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Questions tagged [convergence]

Convergence of sequences and different modes of convergence.

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20 views

Show that $X_n \to 0$ in probability under given condition.

Let $k > 0$. Suppose that $$\forall \epsilon > 0: \exists N: \forall n \geq N: P(|X_n| \geq \epsilon) \leq \epsilon k$$ Show that $X_n \xrightarrow{P}{ 0}$. Attempt: We have to show: $$\...
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0answers
23 views

Analysis of convergence (pointwise, local uniform, uniform)

How can we check if some series for example $$ \sum_{n=1}^{\infty}\frac{nx^2}{n^3+x^3}\qquad x>0 $$ is convergent pointwise / locally uniformed / uniformed? What are the methods / tools to check ...
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3answers
35 views

Find radius of convergence of $\sum_{n=0}^{\infty} (2n+1)z^n$

Let $$\sum_{n=0}^{\infty} (2n+1)z^n$$ be a power series with $z \in \mathbb{C}$ Find the radius of convergence and show that the series is divergent for all $z \in \mathbb{C}$ on the boundary So by ...
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2answers
27 views

Trying to understand convergence, example problem with getting an ε

So I recently started doing convergence in my Algebra class and I am having some trouble understanding conceptually how to find the $\epsilon$-value in the definition $\left|a_n − a\right| \leq \...
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0answers
11 views

Finding the nth Padé aproximant

I am trying to find the $P_N^N(\epsilon)$ and $P_{N+1}^N(\epsilon)$Padé approximant of this function $$ x(\epsilon) = \int_0^{infinity} \frac{e^{-t}}{1+\epsilon t}dt \sim \sum_{n = 0}^{infinity} n! (-...
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1answer
57 views

if $\sum_{n=1}^{\infty} a_n$ converges absolutely then prove that $\sum_{n=1}^{\infty} a_n^2$ converges [duplicate]

I am self studying real analysis.I have come up with following proof and I know that other proofs exist. But, Can someone just tell me if there is anything wrong with the following proof.Thanks in ...
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2answers
34 views

Let $\bar X_n$ be the sample mean. What is the accurate rate of $\bar X_n-\mu$ convergence to $0$,

Suppose $X\sim N(\mu,\sigma^2)$ and $X_1,\cdots,X_n$ are samples from $X$. Let $\bar X_n=\frac1n\sum_{i=1}^nX_i$. Then it is well known that $$\bar X_n\overset{p}\to\mu\qquad\qquad(1)$$ and $$\sqrt{n}(...
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1answer
18 views

Continuous mapping theorem, multivariate case, joint distribution.

I came across the following problem. Convergence in the following always means weak convergence, i.e. $X_n \rightarrow X$ if and only if $Ef(X_n) \rightarrow Ef(X)$ for all $f$ bounded, continuous ...
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10 views

Uniform convergence of polynomial approximation on Schwartz space

I have a question regarding uniform convergence of basis expansion in Schwartz space. For $L^2(\mathbb{R},\lambda)$, $\lambda$ Lebesgue measure, the partial sums of basis expansion (Hermite functions) ...
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1answer
49 views

Proof that $f$ is differentiable

Let $$f(x) = \sum_{n=1}^{\infty} \frac{x^n}{2^n} \cos{nx}$$ Proof that $f$ is differentiable on $(-2,2)$ my approach let $ m := \frac{x}{2} $ so $m<1$ $$ \left| \frac{x^n}{2^n} \cos{nx} \right| ...
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Problem Concerning Cauchy Principle for Sequences.

I have a question but can't seem to figure out how to solve it. The problem states: Let's consider a sequence $x_n$, such that $x_n\to a$, as $n \to \infty$. Using the Cauchy Principle prove that (a)...
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2answers
58 views

Show power series converges for every $x$.

Let $$f(x) = 1 + a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+...$$ be a solution of the differential equation $f'(x)=xf(x).$ Now I need to explain that the power series that define $f(x)$ converges ...
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1answer
52 views

Convergence/Divergence speed of $u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ given $f, g$ continuous and non-negative

Let be $f, g : [0, 1] \to \mathbb{R}_{+}^{*}$ continuous maps such that: $\forall n \in \mathbb{N}, u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ I want to show that $v = \left(\dfrac{u_{n + 1}}{u_n}\...
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1answer
50 views

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ [duplicate]

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ is convergent if and only if (1). $0<a<e$ (2). $0<a\leq e$ (3). $0<a<\frac{1}{e}$ (4). $0<a\leq \frac{1}{e}$ I tried ...
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5answers
34 views

Sequence divergence test

Could I argue that $\left(1+\frac{1}{n}\right)^{n^2}$ = $e ^n$ therefore the sequence diverges? I am wondering if it is a legal move
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27 views

If something happens in 25 % of all cases in every generation, what will the frequency converge to in the long run?

I just saw this map http://consang.net/images/c/c4/Globalcolourlarge.jpg (available on Archive.org if it would ever disappear) and become curious about how inbred people actually are. 3 scenarios: ...
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2answers
38 views

Why does $\sum_{n=1}^\infty \frac{1}{n(\log(n))^{1+2\epsilon}}$ converge?

