Convergence of sequences and different modes of convergence.

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Periodic extension and uniform convergence

Consider the function $f(x)=x, \ -1<x<1.$ I know that $f(x)$ is continuous and has continuous derivative. My question is, why is the periodic extension of $f$ not continuous therefore the ...
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2answers
50 views

prove a sequence without knowing its convergence

Let $(X_n)$ be the sequence with $X_1=2$ and $X_n=\sqrt{5X_{n-1} + 6}$ for all $n\ge 2$. How can you prove that it is convergent? IF given convergence, I know that its limit is $6$, but the question ...
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1answer
12 views

Finding Convergeance sum for two power-series.

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
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40 views

Convergence of series $\sum\limits_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{\sqrt{n}}$ and approximation with maximum error

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
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1answer
20 views

Comparing the order of convergence $\mathcal{O}( h^2 |\log(h)|)$

I don't have any intuition in judging how fast a term of the order $\mathcal{O}( h^2 |\log(h)|)$ is decreasing as $h \to 0$, so i tried comparing it with terms of the form $\mathcal{O}( h^\alpha )$ ...
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1answer
24 views

Uniform convergence and relative error?

I never took a class where uniform convergence was introduced, so I only know the Definition and not much more about it. I have an Approximation of some sequence of functions $f_n$ by some function ...
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0answers
16 views

Prove that if $\{a_n\}$ converges, then $\{a_{2n+1}\}$ converges, using definition of convergence [duplicate]

Use the definition of convergence to prove that if the sequence $\{a_n\}$ converges, and $b_n = a_{2n+1}$, then the sequence $\{b_n\}$ also converges. Progress I was thinking about using proof by ...
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12 views

how to dertermine this series convergence?

Does the series (n-1)/ ( n sqrt(n)) converge from 1 to infinite. why? Do we have to use comparaison test? I don't try nothing because I don't know how to do that lol thanks
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Ordinary Differential Equation with a trigonometric function: radius of convergence?

For the equation $$x^2y'' + y' + \tan(x)\,y = 0$$ establish lower bounds for the radius of convergence about the point $$x_0 = 1.$$
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37 views

Convergence in probability and convergence of Cesaro means

Consider the random variable $X_n$, not necessarily iid. If $X_n\rightarrow 0$ almost surely, then the Cesaro means $\frac 1n\sum_{k=1}^nX_n$ converge almost surely to 0. This cannot be weakened to ...
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4answers
82 views

Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...
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1answer
20 views

Distance between two compact subsets is always $\geq$ distance between two particular points of the subsets

Show that if $K_1$ and $K_2$ are compact subsets of $\mathbb R^p$, then there exist points $x_1$ in $K_1$ and $x_2$ in $K_2$ such that if $z_1$ belongs to $K_1$ and $z_2$ belongs to $K_2$, then ...
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1answer
31 views

Help Understanding Step in Proof of Convergence

The theorem is If $\sum a_n$ is a series of complex numbers which converges absolutely then every rearrangement of $\sum a_n$ converges, and they all converge to the same value. The proof ...
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1answer
214 views

Approximating hypergeometric distribution with poisson

I'm am currently trying to show that the hypergeometric distribution converges to the Poisson distribution. $$ \lim_{n,r,s \to \infty, \frac{n \cdot r}{r+s} \to \lambda} \frac{\binom{r}{k} ...
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3answers
30 views

Finding the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$.

So I am trying to find the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$. I know the integral converges, and I know the answer as well, but I am confused on how to get the correct answer. My problem ...
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1answer
300 views

How to prove convergence in mean implies uniform integrability?

My class notes and wikipedia both say that $X_n \xrightarrow{L^1} X$ $\Leftrightarrow \; X_n \xrightarrow{P} X$ and $X_n$ are uniformly integrable. I am trying to work through the proof. I am able ...
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2k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
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4answers
62 views

Does the series $\sum \frac{1}{n\ (\ln(n))^{3/2}}$ converge or diverge?

Consider $$\sum \frac{1}{n\ \ln^{3/2}(n)}$$ The ratio test is inconclusive. The root test is inconclusive. And it seems right that $\frac{1}{n\ (\ln(n))^{3/2}}\leq\frac{1}{n}$ which diverges, but ...
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1answer
58 views

Consider $f_n(x)={x^n-x^{3n}}$

A. For what values of x is the function series is point-wise convergent, and to what function? B. For what values of x is the series uniform convergence? My answers in the textbook are: A. As $n\to ...
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3answers
36 views

Does the series $\sum (1+n^2)^{-1/4}$ converge or diverge?

