Convergence of sequences and different modes of convergence.

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Sufficient conditions for Uniform Law of Large Numbers

I would need a Uniform Law of Large numbers for $f_T(\theta)$ over $\Theta$ when $f$ is the indicator function and, thus, not continuous over $\Theta$. Do you know about any sufficient conditions?
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63 views

Prove the convergence of : $\sum \ln(n)/n^{3/2}$

I've been having some issues with what to compare it to. I have a hunch it converges. But I just cannot figure out what I can compare it too. Please help. :)
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1answer
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Uniform convergence and differentiation proof of remark 7.17 in Rudin's mathematical analysis

Rudin page 152 Theorem 7.17: Suppose $\{f_n\}$ a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point x0 on [a,b]. If {f′n} converges uniformly on ...
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37 views

Gauss' test for Convergence

In the text I am using, the hypotheses used for Gauss' test for convergence are different to others I have seen. The text has the hypotheses: if the series $\sum_{n=1}^{\infty} a_n$ is such that ...
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1answer
56 views

Different types of Convergence for a Series Function

I am currently investigating the convergence of the following function, $f(x)=\sum\limits_{k=1}^{\infty} \dfrac{x^{k}+\sin(k)}{k^{2}}$ for different "senses". I have shown that $f(x)$ converges ...
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1answer
39 views

How to prove : If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges.

Could you give me some hint how to prove this statement: If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges. I think, obviously wrong, that if $a_{2n}\to a$ and $a_{2n-1}\to b$ than ...
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27 views

Convergence of a matrix product

Let $A=o_{a.s.}(1)$; $A:k\times k$ matrix and $Vu=O_p(1)$; $V:k\times k$; $u: k\times 1$. Specifically, $Vu$ converges in distribution to $\mathcal N(0,I_k)$. Can we show that $VAu=o_p(1)$ or ...
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32 views

Two series sum, one converging, one diverging

I need to give an example of a two series sum a_n and b_n such that the lim a_n/b_n=1. One series has to diverge and one has to converge. a_n and/or b_n don't necessarily have to be positive. I have ...
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33 views

Connection between series

I have to show that if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent then $\sum_{n=1}^{\infty} a_n^2$ is absolutely convergent too. Please give me some hint, how do I start the excercise. ...
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what is the Convergence radius and what happen at the edges?

What is the Convergence radius and what happen at the edges? $$\sum_{n=1}^{\infty}\frac{(x+2)^{n^2}}{n^n}$$ Thank you
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How to read $\bigcap\limits_{r=1}^{\infty}\bigcup\limits_{k=1}^{\infty}\bigcap\limits_{n,m\ge k}^{\infty}\{x:|f_n-f_m|<\frac{1}{r}\}$

Can somebody explain me step by step why does this $\bigcap\limits_{r=1}^{\infty}\bigcup\limits_{k=1}^{\infty}\bigcap\limits_{n,m\ge k}^{\infty}\{x:|f_n-f_m|<\frac{1}{r}\}$ represents the set of ...
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87 views

Tough Limit Question

Let {$x_n$} be monotone increasing sequence of positive real numbers. Show that if {$x_n$} is unbounded, then $\sum_{n=1}^{\infty}(1-\frac{x_n}{x_{n+1}})$ diverges.
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66 views

The limit of $\ln(n) - \ln(n^2 + 1)$ as $n\to\infty$

As $n\to\infty$, what is the limit of $\ln(n) - \ln(n^2 + 1)$ Using properties of logs and limits, I ended up with: $$ \ln \left(\lim \left(\frac{n}{n^2 + 1}\right)\right) $$ where lim is the ...
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1answer
28 views

Big O notation preserved under convex functions?

Suppose that the random variable $X_T$ is $O_p(1)$ as $T \rightarrow \infty$, i.e. $\forall \epsilon>0$, $\exists M_\epsilon>0$ such that $\mathbb{P}(X_T>M_\epsilon)<\epsilon$ $\forall T$. ...
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1answer
37 views

Divergence of $\frac{1}{a_{1}^{s}}+\frac{1}{a_{2}^{s}}+\frac{1}{a_{3}^{s}}…$

Assume that we know this converges. $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+....$$ Is it possible to detect for which largest $0<s<1$ the sum below diverges? ...
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24 views

Question about bounded sequence with two sub-sequential limits.

Could you please give me some hint how to deal with this question. Suppose $(a_n)$ is bounded sequence with 2 sub-sequential limits. Prove : there are real numbers A and B that ...
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1answer
26 views

weak convergence implies point-wise convergence?