I am looking through examples on convergences of random series, and in one of the proofs the following result is used: If $\epsilon > 0$ then $$\sum_{n=1}^\infty \frac{1}{n(\log(n))^{1+2\epsilon}}&...
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1answer
29 views

Banach space exercise

Suppose $X$ and $Y$ are Banach spaces and $T_{n} \in B(X,Y)$. If $T_{n}x_{n} \to 0$ in $Y$ for any choice of unit vectors $\{x_{n}\}$ in $X$, show that $\|T_{n}\| \to 0$.
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1answer
26 views

pointwise convergence on interval and in $ L^p $

I need to prove that sequence of functions $ (f_n)_{n=1}^{\infty} $is pointwise convergent on $[0,1]$ but it is not convergent in the space $ L_2[0,1] $ If I showed that $ f_n \to 0 $ on $ [0,1] $ is ...
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1answer
24 views

Show: if $\sum_{n>0} f(n)$ is convergent, then $\sum_{n>0} n^{1/n}f(n)$ is convergent [on hold]

If $\sum_{n>0} f(n)$ is convergent, then show that $\sum_{n>0} n^{1/n}f(n)$ is convergent . I am trying using Abel's test , but I can't find my way .
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3answers
78 views

Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n*(n!)^2*4^n}{(2n)!} $ converge?

Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n*(n!)^2*4^n}{(2n)!} $ converge? I have no idea how to do this. I have tried to use any trick I am aware of but can't figure this out. Can anyone help ...
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3answers
91 views

Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
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1answer
32 views

Convergence of a sequence… [duplicate]

Let $\{a_n\}$ be a sequence of real numbers. Define $\sigma_n = 1/n(a_1 + \dots + a_n)$. Suppose that $\lim a_n = a \in \mathbb{R}$. Show that $\lim \sigma_n = a$. Here is my work so far... Fix $\...
3
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1answer
111 views

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. Show $\lim_{n \rightarrow \infty} x_{n}$ exists. [duplicate]

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. show $\lim_{n \rightarrow \infty} x_{n}$ exists. To do this the problem has been broken down into three pieces: a) Show that $x_{n} <...
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$\{f_n\}$ sequence of holomorphic functions converges uniformly.

Let $U$ be an open subset of $\mathbb{C}$ and let $\{f_n\}$ be a sequence of holomorphic functions $f_n:U\to \mathbb{C}$. Suppose this sequence is Cauchy with respect to the $L^2$ norm. Then $\{f_n\}$...
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1answer
55 views

Theorem 30.1 (b) in Munkres' TOPOLOGY, 2nd ed: The sequential criterion for continuity

Here is Theorem 30.2 in the book Topology by James R. Munkres, 2nd edition: Let $X$ be a topological space. (a) Let $A$ be a subset of $X$. If there is a sequence of points of $A$ converging ...
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1answer
43 views

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence?

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence? My attempt via induction: If I prove that the denominator grows faster than the numerator, I can conclude ...
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0answers
16 views

Convergence rates for SQP solver on strongly convex problem

Given the following optimization problem: $ \boldsymbol{x}^* = \arg \min_{ \boldsymbol{x}} g(\boldsymbol{x})\\ s.t. : \boldsymbol{b} = A \boldsymbol{x} \\ $ Where $\boldsymbol{A}$ is full row-...
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Probability limit of log chi sqaured

I have the following: $$\Pr\left(T \ln(1+\chi_{1})>\ln(T)\right) \rightarrow 0$$ where $\chi_1$ is a chi-squared distribution with 1 degree of freedom and $T$ is an arbitrary real number and the ...
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A question about the proof of Theorem 5.21 in Van der Vaar(1998)

If we know that $\hat\theta_n\overset{p}\to\theta_0$, how does the following equation \begin{equation} \sqrt{n}V_{\theta_0}\cdot(\theta_0-\hat\theta_n)+\sqrt{n}o_p(|\hat\theta_n-\theta_0|)=G_n\psi_{\...
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0answers
33 views

If $\sum a_n$ is convergent but not absolutely, then $\sum a_n^+$ diverges

Let $a_n \in \mathbb{R}$, such that $\sum_{n=1}^\infty|a_n|= \infty$ and $\sum_{n=1}^m a_n \to a$, as $m \to \infty$. Let $a_n^+=\max\{a_n,0\}.$ Show that $\sum_{n=1}^\infty a_n^+= \infty$. Approach: ...
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3answers
46 views

Convergence of $\sum_{n=2}^\infty \frac{(\ln n)^3}{n^2}$

I was trying to find if the series $\sum_{n=2}^\infty \frac{(\ln n)^3}{n^2}$ converges or diverges. But I couldn't solve the question and I looked at the solution in here. In that page, Limit ...
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2answers
44 views

Does $\sum_{i=2}^n \frac{3}{n\ln(n)}$ converge or diverge? [on hold]

I came upon this question while working: $$\sum_{n=2}^\infty \frac{3}{n\ln(n)}$$ And I was wondering whether it converges or diverges? A help would be greatly appreciated ! Thank you!
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0answers
13 views

