Convergence of sequences and different modes of convergence.

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convergence of this series with n^a*(\log(n))^b in denominator

How do I proceed with finding a and b such that $\sum_{n=2}^{\infty}\frac{1}{n^a {(\log(n)})^b}$ converges ? Which test is the most appropriate to use and find the values ?
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8 views

Z-transform and región of convergence (ROC) [duplicate]

I need complete this problem. Any help me?
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2answers
21 views

convergence and sum of this series

$\sum_{n=1}^{\infty}(\frac{1}{n}-ln(1+\frac{1}{n}))$ is supposed to be convergent. If I use the integral test, I can prove the second term to be a finite integral while the first term is still ...
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1answer
76 views

Does the series $\sum_{n=1}^{\infty}\sin\left(2\pi\sqrt{n^2+\alpha^2\sin n+(-1)^n}\right)$ converge?

Let $\alpha$ be such that $0\leq \alpha \leq 1$. Since $\sin n$ has no limit as $n$ tends to $\infty$, I'm having trouble with finding if the series $$\sum_{n=1}^{\infty}\sin ...
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2answers
146 views

Regarding the radius of convergence and its equality to a certain limit

Let $f$ be a holomorphic function on the open unit disk $\mathbb{D}$, and suppose that $f$ cannot be extended holomorphically to any open set $\Omega$ containing $\overline{\mathbb{D}}$. Let $f(z) = ...
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29 views
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Example about Dominated Convergence Theorem

So I was reading my textbook about Dominated Convergence Theorem: I have $(X,\mathscr{F},\mu)$ as a measure space I have $f,f_n,: X\to [-\infty, \infty], g:X\to [0,\infty]$ integrable and it is the ...
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0answers
33 views

Does $L^{1}$ convergence implies almost everywhere cesaro convergence?

Let $X$ be a compact metric space, $\mu$ a Borel measure and $f_{n} \in L^{1}(X,\mu)$ continuous functions such that $f_{n}(x) \overset {L^{1}} \rightarrow 0 $. Now can we deduce that ...
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41 views

$P(X_n < a$ i.o. and $X_n > b$ i.o.$) = 0$ for all $a < b$ implies that $lim_{n \rightarrow \infty} X_n$ exists a.e.

Suppose for $a<b$ we have $P(X_n < a$ i.o. and $X_n > b$ i.o.$) = 0$. Then $lim_{n \rightarrow \infty} X_n$ exists a.e. but may be infinite. Here "i.o." means "infinitely often"; for any ...
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2answers
26 views

Monotone and bounded or just bounded

I have to know if is it necessary that a sequence to be bounded and monotone to have a limit or it can be just bounded? For example, $A_{n+1}=1/(1+A_n)$, it is bounded but not monotone. However it has ...
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1answer
20 views

Essential Uniform Convergence Implication

Greetings Mathematics Community. I believe that I am thinking too hard about the following problem and would like some guidance in solving it. Let $X$ have finite measure and let $f_n:X \to ...
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2answers
24 views

study of two sequences

I need to study wether thos two sequences converge or not. 1) $u_n=n\sum_{k=1}^{2n+1} \frac{1}{n^2+k}$ 2) $v_n=\frac{1}{n}\sum_{k=0}^{n-1} \cos(\frac{1}{\sqrt{n+k}})$ For the first i get it ...
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1answer
26 views

Real number approximation by a product of two integer powers

Inspired by the question on which points of the $\mathbb{C}$ unit circle can be reached by arbitrary products of two example points, I came up wtih the following: For given $a, b \in \mathbb{R}^+ ...
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0answers
28 views

Trying to find sum of a complex infinite series with Gamma function and factorials

I am trying to find the sum $S$ of the following series. $$S = \sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}t^{2n}\Gamma\left(\frac{1 + 2nH - ...
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0answers
23 views

Is my convergence proof correct?

It's "obvious" that the following sequence converges. I was asked to prove it on a homework assignment and was given no credit for my proof. I wanted to ask the community here if 1) they think my ...
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3answers
31 views

Show $\sum_n \left(1-\frac{K}{n^{1-\epsilon}\sqrt{\log n}} \right)^n$ converges for $\epsilon>0$.

