Convergence of sequences and different modes of convergence.

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2
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4answers
37 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
votes
2answers
38 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
votes
0answers
21 views

Long-time behavior of the solution of a parabolic partial differential equation

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $f\in C^1(\overline\Omega\times\mathbb R)$ $u_0\in C^1(\overline\Omega)$ with $u_0=0$ on $\partial\Omega$ $u\in C^0(\overline\Omega\times ...
1
vote
1answer
20 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 ...
1
vote
2answers
36 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
votes
1answer
22 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
votes
0answers
16 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
votes
1answer
33 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 $, ...
1
vote
0answers
24 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 ...
1
vote
6answers
152 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
votes
3answers
86 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.
1
vote
2answers
46 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
votes
4answers
68 views

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

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
120 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
vote
1answer
17 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
69 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
64 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 ...
-1
votes
2answers
28 views

convergence in mean when mean is the constant?

For a random sequenc $X_n$, if its expectation $E|X_n|=0$, does that mean it converges in mean to $0$? For convergence in mean to $0$ we need $E|X_n-0|\rightarrow0$
0
votes
0answers
46 views

A divergent sequence of integrals

Let {$s_{n}$} be a sequence of increasing step functions that converges pointwise to a limit function $f$ on an interval $I$ that is unbounded. $f(x)\ge 1$ almost everywhere on $I$ . ...
1
vote
1answer
66 views

Measure converges to zero

I'm trying solving the following problem: Let $f:[0,1]\to \Bbb{R}$ be a measurable question such that $f(x)>0$ a.e. Let $\{E_k\}_{k=1}^\infty\subset [0,1]$, a sequence of set such that ...
0
votes
1answer
20 views

What does the following iteration formula do?

The question is: What does the following interation formula do?: $x_{k+1}=2x_k-cx_{k}^2$. I already tried to identify this with newtons method. I.e. I tried to bring that into the form ...
0
votes
0answers
33 views

For which starting values the iteration convergences?

Given: $g(x)=\frac{1}{2}(x+\frac{a}{x})$ for $a\in \mathbb R_{>0}$ Question: For which starting values $x_0>0$ does the iteration $x_{k+1}=g(x_k)$ converges? My thoughts: Should I find an ...
3
votes
3answers
46 views

A sequence of random variables with bounded variance

If $\{X_n\}$ is a sequence of random variable with bounded variance: $$E|X_n^2|\le M<\infty,$$ and $X_n\to X$ in $L^1$, show that $$E|X^2|\le M.$$ I tried to use Fatou's lemma, ...
1
vote
1answer
28 views

Sequences of functions that converge uniformly, or pointwise, but not in $L^1.$

I'm reading the book Real Analysis of Folland. When I reached chapter 2 about the different modes of convergence, there's an example Folland gave that confused me: The 2 function sequences: ...
4
votes
2answers
118 views

Convergence of sequence: $f(n+1) = f(n) + \frac{f(n)^2}{n(n+1)}$

I am trying to work out a problem on some old analysis qual exam, and managed to reduce it to this question, but I can't seem to figure out this final step: Consider the sequence defined by the ...
0
votes
2answers
74 views

Does anyone understand this proof: If $A$ is closed and bounded $\implies A$ is sequentially compact.

This is how it goes, I will highlight the parts in yellow which I don;t understand why it is , or the idea behind it. $A$ is bounded so $(\forall x \in A)(\exists M > 0)(\|x\|<M)$ Let ...
1
vote
0answers
42 views

Prove the convergence of the integral

I have an article to prove the following lemma, but I don't quite understand. Can you explain more? Lemma. We have $N(t)=N_{\alpha_{n}}(t)=\#\lbrace n:\alpha_{n}\leq t\rbrace$. Let $\{\alpha_{n}\}$ ...
0
votes
1answer
28 views

Asymptotic Inequality in Probability

Given that $P(X>a)\leq f(a)$. Now, $f(a)$ tends to zero faster than $P(Y>a)$. Does it mean that $(1)P(X>a) \leq P(Y>a)$ or $(2)P(X>a) \geq P(Y>a)$ as $a \rightarrow \infty$.
0
votes
1answer
24 views

Prove that each $W_0^{1,2}$-function is weakly differentiable

Let $\Omega\subseteq\mathbb R^n$ be open. $u\in\mathcal L^1_\text{loc}(\Omega)$ is called weakly differentiable $:\Leftrightarrow$ $\exists v\in\mathcal L^1_\text{loc}(\Omega;\mathbb R^n)$ with ...
0
votes
1answer
43 views

Asymptoting to 0: is erfc(z) quicker than exp(-z)?