How to determine convergence rate

An iterative method has been used to solve a non-linear equation f$(x)=0$. The table below show the iterations $x_k$ at $k$. $$\begin{array}{c|c|} & \text{} & \text{} \\ \hline \text{k}...
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1answer
42 views

Determining the Values of $\alpha$ for Which the Series is Conditionally and Absolutely Convergent

The task is to determine for which values of $\alpha$ is the following series is conditionally convergent and absolutely convergent. My attempt is below. $$\sum_{n=1}^{\infty} {n^{-\alpha}\cdot(\ln{n}...
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1answer
22 views

series convergence imply nx approaches 0

Proof: For a decreasing sequence of positive reals, show that if the sum converges, then $nx_n \to 0$ but the converse is not true The first part I just assumed a positive limit the series converge ...
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What method will we use to find the optimal solution when we choose using the penalty function

I learn something about penalty method from these slides http://www.numerical.rl.ac.uk/people/nimg/course/lectures/parts/part5.2.pdf In the second slide,it show this quadratic penalty function $\Phi(...
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4answers
43 views

Does the improper integral $\int_0^{\infty}\frac{2^x+1}{3^x+1}dx$ converge?

Does the following integral converge: $$ \int_0^{\infty}\frac{2^x+1}{3^x+1}dx $$ Using linearity of integrals I did $$ \int_0^{\infty}\frac{2^x+1}{3^x+1}dx = \int_0^{\infty}\frac{2^x}{3^x+1}dx + \...
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0answers
79 views

If $\sum a_n$ converges, then $\sum a_n^{\frac{n}{n+1}}$ converges as well.

If $(a_n)$ be a sequence of positive real numbers such that $\sum a_n$ converges, then show that $\sum a_n^{\frac{n}{n+1}}$ converges as well. Attempt: We try to apply the Limit form of the ...
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2answers
37 views

Converges or diverges $\sum_{n=1}^\infty \frac{\ln n}{\sqrt n}$?

I was trying to find if the series $\sum_{n=1}^\infty \frac{\ln n}{\sqrt n}$converges or diverges. First, I tried ratio test and got the limit as 1. I tried Limit Comparison Test's and I only got 0's ...
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2answers
54 views

Is $\int_0^1 \frac{x^n}{\sqrt{1-x^4}}dx$ convergent?

$$\int_0^1 \frac{x^n}{\sqrt{1-x^4}}dx$$ Near $0$ the expression inside is convergent, that is easy. Near $1$ looks like it approaches infinity when $n \ge 0$ But according to the book when $n \ge -1$ ...
3
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2answers
62 views

$\{a_n\}$ be a sequence such that $ a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in…

Let $\{a_n\}$ be a sequence of positive real numbers such that $a_1 =1,\ \ a_{n+1}^2-2a_na_{n+1}-a_n=0, \ \ \forall n\geq 1$. Then the sum of the series $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in......
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0answers
22 views

Convergence acceleration of a series by using transformation

One of the ways of accelerating the convergence of a series is by transforming into a faster series. Examples of this approach can be found in this paper. A generalization of this method leads to the ...
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2answers
34 views

Determine the series is conditionally or absolutely convergent. [closed]

This is the problem: $$\sum_{n=1}^\infty (-\frac{1}{2})^n$$ How can we decide the series is a conditionally or an absolutely convergent?
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0answers
20 views

Hard demonstration - brilliant minds. I came up with the idea of ​obtaining the limit distribution of an estimator [closed]

I do not know exactly how to get there, I thought to use the estimate of maximum likelihood but I'm not sure... If $X_1,\ldots , X_n$ are i.i.d. according to the uniform distribution ${\cal U} (0, \...
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0answers
10 views

difficult demonstration, Convergence in distribution of subsamples? [closed]

I am trying to solve this convergence but I am not know exactly what is the n1, please give me some ideas: Let $X_1$,...$X_n$ i.i.d with cdf $F(\varepsilon_p)$=$p$ and $f(\varepsilon_p)>0 $ if $\...
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1answer
15 views

Complicated Demonstration - Violation of the theorem that converges in probability and not in distribution

I was thinking that if a sequence of random variables $Y_n$ with c.d.f. $H_n$ which converges to $c$ in probability, such that $H_n(c)$ does not converge to $H(c)=1$. How could I make an example ...
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0answers
27 views

According to the uniform distribution U(0,θ), how to obtain the suitably normalized limit distribution for ((n+1)/n) X(n) ) [closed]

I was thinking of doing it by means of the maximum likelihood estimator but how can I apply it to the case in which I get (n + 1) / n?
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0answers
21 views

A problem on the basics of convergence of a series

I was taught that if a series converge , for an example if {an} converges then lim ∑an = 0 But again I was taught that , an = (-1)^n is absolutely convergent and it converges to 1. I think it ...
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0answers
15 views

Convergence on Geometric distribution [closed]

Suppose that $Xn$ ∼ $Geo( {λ/n+λ} )$ with $n = 1,2,...$ where λ is a positive constant. Show that $Xn/n$ converges on when n → ∞, and determine the parameter of the limit distribution.