This is not a homework problem. It has come up in my research. I am trying to show that $$\sum_n \left(1-\frac{K}{n^{1-\epsilon}\sqrt{\log n}} \right)^n$$ converges for $\epsilon>0$. I have no ...
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2answers
44 views

The convergence of an infinite seqeunce

Suppose that $$ a_n = \prod_{k=n}^{\infty}\left(1 - \frac{1}{k^2}\right), $$ for $n \geq 2$. How can we show that $$ \lim_{n \to \infty} a_n = \lim_{n \to \infty}\prod_{k=n}^{\infty}\left(1 - ...
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23 views

Convergence in probability (in measure) of a weighted sum of random functions

Consider a mesh of points $\pi_{n} = (t_{1n},\ldots,t_{K_{n}{n}})$ with $0 < t_{1n} < \ldots < t_{K_{n}n} < 1$ and weights $(w_{1n},\ldots,w_{K_{n}n})$ such that ...
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1answer
84 views

Find the limit of $\sum\limits_{n=0}^\infty \frac{n}{3^n}$ [duplicate]

Hi all What would the best way/method be to approach this, any advice would be appreciated
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2answers
28 views

Find relative radius of convergence for this seies

Given two series $\sum _{n=1}^{ \infty} a_nz^n$ and $\sum _{n=1}^{ \infty} b_nz^n$ who both have radius of convergence $R$, show that the radius of convergence for $\sum _{n=1}^{ \infty} c_nz^n$ is at ...
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1answer
50 views

What does $ \sum_{i = 1}^{\infty} \frac{1}{i(i-1)!}$ converge to?

What does $ \sum_{i = 1}^{\infty} \frac{1}{i(i-1)!}$ converge to? That is $1 + \frac{1}{2} + \frac{1}{3*2!} + ... + \frac{1}{n(n-1)!}$
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2answers
31 views

Independent integrable random variables with 0 expectation so that $\overline{S}_n$ does not converge to 0 in probability

Give an example of independent integrable random variables $X_n$ such that $E[X_n] = 0$ for all n, but $\overline{S}_n = (\sum_{i=0}^n X_i)/n$ does not converge to 0 in probability. As far as I ...
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2answers
38 views

Cauchy convergence in probability implies the existence of a (finite a.e.) limit X

Cauchy convergence of a sequence $X_n$ of random variables in probability implies the existence of an X (finite a.e.), such that $X_n$ converges to X in probability. The problem's hint suggests ...
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1answer
33 views

Sequences in Hausdorff spaces

As far as I see is the sequence concept in Hausdorff spaces well-defined in the sense that we have unique limits. Thus, I was wondering whether the definition of the closure of a set that contains all ...
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1answer
22 views

Abels theorem/Integral Test to show divergence

Hi is it possible to use Abel's Theorem for series to show the following: If $\{x_{n}\}_{1}^{\infty}$ is a positive sequence such that $\lim\limits_{n \rightarrow \infty}x_{n} = 0$ and ...
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1answer
36 views

Assuming uniqueness in Peano Theorem for ODEs, show that the sequence of approximate solutions is uniformly convergent.

If we know existence of solutions for $$ x'=f(t,x), \quad x(t_0)=x_0, $$ obtained by Peano's Theorem, and furthermore we know that the solution in unique ( we have not assumed satisfaction of ...
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0answers
30 views

Determine the limit distribution

I have this question here that I could use some help with. Let $X_1$, $X_2$, . . . be a sequence of random variables such that $P(X_n=\frac{k}{n})=\frac{1}{n}$, for $k=1,2,...,n$ Determine the limit ...
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1answer
23 views

convergence of $a_n = e^{nzi \pi /4}$

If we have series $(a_n)_{n=1}^{\infty}$ where $a_n=e^{nzi \frac{\pi}{4}}$. Where does this series convergence/divergences? If I do the ratio test I get \begin{align} r&=\lim_{n\to \infty} ...
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26 views

Almost sure downward convergence with some conditions implies convergence in $L^1$

If $X_n\downarrow X$ a.s., each $X_n$ is integrable and $inf_n E[X_n] > -\infty$, then $X_n \rightarrow X$ in $L^1$. As far as I know, "$X_n\downarrow X$ a.s." means that for every n, $X_n ...
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3answers
138 views

Absolute convergence tests

Hi I am interested in the following series: $$\sum_{n=1}^{\infty}(-1)^{n}(\sqrt{n+3}-\sqrt{n})$$ I have been able to show that this series converges by proving the Leibnitz test. Does anyone know ...
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3answers
35 views

Integral convergence and limit question

I have a question that has come up in a group homework project that neither I nor my partner have any idea how to solve. I'm hoping someone can give me a hint or some guidance as to how to go about ...
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3answers
52 views

Leibniz test for convergence of non alternating series

I am aware that one can use the comparison test and the integral test to show that the series $$\sum_{n=1}^{\infty}\frac{1}{n(n+3)}$$ converges. Is it possible to use the Leibniz test to show that the ...
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3answers
54 views

Convergence of a series with alternating denominator - Real Analysis

Decide if the series converges absolutely, conditionally, or not at all. \begin{equation} \sum_{n=1}^{\infty}\frac{(-1)^n}{(2+(-1)^n)n} \end{equation} I'm having a lot of trouble with this one. I ...
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1answer
23 views

Correlation between convergence radius of complex series

What do we know about the convergence between complex power series that look almost the same? For instance, if we have series $\sum_{n=1}^{\infty}a_n z^n,$ $\sum_{n=1}^{\infty}a_n z^{n+1},$ and ...
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2answers
44 views

Does $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ imply $\phi(X_n) \xrightarrow{\mathbb P} \phi(c)$ in this case?

