Convergence of sequences and different modes of convergence.

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3
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1answer
23 views

convergence in probability: speed of convergence

I am not sure if the title appropriately describes the question. I will appreaciate any ideas. Suppose $\{X_n:n\geq 1\}$ is a sequence of random variables defined on a common probability space. ...
0
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0answers
22 views

Convergence of orthogonal basis functions

I'm working on a problem where I've generated a set of basis functions using a Laplace series of spherical harmonics to describe the angular part of a 3D distribution and "custom made" basis functions ...
-1
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1answer
23 views

Another question on finiding special kind of power series [on hold]

Let $\sum a_nx^n$ be a real power series with finite positive radius of convergence $R$ ; then does there exist a non-constant real sequence $\{b_n\}$ such that $\sum b_nx^n$ is convergent for at ...
4
votes
1answer
30 views

On finding special kinds of power series

Let $\sum a_n x^n$ be a real power series with finite positive radius of convergence $R$, then is it true that for every real number $s>0$ , we can find a real sequence $\{b_n\}$ (depending on $s$, ...
0
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0answers
22 views

Why do uniform, strong and weak convergence coincide for finite dimensional vector spaces?

For linear operators $A_n$, $A$ in a finite dimensional vector space $V$, I am trying to prove the equivalence of $\|A_n - A\| = \sup_{x \in \mathbb{C}^n, |x| = 1} |A_nx-Ax| \to 0$ as $n \to ...
1
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1answer
34 views

What is the difference between the following $2$ sets?

What is the difference between the following two sets? $\{s\in\mathbb C:\Re(s)\ge1+\delta\},\quad\delta>0$ $\{s\in\mathbb C:\Re(s)>1\}$ I read that $\displaystyle\sum\limits_{n\in\mathbb ...
2
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3answers
58 views

Prove that series $ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $ and $ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $ have the same radius of convergence.

I want to prove that these two power series $$ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $$ and $$ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $$ have the same radius of convergence. What I've done so far is: ...
3
votes
1answer
50 views

Prove or disprove that a series is convergent

I was given the following task which I struggle with. Prove the following statement, or disprove it by giving a counter example: if $\sum_{n=1}^\infty a_n$ is convergent then $\sum_{n=1}^\infty ...
2
votes
2answers
80 views

Convergence of Sequence

If I know that the sequence $\{a_n\}$ converges to $a$, then to prove that the sequence $\{ca_n\}$ (for a constant $c$) converges to $ca$, I would basically want $|ca_n - ca| < \epsilon$... So, to ...
5
votes
3answers
225 views

vercongent sequences

Definition- We say a sequence $(x_n)$ verconges to $x$ if there exist an $\epsilon>0$ such that for all $N\in \Bbb{N}$, $n\ge N \implies |x_n-x|<\epsilon$. Loosely speaking, by convergent ...
1
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1answer
72 views

Convergence to $0$ of a certain series.

I was wondering whether or not the following holds - I didn't manage to get anywhere using standard tricks from elementary analysis. ...
2
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2answers
72 views

On the sequence of function $f_n(x)=n^2x(1-x^2)^n $

Is the sequence of functions $f_n(x)=n^2x(1-x^2)^n $ uniformly convergent over $[0,1]$ ? Is the series $\sum_{n=1}^{\infty} f_n(x)$ convergent with $0<x<1$ ? Is the series uniformly convergent ...
1
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1answer
47 views

Condition on p for convergence of $\sum{\frac{1}{n(\log(n))^p}}$

For what values of $p$ is the series $\sum{\frac{1}{n(\log(n))^p}}$ divergent and for what values it is convergent?
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3answers
76 views

Series convergence $x+\frac{2^2x^2}{2!}+\frac{3^3x^3}{3!}+\frac{4^4x^4}{4!}+\cdots$ [on hold]

Choose the right option. The series $x+\dfrac{2^2x^2}{2!}+\dfrac{3^3x^3}{3!}+\dfrac{4^4x^4}{4!}+\cdots$ is convergent if a. $0<x<1/e$ b. $x>1/e$ c. $2/e<x<3/e$ d. $3/e<x<4/e$ ...
1
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0answers
38 views

How to understand the Identity Theorem in complex analysis, from the point of view of power series expansions

The theorem states that if $f$ and $g$ are analytic functions and their values agree on an open set that is contained in a larger, connected domain, then $f$ must equal $g$ on the entire domain. (The ...
4
votes
3answers
53 views

How to evaluate $\lim\limits_{n \to\infty} \left|\frac{(n+1)^n x^{(n+1)}}{n^n}\right|$?

