Convergence of sequences and different modes of convergence.

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Is $\sum_{p\text{ prime, } p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ convergent or divergent??

Is $\sum\limits_{\text{p prime}, p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ convergent or divergent? So far I have that \begin{align} \sum\limits_{\text{p prime}, p \geq 2} ...
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61 views

Question about series convergence $\sum_{n=1}^\infty \frac{1}{n}$ and $\sum_{n=1}^\infty \frac{1}{n^2}$

So I have been playing around with convergent series recently and I still have a hard time understanding why $\sum_{n=1}^\infty \frac{1}{n}$ diverges and $\sum_{n=1}^\infty \frac{1}{n^2}$ converges. ...
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1answer
21 views

Radius of convergence: Why is it $\geq 1$?

Let $X$ denote a random variable with values in $\mathbb{N}_0\cup\left\{\infty\right\}$. Let $r_X$ denote the radius of convergence of $\sum_{n\in\mathbb{N}_0}\mathbb{P}(X=n)z^n$ with ...
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1answer
37 views

An Elementary Convergence Problem in Probability

Suppose that $X_1,X_2,...$ are degenerate random variables such that $f_{X_n}$ denotes the mass function of $X_n$.$$f_{X_n}(x)=P(X_n=x)= \begin{cases} 1, & x=2+\dfrac{1}{n} \\ 0, ...
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50 views

If $\{X_n\}$ converges in probability to $1$, where does $\{1/X_n\}$ converge to?

Without using the continuous mapping theorem, I want to show that, given $\{X_n\}$ is a sequence of random variables converging in probability to $1$, $\{1/X_n\}$ converges in probability to $1$. The ...
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0answers
24 views

Convergence of a recursive sequence of functions

Consider the sequence of functions $f_n:[0,1]\rightarrow[0,1]$ defined recursively: $$f_n(p)= 1-p + p (f_{n-1}(p))^2 \quad f_0(p)=1-p \quad f_n(1)=0$$ Computationally one can check that $\{f_n(p)\}$ ...
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Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?

I think I understand intuitively how we can assign a value to the sum of all natural numbers. But of all the proofs that I've seen that show why $\zeta(-1) = -\frac{1}{12}$, none of them use their own ...
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21 views

Cauchy sequences and convergent sequences?

I am wondering if one can assume that any Cauchy sequence in $(X,d)$ converges to some point in some larger space $(\hat X,d)$ with the same metric. Take for example $(\Bbb Q,d)$. It is not complete ...
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1answer
149 views

A problem on almost sure convergence

Consider a sequence of random variables defined on the standard unit interval probability space : $$X_n = \begin{cases} 2^n & \text{when} \quad \frac{1}{2^n} \leq \omega \leq ...
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13 views

Supremum of cadlag functions

Let $f_n,f$, $n\in\mathbb{N}$, be (real-valued) cadlag functions on $[0,1]$ such that $$\sup_{0\le t\le 1}|f_n(t)-f(t)|\to 0\text{ as }n\to\infty.$$ Does someone have an idea how to prove that ...
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218 views

Sequences for that $\sum_{n} \frac{1}{x_n}$ is divergent and $\sum_{n} \frac{1}{x_n \ln x_n}$ is convergent

We will denote with $(x_n)$ a given sequence and we introduce the following two series. $$S^* = \sum_{n} \frac{1}{x_n} \quad \text{and} \quad S_* = \sum_{n} \frac{1}{x_n \ln x_n}.$$ We know that if ...
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20 views

Book on Convergence Concepts in Probability without Measure Theory

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
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1answer
9 views

convergence at a specific value

I have a series $S_n=\sum_{i=1}^{n} a*(1-i)$ where $a$ is an unknown constant independent of $i$. Is there a way to figure out for which $n$ the above expression converges to the value 0.01? After ...
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What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?

I tried and got this $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an ...
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20 views

Convergence in distribution plus convergence of moments.

