Convergence of sequences and different modes of convergence.

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Right-const function and pointwise/uniform convergence

Let function $f:\mathbb{R} \rightarrow \mathbb{R}$ be right-const iff $\exists_{M \in \mathbb{R}}\forall_{x,y \ge M}f(x)=f(y)$. Consider function sequence $\{f_n\}$ which every term is right-const. ...
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$L^p$ convergence of smooth compactly supported functions

I already checked the similar question here, but want to check slightly different argument. Given $f(x)\in L^p(\mathbb{R}^n)$, can I find $f_n(x) \in C^\infty_0$ s.t. $f_n \rightarrow f$ almost ...
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What should be the value of $\alpha$ for which the series is convergent?

The series $$\sum \frac{\log(1+\frac{1}{n})}{n^\alpha}$$ a. Converges if $\alpha>0$ b. Diverges for all $\alpha\in \mathbb{R}$ c. Converges if $\alpha=0$ d. Converges if $\alpha<0$ ...
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Is $t_n=\frac{1}{n} \left(1+\frac{1}{\sqrt2}+\cdots+\frac{1}{\sqrt n}\right)$ convergent?

Let $$t_n=\frac{1}{n} \left( 1+\frac{1}{\sqrt2}+\cdots+\frac{1}{\sqrt n} \right)$$ Is the following series convergent? If we let $a_n=\frac{1}{\sqrt n}$, then $$\lim_{n\to\infty}a_n=0\rightarrow \...
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Show that the Maclaurin series of $f(x)=1/\sqrt{1-x}$ holds for $x\in[0,1/2]$ using the Lagrange remainder theorem

Show that the Maclaurin series of $f(x)=1/\sqrt{1-x}$ holds for $x\in[0,1/2]$ using the Lagrange remainder theorem I can see that $$f'(x)=\frac12 (1-x)^{-\frac32}\text{ and }f''(x)=\frac12\frac32(1-...
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(Conceptual) Continuity of binary relation $\succsim$ and definition using contour sets

Some background information: $\succsim$ is a binary relation that represents preference between two goods. $\succsim$ means "x is at least as good as y." Continuity of this relation is defined to be ...
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80 views

Examining a solution of a differential equation without knowing the solution

The differential equation is given by $$\dot x=-x \cos x$$ with $x(0)=x_0\in(0,\frac{\pi}{2})$. Now I need to show that for each choice of $x_0$ the domain of the solution $x: I\rightarrow \mathbb{...
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weakly convergence the sequence $f_{n}= n. \chi_{[-\frac{1}{n},\frac{1}{n}]}$

I need to research on the uniform, weak and strong convergence the sequence $$f_{n}= n. \chi_{[-\frac{1}{n},\frac{1}{n}]}$$ for $n\in \mathbb{N},$ in $L^{2}(\mathbb{R})$ equipped with norm $\...
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For what values of $a$ does $\int_0^\infty\left(\frac{x^a}{1 + x^2}\right)^4 \, dx$ converge?

I'm learning about convergence/divergence of improper integrals and need help with the following problem: Find for what values of $a$ does the following integrals exists $$(1) \int_0^\infty\...
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Comparision test for this series?

How do I check divergence of this series? $$\sum_{n=0}^{\infty} \frac{6}{4n-1} - \frac{6}{4n+3}$$ Wolframalpha said it used the comparision test but I don't see what possible smaller sum to use? ...
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1answer
54 views

conditional convergence of $\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$

prove that the series $$\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$$ is conditionally convergent? I tried to prove that it is not absolutely convergent series by trying to prove that $\sum_{n=2}^{\infty} \...
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0answers
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Using ratio test for sequences?