If we have a bounded sequence $\{f_n\} \in L^p[a,b]$ that converges weakly to $f$ does this mean that the converges is also pointwise?? thank you.
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1answer
28 views

Determine the value of r where the series converges

show that $$ \big(r\big)^{ln(n)} = \big(n\big)^{ln(r)} $$ Then determine the values of r (with r>0) for which the series $$ \sum_1^\infty (\big(r\big)^{ln(n)})$$ converges. r must be in what ...
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86 views

If $\sum a_n$ converges then $\sum (-1)^n \frac {a_n}{1+a_n^2}$ converges?

Could you please give me some hint how to deal with this question: If $\sum a_n$ converges, does this necessarily mean that $\sum (-1)^n \frac {a_n}{1+a_n^2}$ must converge also ? Thanks.
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Help with Convergence/Divergence

So I am trying to prove whether the following problem converges or diverges? $$\sum_{n=1}^\infty \left({n\over n+18}\right)^n$$ So I decided to use the Root test. $$ L = \lim_{n\to ...
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Is this a valid proof of Mertens summation theorem?

Consider $\sum a_n$ absolutely convergent and $\sum b_n$ convergent. Then $\sum c_n$ with $c_n = \sum_{k=0}^{\infty} a_n b_{n-k}$ is convergent, too. So my proof attempt is a little short and I am ...
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Prove that $g_n \to g$ uniformly on $\mathbb{R}$ but $f_{n}g_n$ foes not converge uniformly to$fg$ on $\mathbb{R}$

Let $f_n(x) = x$ and $f(x) = x$ for $x \in \mathbb{R}$ and let $g_n(x) = \frac{1}{n}$ and $g(x) = 0$ for $x \in \mathbb{R}$. Prove that $f_n \to f$ uniformly on $\mathbb{R}$ and $g_n \to g$ uniformly ...
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Continuity of convergence vector

Consider a continuous mapping $f: \mathbb{R}^n \rightarrow \mathcal{K} \subset \mathbb{R}^n$, where $\mathcal{K}$ is a compact set. Let $\alpha \in [0,1]$ and, for all $k \geq 0$, define $a_i := ...
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26 views

Weak convergence implies uniform convergence in distribution?

Consider an empirical processes $\{v_T(\theta), T\geq 1 \}$ and assume it is weakly convergent to the stochastic process $\{ v(\theta); \theta \in \Theta\}$., i.e. $$ E^*(f(v_T(\theta)))\rightarrow ...
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24 views

stats - limiting distribution of $X_i$

Suppose $P(X_{n}=i) = \frac{n+i}{3n+3},$ for $i = 0, 1, 2$. Find the limiting distribution of $X_{n}$. Is the cdf $F_{x_{n}}$ like: " if x = 0, $F_{x_{n}} = \frac{n}{3n+3}$; if x = 1, $F_{x_{n}} = ...
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1answer
46 views

Does the sequence converge? To what?

Let α and β be positive real numbers and define a sequence by setting $s_1 = \alpha, s_2 = \beta$ and $s_{n+2} = \frac12(s_n+s_{n+1})\forall n\in \Bbb \ge1$ Show that the subsequences $\{s_{2n}\}$ ...
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stats - limiting distribution

Let $0 < p < 1$, and let $X_{n}$ have p.d.f. $f_{n}(x) = ( 1 – p ) ( n + 1 ) ( 1 – x )^{n} + p n x^{n – 1}$, for 0 < x < 1, zero elsewhere. Find the limiting distribution of $X_{n}$. ...
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what a and b make the integral convergence?

Consider the following integral $$\iint_Ax^\alpha y^\beta \space dA$$ where $A=\{(x,y)\space|\space0\leq y\leq1-x,x\geq0\}$. Find all possible values of $\alpha$ and $\beta$, for which this integral ...
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30 views

stats - determine limiting distribution

Let $Y_{1} < Y_{2} < ... < Y_{n}$ be the order statistics of a random sample from a distribution with pdf $f(x) = e^{-x} , 0 < x < \infty$, zero elsewhere. Determine the limiting ...
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Change of signs in harmonic series

Prove that if we choose signs for individual terms in harmonic series $\sum_{n=1}^{\infty}{1\over n}$ in such a way that $p$ positive terms are followed by $q$ negative terms (without rearranging the ...
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40 views

Convergence of $\sum_{k=1}^n(1-k/n)a_k$

Assume that the series $\displaystyle \sum_{n=1}^\infty a_n$ converges to a finite number, say $S$. Now let's consider a sequence of modified partial sums $\displaystyle ...
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30 views

Convergence of Secant method or IQI

We have drawn a graph that shows that the methods do converge but is it possible to obtain values for the rate of convergence of the secant and inverse quadratic interpolation methods for a particular ...
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23 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
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1answer
59 views

Is $\sum_{n=1}^{\infty}\frac{\log n}{n^{2}}$ convergent? How to show that?

