes of convergence.

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An⇀̸A in L1[−π;π] ( An is partial fourier sum )

Let \begin{equation*} (A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k \cos(kt) + b_k \sin(kt), \\ a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} x(t) \cos(kt) dt, \\ b_k = \frac{1}{\pi} \int_{-\pi}^{\pi} ...
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1answer
20 views

How to make a topology out of $N$ that involves convergent / divergent sets.

Let $N$ be the naturals $1, 2, \dots$ Call a subset $A$ of $N$ convergent if the reciprocal sum $\sum_{a \in A} \frac{1}{a}$ converges. Similarly call as set divergent if the sum diverges. Notice ...
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21 views

If a bounded sequence is equicontinuous, it has a uniformly convergent subsequence

I am currently having some difficulty with problem 2.7.8 in Introduction to Topology by Theodore Gamelin and Robert Greene. The problem goes as follows A family F of real-valued functions on a ...
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Good Problem Book for Convergence Concepts in Probability

I need a good book containing many challenging exercises (or problems) on Convergence Concepts in Probability. The topics I have covered are: Borel-Cantelli Lemmas Modes of Convergence and ...
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1answer
23 views

Normal convergence: $\sum_{n = 1}^{\infty} \frac{x}{(1 + (n-1)x)(1 + nx)}$

I want to prove that: $$\sum_{n = 1}^{\infty} \frac{x}{(1 + (n-1)x)(1 + nx)}$$ is not normally convergent on $[a, \infty)$ for fixed $a>0$. Let $U_n(x)$ denote the general term. We have: ...
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Skorokhod vs Meyer zheng topology

I am new to the Skorokhod space and I want to know why Meyer-Zheng topology on the space of càdàg functions is weaker than the standard Skorokhod topology. Thanks in advance!
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19 views

Uniform integrability and weak L1 convergence

I am working on exercise 4.14 in chapter 3 (on convergence) in the book "Probability and Stochastics" by Erhan Cinlar. The exercise can be found on page 109. First, let me give the necessary ...
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Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
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convergence multivariate normal

If $X$ and $Y$ have asymptotic normal distribution then using Slutsky's theorem $aX+bY$ is also asymptotic normal, can I conclude that the vector $(X,Y)$ is asymptotic bivariate normal? If not, how ...
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1answer
27 views

Can somebody explain the notation $f \in C^4$

To give some context the full question is: Suppose $f \in C^4$ in a interval containing the root $\alpha$ and that Newton’s method gives a sequence of iterates $\{x_k\}$, $k = 0, 1, 2, \dots$ which ...
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3answers
42 views

Can I use the Dirichlet's test to prove the convergence of $\sum_{n=1}^N \frac{e^{in}}{n}$?

I am trying to state that $$\sum_{n=1}^\infty \frac{e^{in}}{n}$$ converges. Is it correct that $|\sum_{n=1}^N e^{in}|\leq M$ for every positive integer $N$? I.e use $e^{in}$ as the $b_n$ term in ...
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2answers
28 views

Series definied by recurrence relation

Define a sequence recursively by $$\large{a_1 = \sqrt{2},a_{n+1} = \sqrt{2 + a_n}}$$ (a) By induction or otherwise, show that the sequence is increasing and bounded above by 3. Apply the monotonic ...
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Elemenatry topological proof of Erdos conjecture on Arithmetic Sequences

Define $C_n(A) = \{a \in A : \forall d \in \Bbb{N}$ one of $a + d, a +2d, \dots a + (n-1)d$ is not in $A\}$ For $n \leq m$, we have $C_n(A) \subset C_m(A)$. Proof. Let $x$ be in the LHS. Then ...
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1answer
25 views

Is it possible to exhibit a collection of sets

Let a subset $D$ of the natural numbers be called convergent or divergent when the associated series $\sum_{d \in D} \frac{1}{d}$ converges or diverges. Define a topology on $\Bbb{N}$ by defining the ...
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Why $N= max(2,\frac {2}{\epsilon})$ for $|a_n -L|<\epsilon $ convergence problem [on hold]

Using the proof development strategy used regarding the proposition (for all $\epsilon \in \mathbb{R}^+$ there exists an $N \in \mathbb{R}^+$ such that $|a_n - L| < \epsilon$ for all $n > N $) ...
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Sums convergent but not uniformly convergent on [0,1]

Show that both $\sum_{n=1}^{\infty} ({1-x}){x^n}$ and $\sum_{n=1}^{\infty} (-1)^n({1-x}){x^n}$ are convergent on [0,1] but only one converges uniformly. Which one? Why? I was playing around with the ...
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2answers
59 views

prove or disprove convergence [duplicate]

Im trying to prove or disprove the following , but I am having a hard time. It seems that the statement is true , but I have no idea how to prove it. If \begin{equation*} \sum_{n} a_{n}^2 ...
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32 views

series and dyadics [duplicate]