Don't mark this as duplicate. The question is to verify whether my method is correct. Prove: $$\lim_{n\to\infty} \frac{x^n}{n!} = 0$$ Method: Let $\sum_{n=1}^{\infty} \frac{x^n}{n!}$. By ratio ...
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Why does Slutsky's Theorem Fail to Generalize? [closed]

What is a counterexample to the claim that $X_n \rightsquigarrow X$, $Y_n \rightsquigarrow Y$ implies that $X_n + Y_n \rightsquigarrow X + Y$? I know that Slutsky's Theorem guarantees the case that $...
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Convergence of minimum of a sequence of functions

Consider functions $f_n, f$ from $\mathbb{R}$ to $\mathbb{R}$. Suppose $f_n(y)$ converges pointwise to $f(y)$ for all $y$ as $n \rightarrow \infty$. I would like to know under what conditions is the ...
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Convergence of integral $\int_{0.5}^{1} \log(|\log(t)|)\ ^ 7\,\mathrm{d}t $

Does this integral converge ? $$\int_{0.5}^{1} \log(|\log(t)|)\ ^ 7 \,\mathrm{d}t $$ I have tried to compare this integral to $$\int_{0.5}^{1} \dfrac{1 }{\sqrt{x-1}} \,\mathrm{d}t $$ but couldn't ...
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Theorem 3.16. in Analytic Number Theory by Apostol

The below texts are from the book Introduction to Analytic Number Theory by Apostol: I have two questions which I couldn't find solutions for them: $1-$ According to Thm 3.16., $\sum_{n\le x} \...
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1answer
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Convergence of the integral $\int_0^\infty f(x)\frac{xf'(x/(1-1/N))}{f(x/(1-1/N))}\ \mathsf dx$ as $N\to\infty$

How can calculate this integral $$\lim_{N \to \infty} \int_{x=0}^{\infty}f(x) \frac{x f'\left(\frac{x}{1-1/N}\right)}{f\left(\frac{x}{1-1/N}\right)} dx$$ where $f(x)$ is a probability density function?...
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Random subseries of harmonic series expected to converge, but how often does it?

Inspired by a previous question which I can't seem to find, what if we have $$X = \sum_{k=1}^{\infty}\frac{1}{k}\cdot P\left(U(0,1)<\frac{1}{k}\right)$$ That is, each term of the series will be $\...
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Summary of results concerning interchange of limits in series

The document http://www2.iugaza.edu.ps/ar/periodical/articles/volume%2014-%20Issue%201%20-studies%20-16.pdf constructs the theory of double sequences and double series very nicely, supplying the ...
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Confusion about the convergence of Riemann zeta function in terms of the integral

Titchmarsh wrote that $$\zeta(s)=s\int_{1}^{\infty}\frac{\left \lfloor x \right \rfloor-x+\frac{1}{2}}{x^{s+1}}\,\mathrm{d}x+\frac{1}{s-1}+\frac{1}{2}\tag{2.14}$$ using the Euler-Maclaurin summation, ...
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Proof of Leibniz $\pi$ formula

I found the following proof online for Leibniz's formula for $\pi$: $$\frac{1}{1-y}=1+y+y^2+y^3+\ldots$$ Substitute $y=-x^2$: $$\frac{1}{1+x^2}=1-x^2+x^4-x^6+\ldots$$ Integrate both sides: $$\...
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Find $\lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2}$

I have problem with finding the following limit. I suspect that it should be easy, but really I don't have a clue. $$ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2} $$
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Converging savings

Let's imagine abstract situation that there is a city where all families have houses (their number is = $1000$) which are located on the edge of a big circle so every house has exactly two neighbors ...
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Determining bounds for a sum with nested infinite series

I am computing the inner product of the characters of the trivial and the $k$-th irreducible two dimensional representations of the dihedral group $D_n$ of order $2 n$ when $n$ is even. The ...
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What condition on the coefficients $a_n$ will guarantee $f(x) = \Sigma_{n = -\infty}^{\infty}a_{n}e^{2\pi inx}$ is k times differentiable?

