Convergence of sequences and different modes of convergence.

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2
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1answer
68 views

For which $\alpha$, $\beta$ does $\int\limits_1^{\infty} x^{\alpha} \cdot (\ln x)^\beta dx$ converge? [duplicate]

For which $\alpha$ and $\beta$ does the following integral converge ?: $$ \int_{1}^{\infty}x^{\alpha}\,\ln^{\beta}\left(x\right)\,\mathrm{d}x $$ Here is my analysis: I noticed that the function ...
3
votes
2answers
79 views

Problem concerning the sequence $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$

The question is : Prove that the sequence $\{s_n\}$ where $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$ is convergent. Hence find $\lim_{n \to \infty} \left(1 - 1/2 + 1/3 - ... - 1/2n\right)$. I have ...
1
vote
0answers
13 views

Why do they add addition $\mathbb{E} (s_1^-)<\infty$ when they have $\{ s_n, \mathcal{F}_n,n\ge 1\}$ be a submartingale?

By definition, if $\{ s_n, \mathcal{F}_n,n\ge 1\}$ is a submartingale then $\mathbb{E} (|s_n|)<\infty$ for all $n\in \mathbb{N}^*$. But according to a arcticle, in a theorem, author add addition: $\...
3
votes
1answer
48 views

A limit using the Euler number: $\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$

What is answer of this limit and how can I get it? $c$ and $i$ are constants. $$\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$$ I guess it will envolve some Neper/the Euler ...
-2
votes
0answers
32 views

convergence of power series, expanded by maclaurin

I'm getting ready for a test and I stumbled upon a question which goes like this: A function f(x) is given and I need to expand it to a power series using Mcloren sequences and then calculate its area ...
-2
votes
1answer
40 views

Convergence test for series [closed]

"Convergence test for series $\sum_{n=0}^{\infty}y=\sum_{n=0}^{\infty}(x^n)/n!$ by applying stirling's approximation $log(n!)=nlog(n)-n+1$". What i have did is, First i took its general term as $y=(x^...
4
votes
2answers
113 views

On calculations for $\prod_{k=2}^\infty \left( 1+\frac{\mu(k)}{k^3} x\right)$, where $\mu(n)$ is the Möbius function

I define for some set of real numbers $x\in S$ (see that it is my Question 1.) the domain of the function $$f(x)=\prod_{k=2}^\infty \left( 1+\frac{\mu(k)}{k^3} x\right) ,$$ where $\mu(k)$ is the ...
3
votes
1answer
29 views

Convergence sequence of mean implies convergence in mean / weakly consistence of subsequence of regression function estimates

Let $(X_n)$ be a sequence of positive random variables. Suppose that the limit of expectation of this sequence $\lim_{n\rightarrow\infty}\mathbb{E}[X_n]=0$. This imply that $(X_n)$ converges to zero ...
0
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0answers
17 views

Convergence for $\sum_{\text{m composite}}\frac{e^{2\pi\sigma(m)i/(m+1)}}{m^s}$, where $\sigma(n)$ is the sum of divisors function

Let $\sigma(n)=\sum_{d\mid n}d$ the sum of divisors function. When one writes informally the identity $$\sum_{n=1}^\infty \frac{e^{\frac{2\pi\sigma(n)i}{n+1}}}{n^s}=1+\mathcal{P}(s)+\sum_{\text{m ...
1
vote
1answer
18 views

Interchange of limit and sum justification

From the book I have been reading, it seems the following result is implicitly used: If $f_{a}(x):=\sum_{n=0}^{\infty} s_{n}(a,x)$ converges uniformly with respect to (large) $a$, for example $\...
0
votes
1answer
34 views

Continuity and convergence everywhere.

Suppose a sequence of continuous functions $(h_n)$ converges almost everywhere to another continuous function $h$ . Is it possible to infer that $h_n$ infact converges everywhere? If not, under what ...
1
vote
1answer
70 views

Prove that $\int\limits_1^{\infty} \frac{\cos(x)}{x} \, \mathrm{d}x$ converges

Prove the convergence of $$\int\limits_1^{\infty} \frac{\cos(x)}{x} \, \mathrm{d}x$$ First I thought the integral does not converge because $$\int\limits_1^{\infty} -\frac{1}{x} \,\mathrm{d}x \...
0
votes
0answers
6 views

What is the rate of convergence of Brownian motions Increments?