I'm in Calc II and am unfamiliar with what dyadics is but my teacher said that it's possible to find the sum of $\frac{\cos n}{n^2}$ by using dyadics. Would you mind laying out a step by step? ...
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convergence of $\int_{x=0}^{\infty}x^{-\frac{M-1}{2}-N}(1-e^{-x})^{M-1}e^{-x}dx$

I am trying to study the convergence of $$\int_{x=0}^{\infty}x^{-\frac{M-1}{2}-N}(1-e^{-x})^{M-1}e^{-x}dx,$$ where $M$ and $N$ are positive integers. I've tried some $M$ and $N$, and it seems that ...
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4answers
50 views

Determine the convergence of sequence $\tan^2(1/n)$

How to determine the convergence of sequence $$ \sum^\infty_{n=1} \tan^2(1/n) $$ I have used the divergence test $$ \lim_{n\to\infty} \tan^2(1/n) = \tan^2(0) = 0 $$ So we can't say if it diverge ...
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Convergence of a Markov Chain to the normal distribution

If $i$ is a state of an irreducible, postive recurrent Markov chain $X$, and $V_n$ is the number of visits to $i$ between times $1$ and $n$, and further $\mu=\mathbb{E}_i(T_i)$ and ...
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1answer
25 views

Alternating series convergence test

I have a series $\sum_{n=1}^\infty c_n x^n$ where $c \le c_n \le C$. I can determine radius of convergence easily by the root test, but how does one determine convergence for $x = -1$? It is not a ...
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2answers
35 views

Finding the radius of convergence of a power series, $\sum_{n=1}^{\infty} a_n x^n$.

I have to detemernine the radius of convergence of the power series $\sum_{n=1}^{\infty} a_n x^n$, where $(a_n)_{n=0,1,2,...}$ is given by $a_n=2-\dfrac{1}{2}a_{n-1}$ with $a_0=2/3$. So far I've ...
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1answer
28 views

How to find out if a sequence with exponentiation in fraction is convergent

I need to find the convergence of this function: $\sum^{\infty}_{x=1}{\frac{(x+1)^{x^2}}{x^{x^2}2^x}}$ Now my problem is, I have no clue how to do this (I tried the root-test and it did not work ...
2
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1answer
38 views

Prove that $\lim_{n\to \infty}\langle \operatorname{erfc}(-nx), \phi\rangle =\langle H_0, \phi\rangle $

Define the error function $\operatorname{erf}(x)$ as: \begin{equation} \operatorname{erf}(x):=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-y^2}dy \end{equation} and ...
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15 views

uniform converging functions

If I know that $\sum_M^\infty v(x)$ converges towards $f$, then $$\left| \left( f + \sum_{1}^{M-1} v(x) \right) - \sum_{1}^N v(x)\, \right| = \left|\;f - \sum_{M}^N v(x)\,\right| < \epsilon$$ for ...
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52 views

Compute power of a matrix $A$ as $n\rightarrow \infty$

We are given $A^p=A ...A$(p times) And we are given matrix A: $A=\begin{vmatrix}0.6&-0.4&0\\-0.4&0.6&0\\0&0&0.5\end{vmatrix}$ I need to compute $A^p$ as p approach Infinity. ...
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4answers
76 views

Convergence of $\sum_{n=0}^{\infty} \left(\sqrt[3]{n^3+1} - n\right)$

I have encountered the following problem: Determine whether $$\sum \limits_{n=0}^{\infty} \left(\sqrt[3]{n^3+1} - n\right)$$ converges or diverges. What I have tried so far: Assume that $a_n = ...
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Is there something wrong with this proof? Convergence of m-tail of series

I wanted to formally proof that the uniform convergence of the $m$-tail of a series of functions implies uniform convergence of the entire series. It made intuitive sense but I did not know any ...
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1answer
28 views

Uniform convergence of real function

Let $f:[a,b] \to \mathbb{C}$ where $0 < a < b < 2 \pi$ be defined by $f(x) = \sum_{k=1}^{\infty} \frac{exp(ikx)}{k}$. Show that $f$ converges uniformly in $[a,b]$. The problem is that I ...
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Find the radius of convergence of a power series.