$f(x) = \Sigma_{n = -\infty}^{\infty}a_{n}e^{2\pi inx}$ What condition on the coefficients $a_n$ will guarantee $f$ is $k$ times differentiable? I'm not sure where to begin with this, because it ...
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Determine whether the following integral is convergent or divergent $\int^{2}_{0} \frac{1}{\sqrt[3]{x} (x+\sqrt{x})}dx$

Please, help me to determine the following integral: $$\int^{2}_{0} \frac{1}{\sqrt[3]{x} (x+\sqrt{x})}dx$$ As we know, this is the II order indeterminant integral. I've tried to use comparison test, ...
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1answer
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Does iterating $x \cdot \sin(\frac 1 x) + x$ near $0$ approach $0$?

Let $f^1(x) := x\,\sin(\frac{1}{x})+x$ and define $f^N (x):= f(f^{N-1}(x))$ for $N\in \mathbb{Z},\ N>1$. For which $x \in \mathbb{R}$ does $\lim_{N\rightarrow\infty}{f^N(x)}=0?$ Clearly, for $x\...
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Summing power series $\sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n$

Lets have series $$\sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n$$ Obviously, its convergence radius is 1. I should sum it, but don't know what's up with the double factorial. There is a hint in the ...
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1answer
51 views

Does the series $\frac{1}{(\log{n})^4}$ converge?

I have used comparison test to show it diverges: $$\frac{1}{n}<\frac{1}{(\log{n})^4}$$ But is this even right?
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Uniform convergence of following series [closed]

Prove that $\sum_{n=1}^\infty \frac{x^{2n}}{(1 + x + \dots + x^{2n})^2}$ converges uniformly when $x \geq 0$.
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Testing convergence of series $\sum_{n=3}^\infty\frac{1}{n (\ln(n))^p(\ln\ln(n))^q}$

Lets have this problem. $$\sum_{n=3}^\infty\frac{1}{n (\ln(n))^p(\ln\ln(n))^q}$$ I have rewritten this to a form $$\sum\frac{1}{np'^{\ln\ln(n)}q'^{\ln\ln\ln(n)}}$$ For $p,q\in\mathbb{R}$. Obviously, $...
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1answer
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Do these series vanish asymptotically?

Let's consider a monotone increasing sequence $(K_N) \subseteq \mathbb{N}$ with $(K_N) \xrightarrow[N]{} \infty$ and $(K_N) = \mathrm o(N)$ (less increasing than $(N)$). Question: $\sum_{j=K_N + 1}^...
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How do I find the radius of convergence for $\sum_{n=0}^{\infty}\frac{1}{\sqrt{n}}z^n$?

I'm a little unsure about methods on finding the radius of convergence of a function. It would be great to get some help on how to approach these kinds of problems.
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$Y_n\xrightarrow{P}E(X_i)$

I have this problem at hand $X_1,X_2,\cdots $ are iid random variables with finite second moments.Define$$Y_n={2\over n(n+1)}\sum_{i=1}^n iX_i$$ Show that $Y_n\xrightarrow{P}E(X_1)$. I know ...
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How to find Taylor series when $x_0=0$ and radius of convergence for $\frac{x}{1+x}$ for $f:(-1,\infty)$

Through the taylor series formula: $$f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\dots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$ I've got that $f(x)=x-x^2+x^3-x^3\dots$ however my teacher claimed ...
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finding the radius of convergence of $\sum_{n=1}^{\infty} n^2x^n$ [closed]

How does one find the radius of convergence of: $\sum_{n=1}^{\infty} n^2x^n$ using the fact that it's possible to differentiate every term. I have no idea how to go about with this
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Divergence of $\sum \limits_{n=1}^{\infty}\frac{a_n}{a_1+a_2+\dots+a_n}$ [duplicate]

Suppose that $\sum \limits_{n=1}^{\infty}a_n$ series with positive terms which diverges then series $\sum \limits_{n=1}^{\infty}\dfrac{a_n}{a_1+a_2+\dots+a_n}$ also diverges. Can anyone show how to ...
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Geometric mean of random geometric variables converging with probability to a constant

I have the following question at hand.. Let $X_1,X_2,\cdots, X_n$ be a sequence of iid random variables with common uniform distribution on $[0,1]$. Define $$Z_n=\left(\prod_{\ i=1}^{\ n}X_i \...
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5answers
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If a limit is finite does it have to be of the form $0/0$?