The integral is $\int { (1+n^2)^{-1/4}}dn$ is not quite possible, so I should make a comparison test. What is your suggestion? EDIT: And what about the series $$\sum (1+n^2)^{-1/4} \cos ...
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1answer
33 views

what is the difference between bounded and convergent?

I know that bounded means to have an upper or lower bound. Let $E \subset \mathbb{R}$ be nonempty. The set $E$ is said to be bounded above if and only if there is an $M \in \mathbb{R}$ ...
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1answer
41 views

Will this series converge? If so, what is its limit?

If $a_n=(a_{n-1}+a_{n-2})/2$ and $a_1, a_2$ are given, will this series converge? And if so, what is the limit? By intuition I think it converges to $(a_1+2a_2)/3$ , but I am not able to prove it.
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30 views

Convergence of sequence of function

I need to check if sequence of functions $f_n(x):=\sqrt{x^2+\frac{1}{n}}$, $n\in \mathbb{N}$ converges (pointwisely, uniformly) in intervals:$[-1;1]$ and $\mathbb{R}$. Is there any algorithm how to ...
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1answer
33 views

Convergence of proposed approximations to conditional expectation

Let $X,Y$ be real random variables on $(\Omega, \mathscr{F}, P)$ with $E[|X|] < \infty$. Let $Z_1, Z_2, \dots$ be a sequence of proposed approximations of $E[X|Y]$ defined by $$Z_n(\omega) = ...
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2answers
32 views

What does tightness of convergence mean and why do we use this?

In a paper that I am reading it is written: We have $$(X(t_j))_{j=1}^k \xrightarrow{d} (Y(t_j))_{j=1}^k,$$ where $t_0=0, t_1 < \dotsb < t_k$. It further is written: In order to show that ...
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20 views

Proving convergence using epsilon N definition [closed]

jesuserent answers (and they were negative), so I must've been doing it wrong or something.
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2answers
71 views

The limit of $\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$ as $n\to\infty$

The task is to calculate $$\lim_{n\to\infty}\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$$ I tried various estimates I know to find the dominating integrable function and nothing worked. Does anyone ...
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36 views

proving convergence and finding the limit of this sequence if it is convergent?

$a(n)=\sin(1+a(n-1))$ where $a(0)=0$ and $n$ belongs to integers , prove or disprove whether the sequence converges and find the limit . well i understood that this is bounded but i am not able to ...
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40 views

Will the Newton's method be convergent to the root of the following function: $f(x)=\frac{-x}{x^2-1}$?

Will the Newton's method be convergent to the root of the following function, if the starting point $x_0>1$ will be chosen? $$ f(x)=\frac{-x}{x^2-1} $$
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21 views

Is the series convergent or divergent [closed]

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum \sqrt{a_na_{n+1}}$ always convergent?Either prove it or give a counterexample.
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15 views

scalar dimension to the approximation of an integrable function

Let $(M,\mathcal{A},\mu)$ space of probability. If $f\in L^1(\mu)$ then $f=\displaystyle{\lim_{n \rightarrow +\infty}\sum_{i=1}^n}\alpha_i\chi_{C_i}$ where $C_i\in \mathcal{A}$ and $\alpha_i \in ...
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1answer
26 views

Assumption in a convergence proof

Im in the middle of a proof of the fact that for a>0, if lim $x_n$ = a, then lim $\sqrt{x_n}$ = $\sqrt{a}$. I'm in the step that i use $| \sqrt{x_n} - \sqrt{a} |$ = $ \frac{|x_n - a|}{ ...
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2answers
40 views

If $x = lim (x_n)$ and if $|x_n - c| < \epsilon~~ \forall ~n \in N$, then it is true that $|x-c|< \epsilon$

If $x = lim (x_n)$ and if $|x_n - c| < \epsilon~~ \forall ~n \in N$, then is it true that $|x-c|< \epsilon?$ I am a little confused about this question ( it appears in the Bartle's elements of ...
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1answer
25 views

How does $\mathcal{L}^1$-convergence of a series of $\mathcal{L}^1$ random variables imply that $\sup_{n \in \mathbb{N}} \mathbb{E}[|X_n|] < \infty$?