Suppose that the sequence of r.v $\{X_n\}_{n\geq 1}$ has all the moments, and $X_n\stackrel{D}{\longrightarrow} X\sim N(0,\sigma)$. Assume that $E\{(X_n)^K\} \stackrel{n} {\longrightarrow} E(X^K)$, ...
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270 views

Does strong law of large numbers hold for an average of this triangular array containing “almost i.i.d.” sequences?

If a sequence of $n$ i.i.d. positive random variables $X_1,\ldots,X_n$ has the expectation $\mathbb{E}[X_1]=\mu<\infty$, then it is known that the strong law of large numbers (SLLN) holds: ...
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192 views

Prove or disprove: $\sum a_n$ convergent, where $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$.

Let $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$. Show that $a_n>0\ \forall\ n\ge1$. Prove or disprove: $\sum\limits_{n=1}^\infty a_n$ is convergent. I can't show that $a_n > 0\ \forall ...
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30 views

borel-measurable function is pointwise limit of a sequence of continuous functions, wich is uniformly bounded

Let $H$ be a Hilbert space over $\mathbb{C}$, $A\in L(H)$ ( $A:H\to H$ is linear and continuous) and let $A$ be self-adjoint. Consider the spectrum of A, $\sigma(A)$ and $f:K\to \mathbb{K}$ a bounded, ...
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1answer
34 views

Limit of products in $L^p(\mathbb R^d)$

Fix $1 \leq p < \infty$. If $f_n \to f$ in $L^p(\mathbb R^d)$, $g_n \to g$ pointwise, and $\| g_n \|_{\infty} \leq M < \infty$ for all $n$, prove that $f_ng_n \to fg$ in $L^p(\mathbb{R}^d)$. ...
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86 views

$\sum_1^n 2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1} $ converge or not? [duplicate]

how to check if this converge? $$\sum_{n=1}^\infty a_n$$ $$a_n = 2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1}$$ what i did is to show that: $$a_n =2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1} > 2\sqrt{n} - ...
3
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1answer
179 views

How to compute this kind of limit? [duplicate]

Let $x_0=a,x_1=b$ $$x_{n+1}=\Big(1-\dfrac{1}{2n}\Big)x_n+\dfrac{ x_{n-1}}{2n}, n\ge1$$ Find $\lim x_n.$ If limit exist I can plug limit as $l$ to get a equation of $l$, whose root will be the ...
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2answers
31 views

A sequence of Continuous Functions Converges Uniformly over $\mathbb{R}$ if it Converges Uniformly over $\mathbb{Q}$

I'm trying to show that if ${f_n}$ is a sequence of real functions that is continuous over all of $\mathbb{R}$ and that converges uniformly to $f$ over $\mathbb{Q}$, then it converges uniformly to $f$ ...
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2answers
55 views

$x_{n+2}=\frac 13 (x_n+2x_{n+1})$, then lim $x_n$=? [closed]

Let $x_0=a,x_1=b$,if $x_{n+2}=\frac 13 (x_n+2x_{n+1})$, then $\lim x_n=?$ I see here a geometrical interpretation of the points of sequences. It is $ x_2$ the inner point 2:1 of a and b. ...
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What can be said about the limit of a converging infinite polynomial?

Suppose we have the following polynomial of infinite order: $f(x) = a_0+a_1x+a_2x^2+...=\sum_{n=0}^{\infty}a_nx^n$ Also suppose that $f(x)$ converges to some limit $f^*$ as $x\nearrow\infty$, i.e. ...
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3answers
51 views

Proving the reciprocal of a divergent sequence is convergent

I want to prove that given a sequence (an) where the limit as n --> infinity of (an) = infinity, the limit as n --> infinity of 1/(an) = 0. It's introductory real analysis, but I'm not sure where to ...
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1answer
54 views