Would like to know what the rate of convergence of brownian motion is? I know each brownian motion increment is distributed with N(0,t) so do i need to apply a CLT?
1
vote
0answers
42 views

closeness of matrices

I'm really lost in math and would really appreciate any help with the following problem. Denote as $S_{+}(p)$ the set of all positively defined symmetric real-valued matrices of size $p \times p$. ...
1
vote
1answer
39 views

Condition on $(a_n)_{n \in \mathbf{N}}$ for Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ to converge for $Re(s) > 1$.

I'm curious about the conditions on a sequence $(a_n)_{n \in \mathbf{N}}$ of real numbers such that the Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ converges absolutely for $Re(s) > 1$. ...
6
votes
5answers
4k views

Does $\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}+\sqrt{n+1}}$ converge?

Does the following series converge or diverge? $$ \sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}+\sqrt{n+1}} $$ The methods I have at my disposal are geometric and harmonic series, comparison test, limit ...
0
votes
0answers
28 views

On Dirichlet series and Firoozbakht's conjecture

On assumption of the Firoozbakht's conjecture (this is the Wikipedia, but the reference is for Carlos Rivera's Page) one has that can writes informally the Dirichlet series in LHS of this inequality $$...
4
votes
2answers
91 views

The convergence of $\sqrt{x^2+\frac{1}{n}}$

Let $f_n(x)=\sqrt{x^2+\frac{1}{n}}$. i) Determine the limit-function $f$. ii) Does $f_n(x)$ converge uniformly to $f$? For the first: We have $\lim_{n\rightarrow \infty}\sqrt{x^2+\frac{1}{n}}=\sqrt{...
3
votes
1answer
25 views

Investigate convergence with cos/sin

Well I want to investigate the convergence of the following integrals(in the linked picture): $$\int_{1}^{\infty}\cos(x^t)dx\quad,\quad t\in \mathbb{R}$$ $$\int_{1}^{\infty}\sin(x^t)dx\quad,\quad t\in ...
2
votes
1answer
48 views

Is this infinite series of continuous functions $f(x)=\sum_{n=1}^{\infty} \sin(\frac{x}{n^2})$ continuous?

The original question: Consider the function $$f(x)=\sum_{n=1}^{\infty} \sin\left(\frac{x}{n^2}\right).$$ Is $f$ a continuous function on $\mathbb{R}$ ? I know that the infinite sum of continuous ...
1
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0answers
35 views

Convergence of sequence $a_0 := b$ and $a_{n+1} = 2^{a_n}$ in $\hat{\mathbb{Z}}$.

I have a question about the convergence properties of a sequence in $\hat{\mathbb{Z}}$, the completion of $\mathbb{Z}$. It is part of an exercise is due to this syllabus. I got confused somewhere. ...
1
vote
2answers
34 views

Summation of a convergent series

I have the following problem: So I start as follows: $B_{2}M_{2}=\frac{1}{\sqrt{3}}$ and I realize that $B_{2}M_{2} = A_{2}B_{2}$, so $B_{3}M_{3} = \frac{1}{\sqrt{3}}^{2}$. Next, I compute $A_{1}...
2
votes
3answers
57 views

Convergence test of $\sum\limits_{n=3}^{\infty} \frac{1}{n\log(n)\log(\log(n))}$

I know, there are some threads dealing with this sum but I want to solve it with the integral test for convergence(more) $$\sum\limits_{n=3}^{\infty} \frac{1}{n\log(n)\log(\log(n))}$$ I can't ...
0
votes
0answers
16 views

On a second set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of these claims and show where were my mistakes or inaccurancies? Also ...
0
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0answers
14 views

On a first set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of this claims and show where were my mistakes or inaccurancies? Also ...
3
votes
3answers
106 views

examine the convergence of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{n+(-1)^{n+1}}$