Let $(X_n)_{n \in \mathbb{N}}$ be a series of random variables with $\forall i: X_i \in \mathcal{L}^1(\Omega, \mathfrak{F}, P)$ and $X_n \rightarrow^{\mathcal{L}^1}X$. How do I show then, that ...
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169 views

Proving that the following series is convergent

Can someone please help me prove that this series is convergent? $$ \sum_{i=1}^\infty \left({n^2+1\over n^2+n+1}\right)^{n^2} $$ I guess I'm supposed to show that the limit of the sequence is an "e" ...
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1answer
53 views

Find a Cauchy sequence that does not converge

I am supposed to look at $l_0$, the set of all sequences with finitely many non-real elements in $(l_0,d_{\infty})$. It is just that I don't quite understand how the $d_\infty$-metric is defined on ...
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1answer
40 views

Proving $\lim_{n\rightarrow\infty} \sin(n)/n = 0$ using epsilon definition

So the limit as $n \rightarrow \infty$ of $2\sin(n) / n$ is $0$. How do I prove this? I say; $$|2\sin(n) / n - 0 | = |2\sin(n) / n| < \epsilon$$ for which $n$'s ? Not sure how to get rid of the ...
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1answer
16 views

Convergence of sequences and limits

Let $(f_n)$ be the fibonacci sequence and let $x_n = \dfrac{f_{n+1}}{f_n}$. Given that $\lim_{n \to \infty}(x_n) = L$ exists, determine the value of $L$.
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approximate a Borel set by a continuous

I wonder if it is possible to approximate a Borel set by a continuous function i.e. Let $B$ a Borel set in $(X,d)$ (compact separable metric space) I wonder if there continuous functions ...
2
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3answers
43 views

$\sum_{n=1}^{+\infty} \frac{n^{n-1}v^n}{n!}$ for what value of $v$ this series will be convergent? How to proceed for it?

I am interested in the convergence of the series $$\sum_{n=1}^{\infty} \left( \frac{n^{n-1}}{n!}v^{n} \right).$$ This series defines the tree function.
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1answer
53 views

extending the convergence of measures

Let $F$ the real vector space of all applications $\phi: X \times Y \rightarrow \mathbb{R}$ where $(X,\mathcal{B}_1, \mu)$, $(Y,\mathcal{B}_2 )$ measurable spaces with $X$ and $Y$ are compact metric ...
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3answers
113 views

Convergence of alternating nested radicals

Last evening, after reading a couple of questions about nested radicals, I started to wonder about problems involving what I will term "alternating nested radicals;" below is an example, which I found ...
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1answer
34 views

Why does these complex sequences converge uniformly?

I have one complex series and one sequence. It is used in complex analysis in a part of my book where they are integrated. However, as you know in order to change limit and integration order it has to ...
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30 views

solution of the set of real non-linear equations

I have a set of real non-linear equations as following: \begin{equation} y_0 = f(y_0,y_1) \\ y_1 = g(y_0,y_1,y_2) \\ y_2 = g(y_1,y_2,y_3) \\ \vdots \\ y_{n-1} = g(y_{n-2},y_{n-1},y_n) \\ y_n = ...
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Decide whether the following series converges $\sum_{n=1}^{\infty}\dfrac{(\ln n)^2}{n^{3/2}}$

Looking for a neat and smart way to solve this. I am having a tough time with this
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1answer
41 views

Does uniform convergence depends on the metric?

Definition: Let $f_n:X\to Y$ be a sequence of functions from a set $X$ to the metric space $Y$. Let $d$ be the metric for $Y$. The sequence $(f_n)$ d-converges uniformly to the function $f:X\to Y$ if ...
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2answers
30 views

A query in the proof of convergence of the set $\{1/n\}$

I have a query regarding the proof of the statement that the set $S = \left\{ \dfrac {1}{n} \right\}$ has limit point $0$. I am studying an introductory course in Analysis. Proof: From the ...
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1answer
44 views

Is this series divergent or convergent?

I've been stuck with this problem for a couple of days trying to solve it but got no where till now. The problem states that we have to prove if the series given below is convergent or divergent, if ...
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20 views

Sequence of Functions that does not Converge

I'm asked to show that the sequence of functions $f_n(x) = n^2x^n$ defined on the closed interval $[0,1]$ does not converge pointwise to any function as $n \to \infty$. For $0 \le x \lt 1$ I think I ...
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1answer
29 views

Use of taylor series in convergence

Homework problem here, would appreciate an explanation to the answer of this question. Problem: Find the rate of convergence of $$ \lim\limits_{h \to 0} \frac{\sin(h)}{h} = 0 $$ The book solves ...