I'm trying to find the radius of convergence for the series $\sum_{n=0}^{\infty} \frac{n^n}{n!}x^n$ and have used Wolfram Alpha to find that it is $|x| < \frac{1}{e}$ and am trying to show that ...
11
votes
2answers
316 views

Find a closed form of the power series

Let a power series $$S(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{4n+1},$$ then $1$ is the radius of convergence of $S$ .In fact $S(x)$ convergens for each $x\in[-1,1).$ My work is to find a closed form of ...
1
vote
2answers
72 views

Convergence, Integrals, and Limits question

Let $f: [0,\infty)\to \Bbb R$ be a positive,decreasing monotonic function. Prove the following statement for every a>0 providing the integral on the right side converges. First I managed to ...
7
votes
1answer
127 views

Proving convergence for a series.

Let there be given that $\displaystyle\sum_{n=1}^{\infty}a_{n}$ is a convergent series and $a_{i}$ is not negative for every $i\in\mathbb{N}$. And i want to prove that this series is also convergent ...
3
votes
1answer
65 views

How to resolve the issue of two sequences converging to zero for $n, m \to \infty$?

My question is motivated by the following exercise in probability theory: Let $X_n \to X$ in probability and $X_n \geq Y$ a.s. Show that $X \geq Y$ a.s. I noticed that for all $n, m \in ...
0
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1answer
43 views

If $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$ then $\displaystyle\lim_{n\to\infty}|x_n|=0$. [on hold]

Let $(x_n)$ be a sequence of real numbers such that $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$. Show that $\displaystyle\lim_{n\to\infty}|x_n|=0$. Remark: Not use that exist $0<r<1$ ...
3
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1answer
39 views

If ${a_i} \to 0$ and $\{ {X_i}\} _{i = 1}^\infty $ is a sequence of iid random variables with zero mean, does ${a_i}{X_i} \to 0$ almost surely?

My problem is slightly more specific than the title of this question: Let $0 < \beta < 1$ and let $\{ {X_i}\} _{i = 1}^\infty $ be a sequence of i.i.d. random variables with $E({X_i}) = 0$. In ...
7
votes
1answer
51 views
+50

When does interchangibility of limit and Riemann integral imply uniform convergence?

Let $\{f_n\}$ be a sequence of real-valued functions defined on an interval $[a,b]$ such that each $f_n$ is Riemann integrable, $\{f_n\}$ converges point-wise to $f$, $f$ is Riemann integrable and ...
-1
votes
1answer
58 views

What is the limit of this product? (SOLVED)

What does this limit equal? $$\lim\limits_{k\to\infty}\left(\prod_{n=1}^kn^{2^{k-n}}\right)^{\frac{1}{2^{k-1}-1}}$$ All that I have tried so far is computation and it does seem to converge. I ...
0
votes
2answers
24 views

Convergence when the derivative is uniformly continuous

Let $f: \Bbb R \to \Bbb R$ be a derivable function. $f'$ is uniformly continuous in $\Bbb R$ Prove that $[n(f(x+1/n)-f(x))]$ converges uniformly to $f'(x)$ I'm having a hard time seeing why does ...
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1answer
33 views

Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$?

Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$? Its clear to see that the point-wise convergence is to $0$. By finding the derivative I obtained that the maximum of ...
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6answers
74 views

For which values of $x$ does this series converge?

For which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for ...
0
votes
1answer
37 views

If independent r.v. converge in probability to a constant, do they converge almost surely?

I've seen several examples when a sequence of r.v. converge in probability but not almost surely, yet none of them had the sequence to be independent. Would additional conditions of independence and ...
0
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1answer
24 views

A question about a proof in nonlinear programming book

I have a question about the proof of Proposition 1.2.1 (Stationarity of limit points for gradient methods) in the nonlinear programming book (2nd edition) by Bertsekas. At the beginning of the proof ...
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1answer
37 views

Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \le 0$. Proof: I will attempt to show that the ...
1
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1answer
33 views

Simple Question about Monotone Convergence Theorem

Suppose we have a sequence of (discrete) random variables $X_0, X_1, \dotsc$ over $E$ and $A \subseteq E$. Let $Y$ be some other random variable. Moreover, let $Z$ be a random variable with values in ...
0
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2answers
12 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
2
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4answers
46 views

Use comparison test to determine convergence

$$\int_{1}^{\infty}\frac{\ln x}{\sinh x}dx$$ I tried several functions and failed to get integrable convergent bigger function. Thanks for help.
4
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2answers
44 views

Why can the elements of $L^\infty$ be approximated in $L^p$ by $C^1$-functions?