I have no idea how to examine the convergence of the series: $$\sum_{n=0}^{\infty} \frac{(-1)^n}{n+(-1)^{n+1}}$$ We can see that $\frac{(-1)^n}{n+(-1)^{n+1}} \to 0$. However we can't use criterium ...
1
vote
2answers
37 views

Convergence of given sequence

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function with: $$f(x) = x - \arctan{x}$$ We consider the sequence $(x_{n})$ with $x_{0} > 0$ and $x_{n + 1} = f(x_{n})$, for any $n \in \mathbb{N}$...
0
votes
2answers
19 views

convergence of improper Integral..

I need help finding if the improper integral below converges. $$\int _{ 2 }^{ \infty }{ \frac { dx }{ \sqrt [ 3 ]{ 1-{ x }^{ 4 } } } } $$. we learnt at class: comparison test ratio test Thanks ...
4
votes
5answers
390 views

Negation of the definition of limit

A sequence $(x_n) $ of real numbers converges to a real number $ x $ if For all $\epsilon> 0 $ there exists a natural number $ n_0 $ such that for all $ n \ge n _0 $, $|x_n - x| < \epsilon $. ...
1
vote
1answer
39 views

Finding a limit of a two variable function: $f(x,y)=\frac {\sin(x^2-xy)}{\vert x\vert} $

I have this exercise but not sure if I'm doing it right $$\lim_{(x,y)\to (0,0)} \frac {\sin(x^2-xy)}{\vert x\vert} $$ I assume $\frac {\sin(x^2-xy)}{\vert x\vert}\le\frac {1}{\vert x \vert} $ then ...
2
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2answers
22 views

Convergence of the series $\sum_{n=1}^{\infty}(-1)^n\ \frac{n\log(n)}{e^n}$

Is the series $\sum_{n=1}^{\infty}(-1)^n\ \dfrac{n\log(n)}{e^n}$ convergent or divergent? How can I solve this question? Please Help. Thank you.
2
votes
1answer
28 views

Integral operator convergence study in $L^2(\mathbb{R})$

Exercise I want to study in $\mathcal{H}=L^2(\mathbb{R})$ the convergence of $$A_nf(x)=\log n \int_\mathbb{R} \frac{1}{1+n(x-y)^2}f(y)dy. $$ Solving a) Pointwise convergence : For $n \to \infty$ ...
2
votes
1answer
36 views

Find the interval of convergence for the series $\sum_{k=0}^\infty a_kx^k$ with $a_k = \alpha a_{k-1} + \beta$

Consider the series $\sum_{k=0}^\infty a_kx^k$ with $a_0=1, a_k = \alpha a_{k-1} + \beta, k\geq 1, \alpha,\beta \geq 0$. Determine the interval of convergence of the series. I've tried looking at ...
2
votes
2answers
42 views

$n\lvert a_n\rvert \to 0$, then $\sum_{k=1}^n \frac k n\lvert a_k\rvert \to 0$?

I am now working on the converse of Abel's THM and found out one proof of the conditional converse of the theorem. The proof says, Suppose $$n\lvert a_n\rvert \to 0$$ as $$n \to \infty$$ Then, $$\...
1
vote
1answer
69 views

What is the interval of convergence of : $\sum_{n=1}^\infty\frac{n^n}{n!}x^n$?

$x+ \frac{2^2x^2}{2!}+ \frac{3^3x^3}{3!}+ \frac{4^4x^4}{4!}+...$ Possible answers- 1.($0,1/e$) 2.(1/e, $\infty$) 3.(2/e, 3/e) 4.(3/e, 4/e) ...
7
votes
2answers
580 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: $\frac{1}...
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votes
1answer
33 views

Convergence of a series, does it converges? [closed]

Determine if this series converges, and if converges, compute its sum: $$ \sum_{n=1}^{\infty}2^{3n}4^{2-2n}. $$ Determine if the series converges or diverges: $$ \sum_{n=1}^{\infty}\cos\frac{n^{2}+1}...
1
vote
1answer
41 views

Convergence of modified series with $e^{-x^2}$

Let $f(x) := e^{-x^2}$ and I want to check, if $\sum_{n=0}^{\infty} f(x-n)$ convergence to some function $F(x)$. Also I want to check type of convergence and if $F \in C^1$ and if it is Lipschitz. So,...
0
votes
3answers
54 views

What does the derivative have to do with slow convergence in Newton's Method?

