Convergence of sequences and different modes of convergence.

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Prove Uniform Convergence of Series of functions-Help?

Let F0 be a bounded Riemann integrable function on [0, 1]. For n ∈ N, define $F_n(x)$ on [0,1] by $F_n(x)$ = $\int_{0}^{x}$ $F_{n-1}(t)$ dt 1) Prove that for all n∈ N and x∈ [0,1], we have ...
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93 views

Almost Sure convergence.

Given $(X_n, n\in\Bbb N)$ and $(Y_n, n\in\Bbb N)$ sequences of random Variables. For all $n\in\Bbb N$ it is : $X_n=Y_n$ almost sure. Now the question: Is then $P(X_n=Y_n \forall n\in\Bbb N)=1$? ...
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29 views

Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution: In my syllabus ...
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52 views

Need to know how: $L^{p}$ convergence for function.

I feel terrible at the moment as I have exhausted everything possible to understand this and I am still stuck. I have looked everywhere for some sort of resource to the concept at hand, and yet, they ...
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1answer
96 views

Convergence in measure of integrable functions implies limit is integrable?

I'm going through my handwritten notes for my upcoming exam (so not homework) and the above was stated but not proven in class. The full statement is a little different, but the above part is the only ...
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155 views

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

Find if the following series converge: $$\displaystyle \sum_{n=1}^{\infty} \left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$ What I did: $$a_n=\left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$ $$b_n=\frac ...
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48 views

Let $ \sum_{n=1}^{\infty}a_n$ be a positive convergant series, prove that $ \sum_{n=1}^{\infty}2(a_n)^3$ converges as well

Let $ S_1=\displaystyle\sum_{n=1}^{\infty}a_n$ be a positive convergant series, prove that $ S_2=\displaystyle\sum_{n=1}^{\infty}2(a_n)^3$ converges as well. We have $\exists l :\forall ...
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277 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
352 views

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 $x_0$ on [a,b]. If $\{f_n'\}$ converges ...
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0answers
87 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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194 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
47 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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36 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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59 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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38 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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33 views

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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95 views

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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161 views

If $(x_n)$ is an unbounded increasing sequence then $\sum (1-x_n/x_{n+1})$ diverges

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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97 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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44 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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38 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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32 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
258 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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132 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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147 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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45 views

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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50 views

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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178 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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36 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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50 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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39 views

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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23 views

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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58 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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61 views

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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Convergence of $\sum\limits_{n=1}^{\infty}\left(\frac{\sqrt n-1}{\sqrt n}\right)^n$

$$\sum_{n=1}^{\infty}\left(\frac{\sqrt n-1}{\sqrt n}\right)^n$$ It kind of looks like the Euler's number limit, but I didn't succeed in proving that it converges. Anyone? This was an exercise at the ...
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2answers
57 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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144 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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57 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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69 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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80 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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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
87 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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21 views

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
68 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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71 views

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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2answers
61 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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19 views

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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99 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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2answers
276 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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1answer
645 views

Is $\sum_{n=1}^{\infty} \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$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 ...