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $f\in L^\infty(\Omega)$. From which theorem does the existence of $(f_k)_{k\in\mathbb N}\subseteq C^1(\overline\Omega)$ with ...
1
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1answer
25 views

Weak convergence in the Sobolev space and compact embeddedness

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sopolev space $u\in C^0\left(\overline\Omega\times [0,\infty)\right)\cap C^{2,1}\left(\Omega\times ...
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2answers
37 views

Show that the series converges ($l^2$)

I know that $\sum_{k=1}^\infty|y_n|^2=S<\infty$. I also have that $\lambda >1$. I need to show that $$ \sum_{k=1}^\infty \left| \frac{y_1}{\lambda^k} + ...
2
votes
1answer
23 views

If $\sum_{k=0}^{r-1} c_k =0 $, and $a_n \to 0$, does $\sum_{n=0}^{\infty} \sum_{k=0}^{r-1} c_ka_{nr+k} $ converge?

This is a generalization of the alternating series convergence result and this: Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series? Here is my question: If ...
0
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1answer
27 views

Slow convergence simulating log-normal sample from the normal

I am trying to simulate a log-normal random variable $Y$ with mean $m = \mathbb{E}[Y] = 0.001$ and standard deviation $s = 0.094$ by simulating a normal sample instead, and then exponentiating it. ...
2
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0answers
24 views

Central limit theorem in multidimension with unknown covariance

Let $X_1,\dots,X_n$ be samples from a distribution on $\mathbb{R}^d$ that has a finite second moment. If $d=1$, $\bar{X}_n=1/n\sum_{i=1}^nX_i$ and $S_n=1/(n−1)\sum_{i=1}^n(X_i−\bar{X}_n)^2$ then ...
0
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1answer
35 views

Show that if $\sum_{k=1}^m c_k =0 $, $\sum_{n=0}^{\infty} \sum_{k=1}^m \frac{c_k}{nm+k} $ converges.

This is a generalization of this: Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series? Here is my solution. To show that if $\sum_{k=1}^m c_k =0 $, ...
2
votes
1answer
44 views

Does convergence in probability preserve the weak inequality?

Suppose I have two sequences of random variables $\{x_n\}$ and $\{y_n\}$ such that $x_n\leq y_n$ and $\text{plim}\;x_n=L_x$ and $\text{plim}\;y_n=L_y$, can I say $L_x\leq L_y$ (almost surely)? Does ...
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6answers
154 views

Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series?

Is there someone who can show me how do I evaluate this sum :$$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$$ Note : In wolfram alpha show this result and in the same time by ratio test ...
3
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3answers
103 views

Determine if this series $ \sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}$ converges

Determine if the following series converges: $$ \sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}. $$ (http://i.stack.imgur.com/qWiuy.png) I don't know how to start.
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2answers
48 views

Determine if $ \sum_{n \geq 2} \frac{1}{\sqrt[n]{\ln n}}$ congerges

Determine if the following series converges: $$ \sum_{n \geq 2} \frac{1}{\sqrt[n]{\ln n}}.$$ I'm supposed to use here the limit comparison test, but I don't know how to choose the second series.
0
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4answers
69 views

Is it always possible to converge from an integer to another integer? [closed]

Let's say I'm given a fixed integer, I. I'd like to know if it is always possible to find a function, that starting from any random integer J will converge to or oscillate reasonably close (let's say ...
3
votes
2answers
122 views

Why doesn't this work for Rudin Exercise 3.8

The problem is 3.8 exercise in baby Rudin: If $ \sum{a_n} $ converges and $\{b_n\}$ is bounded and monotonic, prove that $\sum{a_nb_n}$ converges. Why can't I just do this?: Let $M$ be an ...
1
vote
3answers
77 views

Integral Convergence $\sin{x}/x^{3/2}$

Does the following integral converge: $$\int_0^\infty{\frac{\sin x}{x^{3/2}}}dx$$ I have tried to integrate this by parts and arrived at: $$-x^{-3/2}\cos x -\int \frac 12{x^{-1/2}}\cos{x} dx $$ ...
1
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1answer
18 views

Convergence of a sequence of Hölder continuous functions with respect to the Sobolev norm

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $W_0^{1,2}(\Omega)$ be the Sobolev space and $C^{2,\alpha}$ be the Hölder space for some $\alpha\in (0,1]$. Suppose $(u_k)_{k\in\mathbb ...
6
votes
2answers
72 views

How to prove $f_n$ converge uniformly?

For each $n \in \mathbb{N}$ consider the function $f_n : [0,+\infty) \to \mathbb{R}$ given by $$f_n(x) := \sin\left(\sqrt{4\pi^2n^2+x}\right), \ \ \ \ \forall x \ge 0.$$ Prove that ...
1
vote
3answers
65 views

Finding convergence of a series using integral test

The series:$$\sum_{n=1}^{\infty}\left(\frac{\ln(n)}{n}\right)^{2}$$ Question: a) show that it converges b) find the upper bound for the error in approximation $s\approx s_{n}$ Trial: The section ...