Let $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ and $\phi: \mathbb R \to \mathbb R$ be bounded, continuous in $c$, and $\phi(c)=0$. Show that $\mathbb E\left[\phi(X_n)\right]\to0.$ I was going ...
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1answer
45 views

Confusion regarding almost sure events. If given infinite time, will a discrete-time gaussian process cover the entire real line?

This question really pertains to any discrete time continuous-valued, stationary stochastic process on the real line, but the Gaussian process will be adequate for this question. I have this ...
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13 views

Convergence of a monotonically increasing sequence

Given $a_n$ monotonically increasing and $a_n>0$. Which of the following converges? (1) $\frac{1}{a_n^2}$ (2) $e^{-a_n}$. I could not see any reason why both (1) and (2) will not converge. If ...
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2answers
43 views

Does $X_n \xrightarrow{\text{in distr.}} X$ and $|X_n|\leq Y$ imply $|X|\leq Y$?

We know that $$X_n \xrightarrow{\mathbb P} X \text{ and } |X_n|\leq Y \implies |X|\leq Y \text{ a.s.}$$ I was wondering if the same holds in case of convergence in distribution. So far, I've shown ...
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0answers
31 views

Discretization of the standard uniform dist.

I need some help. Sorry for the poor use of LaTEX... a) $U_{n} = \lfloor{nU}\rfloor/n$ prove that $\lim_{n \to +\infty} {U_{n}} = U$ where $U \sim unif[0,1]$ and thus that $lim_{n \to +\infty} ...
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1answer
23 views

Prove that the square root and exponent of a function in a $\limsup$ equals the the square root and exponent of a $\limsup$ of the function?

By what property do the following equalities hold? \begin{align*} \frac{1}{\limsup\limits_{n \to \infty} \sqrt[2n]{\left|a_n\right|}} = \left( \frac{1}{\limsup\limits_{n \to \infty} ...
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4answers
27 views

Why a sequence $\{x_k\}$ in $S$ such that $||x||>k$ cannot be a convergent subsequence?

This is an excerpt from my text: A set $S \subset \mathbb{R}^n$ is said to be compact if for all sequences of points $\{x_k\}$ such that $x_k\in S$ for each $k$, there exists a subsequence ...
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1answer
52 views

Does the series $\sum_{n=1}^{\infty} \sin\left((-1)^{n}\over n\right) $ converges conditionally? [closed]

Which convergence tests can I use in order to show that the series $$\sum_{n=1}^{\infty} \sin\left((-1)^{n}\over n\right) $$ converges conditionally.
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3answers
50 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
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3answers
68 views

$\sum (\frac{1-2n}{6+2n})^n $ converges?

Verify if $$\sum_{n=0}^{\infty} \left(\frac{1-2n}{6+2n}\right)^n $$ converges The root test is inconclusive and the limit of the general term is 0. I think I should use the comparison test, in this ...
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1answer
35 views

Sequence of functions that converges a.e. but not in the $L^1$ norm [closed]

How can I construct a sequence of functions that converges a.e. but it does not in the $L^1$ norm?
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5answers
126 views

Convergence of $\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$

Does the series $$\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$$ converges? My attempt: Since the ratio test is inconclusive, my idea is to use the Stirling Approximation for n! $$\frac{(2n)!}{n!n!4^n} ...
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2answers
374 views

Convergence of a series 1/(2n+1)

I'm looking for a way to get an estimate on a sum of the following series: $$\sum_{i=1}^{n} \frac{1}{2i-1}$$ My exact question would be the solution for $n=500$ but I'd be interested in the generic ...
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1answer
34 views

The characteristic function induced by L^1 convergence function

Assume $f_n\in L^1(\Omega)$, $f\in L^1(\Omega)$, and $f_n\to f$ in $L^1(\Omega)$ where $\Omega\subset R^N$ is open. Define $$ E_t^n:=\{x\in\Omega, f_n(x)>t\}.$$ Hence we have \begin{equation} ...
0
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1answer
15 views

How to use the triangle inequality to get $d(x_k,y)\ge d(x,y)-d(x_k,x)$ (Prove limit of a sequence is unique)

Here, $d$ is the Euclidean distance: And the triangle inequality in terms of $d$ is: I have no idea how to come up with the $d(x_k,y)\ge d(x,y)-d(x_k,x)$ inequality. What I've tried: ...
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2answers
25 views

Convergence/divergence test for infinite integrals

What would be a suitable test for convergence or divergence of the series: $$\sum_{k=1}^{\infty} \frac{k}{k^3+1}?$$