Convergence of sequences and different modes of convergence.

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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 ...
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 ...
3
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1answer
49 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 ...
1
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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 $\...
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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
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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 \...
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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
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0answers
45 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
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2answers
58 views

When a Markov chain converges to a steady state, what kind of convergence is it?

Let $A$ be a transition matrix, the steady state distribution $x$ satisfies the distribution $Ax = x$. One can prove that under certain circumstances, $$\lim_{n\rightarrow\infty}A^n q=x$$ where $q$ is ...
0
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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 $$...
1
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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 $Re(s) > 1$. ...
3
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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
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1answer
49 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 ...
4
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2answers
93 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{...
1
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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}...
1
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0answers
36 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. ...
0
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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 ...
2
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3answers
59 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 ...
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1answer
51 views

Can you prove the convergence of $ \int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx $?

Can you prove the following improper integral is convergent? $$ \int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx. $$
3
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3answers
107 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 ...
0
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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 ...
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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 $n \in \mathbb{N}$...
4
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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 ...
1
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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 ...
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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.
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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) ...
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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
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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
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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 ...
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3answers
55 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-\...
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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}{...
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1answer
41 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
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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)^{...
4
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5answers
391 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 $. ...
0
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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\...
2
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0answers
44 views

Almost sure convergence (correctness of an argument)

Is this statement correct? If $X_n \xrightarrow{a.s} c$, where $X_n$ is a sequence of random variables and $c$ is a constant, then we can conclude that since almost sure convergence implies on ...
3
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1answer
30 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 ...
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1answer
69 views

Sequence of polynomials converging to zero, but not uniformly on unit disc

I have been trying to solve the following without success so far: Show that there exists a sequence of polynomials satisfying $P_n(z)\rightarrow 0$ for every $z\in \mathbb{C}$, but the convergence is ...
4
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3answers
70 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 $w \colon V \to \{0, \ldots, 3\}$. ...
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1answer
42 views

Limit of a Function with help of Euler's Formula

I've been trying to get the limit of a function, but I don't know how. The function is $\displaystyle{10^{n}\left(1 - \mathrm{e}^{\mathrm{i}t/10^{\,n}}\,\right)}$ and the solution says this ...
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1answer
41 views

the proof for weak law of large numbers

This is the text of proof of the law I don't understand why when $n\to \infty$, $\frac{\sigma^2}{n\epsilon^2} \to +\infty$? isn't $\frac{\sigma^2}{n\epsilon^2}=\frac{1}{k^2}$? how come $\frac{1}{k^2}...
3
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1answer
77 views

Behavior of fundamental solution to heat equation after projection

I am considering the behavior of $$\frac{1}{h}\|(1-P_h)S(h)v\|,\tag{1}$$ and $$\frac{1}{h}\|(1-P_h)S(h)P_hv\|,\tag{2}$$ as $h\to 0^+$ for a fixed good enough $v$. I hope to show one of them converges ...
4
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2answers
55 views

Show that the sum of a sequence of random variables converges almost surely

Let $(X_n)_{n\in\mathbb N}$ be a sequence of non-negative iid random variables with $\mathbb E[X] < \infty$. How could one go about showing that $\sum^{\infty}_{k=0} e^{X_k} c^k < \infty$ ...
0
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1answer
14 views

Reference needed for properties of Convergence of Random Variables

Does anybody know a good reference for properties of convergence of random variables? For example, if $X_n$ converges almost surely (a.s) to $X$ and if $Y_n$ converges a.s to $Y$, then $X_n Y_n$ ...
2
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1answer
39 views

Convergence of $V_n=\prod\limits_{i=1}^n U_i$

I struggle to do this exercise: Let $U_1,U_2,\dots$ be a sequence of i.i.d. random variables. We define $$V_n=\prod\limits_{i=1}^n U_i$$ Show that $V_n^{1/n}$ converges almost sure and calculate the ...
2
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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 $n \to \infty$ ...
3
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2answers
65 views

radius of convergence of $1/(1+z^2)$ about $z=2$ using geometric series approach

I would like to calculate the radius of convergence of $f(z)= 1/(1+z^2)$ about $z=2$ using the geometric series approach. Let me first state that according to a theorem, the radius of convergence ...
2
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1answer
90 views

Evaluate $\sum_{k=2}^\infty\frac{1}{k^3-1}$

I was considering a specialization of the Cauchy product $$ \left(\sum_{n=1} ^\infty x^n \right) \left(\sum_{n=1}^\infty (-1)^n x^n \right)=\frac{-x^2}{1-x^2},$$ that converges for $0<x<1$. ...