In my text book it is written that if $$\lim_{x\to0}\;\frac{\cos(4x) + a\cos(2x) + b}{x^4}$$ is finite then $\frac{\cos(4x) + a\cos(2x) + b}{x^4}$ should be of the form $0/0$ and therefore $\cos4x + ...
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Convergence of $\sum_{n=0}^\infty (\sin{\frac{1}{n}})(\sinh{\frac{1}{n}})(\cos n)$ [duplicate]

$|(\sin{\frac{1}{n}})(\sinh{\frac{1}{n}})(\cos n)|\le|(\sin(\frac{1}{n}))(\sinh{\frac{1}{n}})|$ Since $\lim_{x\rightarrow 0} \frac{\sin(x)}{x}=\lim_{x\rightarrow 0} \frac{\sinh(x)}{x}=1$ Thus $$\...
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Is this series divergent or convergent?

Please explain what method you used to prove so. $$\sum_{n=3}^\infty \frac{\tan\left(\frac{\pi}{n}\right)}{n}$$
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Does this sequence of random variables converge almost surely?

I was trying to understand why almost sure convergence doesn't imply convergence of the mean and I encountered this answer. However, I do not understand why this sequence of random variables ...
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1answer
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Prove/disprove radius about radius of convergnce

I have the following statement - The Taylor series of $\frac{x}{x+2}$ around $X = 1$ has a radius of convergence of $R = 4$. Is it right to say that this statement is false because a function is ...
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1answer
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Prob. 4 (b), Sec. 20 in Munkres' TOPOLOGY, 2nd ed: Which of these sequences are convergent w.r.t. the product, uniform, and box topologies?

Let $\mathbb{R}^\omega$ denote the set of all the (infinite) sequences of real numbers. Then which of the following sequences in $\mathbb{R}^\omega$ are convergent (and if so, then to which points(s)) ...
2
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2answers
32 views

Pointwise limit of the sequence of functions $h_n(x)=1,\, \forall x\ge 1/n$ and $h_n(x)=nx,\,\forall x\in[0,1/n)$

Pointwise limit of the sequence of functions $$h_n(x)=\begin{cases}1,&\text{if }x\ge \frac1n\\nx,&\text{if }x\in[0,\frac1n)\end{cases}$$ The trouble with this question is that I think that $...
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1answer
133 views

Why does $\sum_{n=2}^\infty \frac{1}{\ln(n!)}$ diverge?

$$\sum_{n=2}^\infty \frac{1}{\ln(n!)}$$ I tried by comparing it to $\sum_{n=1}^\infty \frac{1}{n}$ but i seem to fail. I think I need to compare with series that are smaller and diverge. Help.
3
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77 views

Weak convergence with uniformly convergent functions.

Suppose $F_n$ are a sequence of distribution functions with the property that for any measurable $g$ , we neccesarily have that $$ \int_{\mathbb{R}} g \; dF_n \xrightarrow{n \rightarrow \infty} \int_{\...
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1answer
52 views

Can every element in the arbitrary space be converged to?

When I have for example $\mathbb R$ then I'm able to create a sequence which will converge to any of the elements in $\mathbb R$: \begin{align} \frac{1}{n} &\rightarrow 0\\ \frac{1}{n} + 1 &\...
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0answers
18 views

Weak Law of Large Numbers, biased expectation?

I want to show that: $$\hat{\sigma^2}=(1/n)\sum^{n}_{i=1} ( X_i-\bar{X} )^2$$ is a consistent estimator of $\sigma^2$. I was using the Weak Law of Large Numbers in the sense that: $$E(X_i-\bar{X })...
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1answer
74 views

Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$ Y(k)=\frac{1}{k}\sum_{i=1}^k X_i $$ Notice that $Y(k)$ is ...