Is $\sum_{n=1}^{\infty}\frac{\log n}{n^{2}}$ convergent? How to show that? I was trying to prove Mertens third theorem and i got stuck at this.
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Almost sure convergence of a sum of independent exponential random variables?

I'm in difficult with this exercise... I hope someone can help me. Let $X_1,X_2,...$ be independent random variables, $X_n\sim \exp(\lambda_n)$, where $$0 < \lambda_n\rightarrow \lambda , \lambda ...
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1answer
37 views

weakly convergent subsequence implies strongly convergent

Statement: Let $X$ be a Banach space If $x_n \rightarrow x$ weakly and every subsequence of $\{x_n\}$ has a strongly convergent subsequence, then $x_n\rightarrow x$ strongly in $X$ Attempt: ?
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52 views

What is the convergence or divergence of $\sum _{n=1}^{\infty \:}\left(\sqrt{n+\sqrt{n}}-\sqrt{n}\right)$ [closed]

$\sum _{n=1}^{\infty \:}\left(\sqrt{n+\sqrt{n}}-\sqrt{n}\right)$ Can you show me the work for this question
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2answers
59 views

A convergent series property [duplicate]

This came up in a friend's exam and it must be one of those ${\epsilon},N(\epsilon)$ arguments I could do in a snapshot in my twenties but now I can't figure out how the proof should go: For a ...
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Expand a function in Maclaurin's series

Please help me expand this function in Maclaurin's series to find an interval of convergence. I tried to turn it into this expression: but it doesn't give me the result. Help me please it's ...
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1answer
42 views

Infinite Series: Convergent?

I just came across the following question from an old book: $$S=\sum_{k=1}^{\infty} \frac{\cos(\sqrt{k} \pi)}{\sqrt{k}}.$$ Is it convergent? My guess is that it is but how to show it? Thanks in ...
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Convergence of series $A_n = \sqrt{\sum_{k=n}^{\infty} a_k} - \sqrt{\sum_{k=n+1}^{\infty}a_k} $ if series $a_n$ converges

I must show that if a series $\sum_{n=1}^{\infty} a_n$ with positive terms converges, then the series $\sum_{n=1}^{\infty} A_n$, where $A_n = \sqrt{\sum_{k=n}^{\infty} a_k} - \ ...
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1answer
20 views

Absolute and conditional convergence of function series

I have a problem. I have to explore absolute and conditional convergence of this function series Unfortunately. I didn't find in my reference any words about "absolute and conditional", Instead ...
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1answer
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How to show $\Gamma (n)$ is convergent if and only $n>0$?

How to show $\Gamma (n)$ is convergent if and only $n>0$ where $\Gamma (n)$ is the gamma function.
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48 views

Find a radius of convergence of power series

I have to Find a radius of convergence of this power series I' ve decided to use D'alambert indication: Looking for a limit i meet a problem with a factorial Please. help me finish this ...
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137 views

explore the convergence of series with ln(n)

Help me explore the convergence of red color rounded series. On this photo (the equation below) I used radical indication but it doesn't show me the result. What would be better to use? ...
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60 views

Is this convergent or divergent

I have worked on this my answer is L= Div and B Consider the series $\displaystyle \sum_{n=1}^{\infty} a_n$ where $$a_n = \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$$ In this problem you must ...
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65 views

Decide convergence of the series

Using Taylor expansion decide convergence of the series: $$\sum_{n=1}^{\infty}(e-(1+{{1}\over{n}})^n)^p = \sum_{n=1}^{\infty}a_n$$ I expanded $a_n$ like this $a_n = (e-(1+{{1}\over{n}})^n)^p = ...
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124 views

Doubly infinite sequence limit

I have a 2-indexed sequence $F^n_m$ where $n,m$ are natural numbers and I am concerned about the behavior as $n\to\infty$ and/or $m\to\infty$. The sequence is expressed as ...
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15 views

Expectation of O_p(1) process

Suppose $\{X_n \}$ is bounded in probability, i.e. $Prob(|X_n| > M_\epsilon) = \epsilon$ for all $n > N_\epsilon$, $M_\epsilon < \infty$. Is there any condition(s) to guarantee that ...
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52 views

Convergence of series with sum

Consider a continuous mapping $f: \mathbb{R}^n \rightarrow \mathcal{K} \subset \mathbb{R}^n$, where $\mathcal{K}$ is a compact convex set. Let $\alpha \in [0,1]$ and, for all $k \geq 0$, define $a_i ...