Convergence of an integral $\int_0^\infty\frac{dt}{1+t^\alpha\sin^2(t)}$

For what $\alpha\in\Bbb{R}$ does $\displaystyle\int_0^\infty\frac{dt}{1+t^\alpha\sin^2(t)}$ converge ? The $0$ bound doesn't seem to be much of a problem, but I don't see how to deal with the ...
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Stone-Weierstrass: Examples

By Stone-Weierstrass one has: $f\in\mathcal{C}(K):\quad p_n\to f$ Now, for analytic functions this is just Taylor: $$f\in\mathcal{C}^\omega([a,b]):\quad f(x)=\sum_{k=0}^\infty a_kx^k$$ But, how does ...
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39 views

What is the limit value of this function (sin..)

I wanted to ask you a question about this specific function: $$\lim\limits_{x\to\infty}\left( \sin(\sqrt{x})-\sin(\sqrt{x+1})\right)$$ Somehow I can't comprehend how to do this task On one hand sin ...
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12 views

does function converge uniformly

Does the sequence $(f_n)$ of functions $f_n:(0,\infty)\to\mathbb R$ $$f_n(x) = \begin{cases} 0 & x \in (0,\frac1{n+1}) \\ \sin^2 \frac\pi x& x \in [\frac1{n+1},\frac1n]\\ 0 ...
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0answers
35 views

Test $\sum_{n=1}^{\infty}\Big(1-n \sin(\frac{1}{n})\Big)$ for convergence with asymptotic comparison

Suppose we want to test the following series for convergence $$\sum_{n=1}^{\infty}\Big(1-n \sin(\frac{1}{n})\Big)$$ I have found a solution that uses the asymptotic comparison test and uses the ...
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1answer
76 views

Is this series: $\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}}$ convergent?

How can I show that the following series is convergent or divergent ? $$\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}},x\in \mathbb{R}$$ I want to use Abel-Dirichlet criteria. I've ...
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5answers
99 views

How to prove this limit: $\lim\limits_{n \to \infty} \frac1{\sqrt[n] n}=1 $? [duplicate]

This is what I am trying to prove: $$\lim_{n \rightarrow \infty} \frac1{\sqrt[n] n}=1 $$ How do I go about this?
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1answer
52 views

Does $\int_0^1 \sqrt{\frac{1+x}{\sin{x}}}$ converge?

Does $$\int_0^1 \sqrt{\frac{1+x}{\sin{x}}}dx$$ converge? I have tried to substitiute $x$ in nominator as $\tan{x}$ and simlify it using trigonometric formulas, but the integral was still too ...
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80 views

Determine if the following series is convergent or divergent $\sum_{n=1}^{\infty}\frac{ \sin{nx}}{n}$

I've been doing exercises with series for some time and now I've got an exercise that I'm supposed to solve using Abel or Dirichlet criteria. The problem is : Determine if the following series ...
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27 views

Converges or diverges: $\sum_{n=1}^{\infty}\left [\arctan{\frac{(-1)^n}{n}}+{(-1)^n\over n^2}\right]$

How can I show that the following series converges or diverges ? $$\sum_{n=1}^{\infty}\left [\arctan{\frac{(-1)^n}{n}}+{(-1)^n\over n^2}\right]$$ $\sum_{n=1}^{\infty}\left ...
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23 views

Convergence rate of the power method for finding eigenvectors

Let $M$ be a real-valued square matrix with an eigenvector $w$ strictly larger (in absolute value of the corresponding eigenvalue $\lambda$) than all others, and let $v$ be any vector not orthogonal ...
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69 views

Prove a convergent sequence has either a minimum, a maximum or both.