I am looking for an intuition for the following behavior: Let $f(x)=x^2$ Apparently the Newton's Method iterations to find the root (at $x=0$) converge in this case relatively slow: $x_{x+1}=x_n-\...
0
votes
1answer
21 views

Convergence of random variable 5

If $Q < \frac{n}{m^2} X_n$ where $X_n$ is a sequence of random variables, $X_n \xrightarrow{a.s}1$, $0\leq Q \leq1$, $m=\omega(\sqrt{n})$ (The $\omega$ denotes the order, see here). Then, how can ...
1
vote
1answer
39 views

Prove the convergence of sequence and find its limit

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ with: $$f(x) = \frac{1}{x^2}e^{\frac{1}{x}}$$ We consider the sequence $x_n$, having $x_0 \in \left ( 0, \frac{1}{2} \right )$ and $x_{n+1} = f(\frac{1}{...
2
votes
1answer
52 views

Almost sure convergence equivalence

Are the following statements equivalent? $$a) X(t)/t\xrightarrow{a.s} c $$ $$b) X(t)\xrightarrow{a.s} t c $$ where $c$ is a constant and $X(t)$ is a sequence of random variable. By ...
0
votes
1answer
38 views

Does the Taylor series of $e^{f(x)}$ converge everywhere?

In STAT 110, the professor says "the Taylor series of $e^x$ converges everywhere, and then proceeds to convert: $${e}^{t^2/2} = \sum_{i=0}^\infty \frac{{(t^2/2)}^n}{n!}$$ I understand that the ...
1
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0answers
24 views

Martingale Convergence Theorem

I have a Question regarding MCT which I am stuck in, the question goes like this: Let $X_0 = 1$ and assume that $X_n$ is distributed uniformly on $(0,X_{n-1})$. and $Y_n = 2^nX_n$. the questions ...
0
votes
1answer
26 views

Convergence of $\int_1^\infty\int_1^\infty(x+y)^{-a}dxdy$

My question is as follows: Find $a>0$ so that $I=\int_1^\infty\int_1^\infty(x+y)^{-a}dxdy$ converges. My attempt: Assume that $I$ converges. $I=\int_1^\infty x^{-a}\int_1^\infty(1+y/x)^{...
0
votes
1answer
35 views

Two sequences have the same limit

Let $f$ and $g$ be real-valued continuous functions on $\Bbb R^2$ that satisfy the following condition: $$ x<y \implies x< f(x,y) < g(x,y) <y $$ Assume that there are two sequences $\{a_n\...
0
votes
2answers
684 views

Confused about series and testing for convergence/divergence?

I'm finding it quite difficult to understand the idea of series and limits to test for divergence or convergence. Perhaps more so in finding such a limit. I have the series $$\sum_{n=1}^\infty \...
13
votes
3answers
323 views

Where has this common generalization of nets and filters been written down?

It is well-known that there are two different ways to generalize the theory of convergence of sequences to arbitrary topological spaces: nets and filters. They are of course essentially equivalent, ...
6
votes
1answer
2k views

Basic facts about ultrafilters and convergence of a sequence along an ultrafilter

Could you help, please. I need the information about the ultrafilters, namely, any ideas how one can see that they exist and a proof of the fact that for any ultrafilter every sequence on a compact ...
4
votes
3answers
69 views

Are Pandemic chain reactions confluent? (vertex spills weight to neighbors at threshold, once)

Are resolutions of chain reactions order-independent in the board game Pandemic? More formally: You're given an undirected graph $G = (V, E)$ and a vertex weight $w \colon V \to \{0, \ldots, 3\}$. ...