Consider the power series $$\sum_{n=0}^{\infty}a_nx^n$$ where $a_0=0$ and $a_n=\frac{\sin(n!)}{n!}$ for $n≥1$ Let $R$ be the radius of convergence then $R≥1$ $R≥2π$ $R≥π$ My attempt: I used the ...
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1answer
20 views

Convergent Sequence and its limit

Can anybody help me out in this problem: I am not able to figure out how the value of lambda in the 2nd problem comes out to be 2 ? In the first problem value of lambda came out to be 2 after ...
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1answer
29 views

Explicit example of normal family

Suppose $\mathscr F \subset H(\Omega$) for some region (i.e. open connected) $\Omega$. ($H(\Omega)$ means the set of all holomorphic function in $\Omega$) We call $\mathscr F$a normal family if every ...
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Determine wether the series: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ diverges or converges? - I want to check if my reasoning is correct

I got that the series: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ converges by doing the following: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ = $\sum_{n=0}^\infty ...
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2answers
26 views

Sequence of monotone functions problem

Found a problem on "The Way of Analysis" by Strichartz. Say $f_n$ converges to $f$ (pointwise) and each $f_n$ is increasing. (a) Must $f$ be increasing? (b) What happens if each $f_n$ is strictly ...
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19 views

Better chances on a gambling game [on hold]

I'm working on a program that I hope it will help me to get better chances on a gambling game. I have a very large database of numbers how occur in this game and my program pass through those numbers ...
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1answer
36 views

$\sum_{n=1}^\infty a_n<\infty$ if and only if $\sum_{n=1}^\infty \frac{a_n}{1+a_n}<\infty$

For $a_n$ positive sequence. I think I can prove one direction, but not both.
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When using the Integral test, why is the value of the integral different from the sum of the series?

According to my textbook, the value of the improper integral is not always equal to the sum of the series. But why is that?
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Show that $\sum_{n=1}^{\infty} \dfrac{1}{n} \ln \left( 1+\dfrac{x}{n} \right)$, with $x>-1$ is pointwise convergent.

Title says it all, really. I have to show, that: $\sum_{n=1}^{\infty} \dfrac{1}{n} \ln \left( 1+\dfrac{x}{n} \right)$, with $x>-1$ is pointwise convergent, but I have no idea where to start. I am ...
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Weak convergence in probability implies uniform convergence in distribution functions

This is a problem that I am totally stuck at. I know the fact that $F_n$ converges pointwise to $F$ in this question. Also, I looked through Google and found out that I have to show first that ...
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24 views

Prove that the given sequence is convergent

prove that the sequence $\frac{2n^2 + n + 10}{n^2 + 5}$ is convergent I've come down as far as $\left|\frac{n}{n^2 + 5}\right| < \epsilon$, however i don't know what the next step is.
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64 views

How $(f_n)$ converges uniformly on $[a, b]$

Let $(f_n)$ be defined and continuous on an interval $[a, b]$, and differentiable on $(a, b)$. Let $c \in [a, b]$. Assume that $(f_n(c))$ converges and that $(f'_n)$ converges uniformly on $(a, b)$. ...
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1answer
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Uniform convergence for the sequence $\psi_n(x)=n^{-1}e^{-n^2/(n^2-x^2)}$

How can I prove that the sequence of functions: \begin{equation} \psi_n(x)=\begin{cases} n^{-1}e^{-n^2/(n^2-x^2)}, & |x|\leq n \\ 0, & |x|\geq n \end{cases} \end{equation} convergences ...
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1answer
28 views

How to prove that a $\phi \in C^{\infty}(\mathbb{R})$.

I would like to prove that the function, defined as: \begin{equation} \phi(x)=\begin{cases} e^{-1/x}, & x>0 \\ 0 , & x \leq 0\end{cases} \end{equation} is a $C^{\infty}(\mathbb{R})$. So ...
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27 views

Convergent series with condition

There exists a sequence of strictly decreasing real positive numbers $x_n$ such that its series converges, but the quantity $$\frac{x_n^2}{x_n-x_{n-1}}$$ doesn't converge to zero? All the famous ...
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46 views

Radius of convergence of $\sum_{n\geq 0}a_{n}x^{n}$.

Consider a series $\sum_{n\geq 0}a_{n}x^{n}$ where $a_{0}=2/3$ and $a_{n}=2-(1/2)a_{n-1}$ for all $n$. It is assumed that $2/3\leq a_{n}\leq 5/3$ for all $n\geq 1$. My problem is about determining its ...
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3answers
35 views

convergence of series $\sum_{n=1}^{\infty}\frac{4n^2+5n}{n(n^2+1)^{\frac{3}{2}}}$

I can't find a way to decide whether it convergent or not... tried the root test, but the result is 1(which means nothing)
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2answers
40 views

for what valus of p, the series converge [closed]

For which values of $p>0$ does the series $\sum_{n=2}^\infty \frac{1}{n(\ln(n))^p}$ converge?
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1answer
16 views

Show that the radius of convergence of a sum of series is at least as big as minimum of radii of these series.

I am struggling with the following task. Suppose $\sum^{\infty}_{n=0}a_nx^n$ has radius of convergence $R$ and $\sum^{\infty}_{n=0}b_nx^n$ has radius of convergence $S$. I want to show that the ...
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0answers
10 views

Convergence in Distribution for two Dirichlet distributions

I'm working on a problem and I wanted to get some hints on how to solve it. To me, it seems like showing convergence to distribution but since it's been a while that I've not worked on these types of ...