Let $a_n$ be a convergent sequence. Prove $a_n$ has a minimum, a maximum or both. I am being prepared for a final exam, which is why it is important to me to know that $I$ am correct in $my$ ...
3
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1answer
31 views

$l_2$ sequence, series with square root

I'm trying to prove that the following functional is continuous: $$\phi : \mathcal{l}_2 \ni \{x_n \} \rightarrow \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}x_{3n} - \sum_{n=1}^{\infty} \frac{1}{n}x_{2n} ...
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Determine if it converges or diverges : $\sum_{n=1}^{\infty} \frac {2^n \cdot n!}{1\cdot2\cdots (2n-1)}\cdot \frac{1}{\sqrt{2n+1}}$

Here's the series: $$\sum_{n=1}^\infty \frac {2^n \cdot n!}{1\cdot2\cdots (2n-1)}\cdot \frac{1}{\sqrt{2n+1}}$$ Does it converge or diverge ? Thanks
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56 views

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

I've tried solving this exercise but got stuck on a big expression that I could not untangle. I've obtainded the following thing: $$\lim_{n \to \infty} n\frac{2 \cdot 2^n +3 \cdot 2^n\cdot ...
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52 views

The convergence of $\sum ( \sqrt{n+1}-\sqrt{n})^ a$. How do I keep it from here?

Check when the series $\sum ( \sqrt{n+1}-\sqrt{n})^ a$, $a\in \Bbb{R}$ converges. $Attempt$: Let us focus on $a_n= ( \sqrt{n+1}-\sqrt{n})^ a$. $a_n=( \sqrt{n+1}-\sqrt{n})^ a=( ...
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4answers
93 views

Does $\int_0^\infty e^{-x}\sqrt{x}dx$ converge? [closed]

Does $\int_0^\infty e^{-x}\sqrt{x}dx$ converge? Thanks in advance.
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85 views

$f(x)\sim 1/x \implies (1+f(x))^x\to e$, but what family of functions maximizes the speed of convergence from below?

This problem is subordinate to finding out if $$\left(1+\frac{\log p_{n+1}}{p_n}\right)^{p_{n+1}/\log p_n},$$where $p_n$ is the $n$-th prime, never stabilizes above or below its limiting value, which ...
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4answers
68 views

Prove $\ln(1+x^{2})=\arctan x$ has two solutions in $\Bbb{R}$. How do I finish my proof?

Attempt: Define a function $g(x)=\ln(1+x^{2})-\arctan x$. Then $g'(x)={2x\over 1+x^2}-{1\over 1+x^2}$ and $x={1\over2}$ is a potential extremum. $g''(x)={-2x^2+2x+2\over (1+x)^4} \Rightarrow ...
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1answer
25 views

My proof of: Given an adherent point, some sequence converges to it.

(I have two specific questions.) Is my proof correct? Are style, wording, and punctuation alright? $\textbf{Definition 1.}$ Let $X \subseteq \mathbb{R}$. Let $x \in \mathbb{R}$. The point $x$ ...
2
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1answer
31 views

Net convergence in metric spaces

This is a question about convergence of nets which I don't quite understand yet. In metric spaces convergence of sequences encodes the topology but suppose we want to study convergence of nets even ...
2
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1answer
33 views

Convergence in distribution conditions

I'm reading the weak convergence section (Ch.$18$) in "Probability Essentials" by Protter and Jacod. Theorem $18.7$ states that "$Xn \overset{D}{\to} X \iff \underset{n\to\infty}{\lim} E[g(X_n)] = ...
3
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1answer
52 views

Solve this limit $\displaystyle \lim_{n\to \infty} \int_{0}^1 nxe^{-nx}dx$

I would like to solve that limit: $\displaystyle \lim_{n \to\infty} \int_{0}^1 nxe^{-nx} dx$ I would like to take the limit inside the integral but for doing this, I have to use Beppo Levi's ...
0
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1answer
26 views

Geometric mean with the law of large numbers

I'm currently studying some probability and I'm stuck with this question. Let $R_1, . . . , R_n$ be independent continuous uniform over [0, 1] random variables. The geometric mean of $R_1, . . . , ...