If I have a function of the form $\mathrm{erfc}\left(z\right)/e^{-z}$, should I expect its limit at large $z$ to be $0$ or $\infty$? My instinct is that it should be $0$, by considering the ...
-3
votes
1answer
21 views

Convgergence of series with different parametes [closed]

I have this series : $$\sum\limits_{n=1}^{\infty} n^{-a}\log(n)^{-b}$$ For what values of $a$ and $b$ does the series converge?
1
vote
2answers
60 views

Does this series converge or diverge and by which test?

$$\sum_{n=1}^\infty (-1)^{n+1} \sin(1/n^3)$$ I tried to apply the divergence test. I know $\lim_{n\to \infty}$ is 0 for $b_n$ but I don't think $b_n$ is decreasing. any ideas on how I can test this ...
0
votes
1answer
15 views

Separability of $l^{p}$ spaces

How can I prove that the space $l^{p}$ equipped with the norm (for $x=(x_{n}) \in l^{p})$: $||x||_{p}=(\displaystyle\sum_{n}|x_{n}|^{p})^{1/p}$ Is a separable space? (i.e. showing that there is a ...
1
vote
1answer
44 views

Why doesn't this work for Rudin Exercise 8 Chapter 3 series proof?

Okay so Here is the problem: If $\sum{a_n}$ converges and $b_n$ is bounded and monotonic, prove that $\sum{a_nb_n}$ converges. So I can follow the long epsilon based proof and I'm good with all ...
0
votes
2answers
73 views

Test the convergence of the series $\sum_{n=1}^{\infty}a_n$

Test the convergence of the series $\sum_{n=1}^{\infty}a_n$ , where $$a_n=\begin{cases}\dfrac{1}{n^2} & \text{ if $n$ is not a square integer},\\[6pt] \dfrac{1}{n^{2/3}} & \text{ if ...
0
votes
2answers
69 views

Show that if $a_n > 0$ for all $n\in\mathbb{N}$ and $\sum_{n=1}^{\infty}a_n$ converges then $\sum_{n=1}^{\infty}\sin{(a_n)}$ converges.

Show that if $a_n > 0$ for all $n\in\mathbb{N}$ and $\displaystyle \sum_{n=1}^{\infty}a_n$ converges then $\displaystyle \sum_{n=1}^{\infty}\sin{(a_n)}$ converges. I conjecture that the final term ...
-1
votes
2answers
53 views

almost sure convergence given density [closed]

my problem: Let $X_n$ be iid random variables with density $f(x)=\frac{1}{2}x^{-2}1_{\{|x|>1\}}$. Show that $\frac{X_n}{n}$ does NOT converge almost surely. Can anybody help me?
3
votes
3answers
133 views

If $ \sum_{n=1}^{\infty}x_na_n $ converges when $x_n\to 0,$ then $ \sum_{n=1}^{\infty}a_n $ also converges. [duplicate]

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence with non negative terms so that for every sequence $\{x_n\}_{n=1}^{\infty}$ with $x_n \geq 0$ and $\lim_nx_n=0$, the series $\sum_{n=1}^{\infty}x_na_n$ ...
4
votes
1answer
72 views

Baby Rudin exercise 3.3 solution, possible typo in solutions manual?

Okay so I'm working through the exercises in Rudin and after checking my solutions manual for 3.3, I found something that seems like it can't be true. Here is the original question in rudin: ...
7
votes
2answers
192 views

Show that the series converges, but not absolutely.

Show that the series converges, but not absolutely. $\sum_{n=1}^{\infty}( $exp$(\frac{(-1)^n}{n})-1)$. My Try: Let $a_n=$exp$(\frac{(-1)^n}{n})-1$. I was going to use alternating series test ...
6
votes
1answer
82 views

Is there continuous $f: [0, 1] \rightarrow [0, \infty)$ such that for all $x$ there is $y$ with $f(y) < f(x)$?

I think there isn't. Here's a sketch of a proof. I'm just not sure whether it really works because I'm not confident with the transfinite versions of the standard theorems about limits and convergent ...
-1
votes
2answers
51 views

Why the length of the zigzag curve approximating the circle does not approach the length of the circle?

I recently bumped into this question which asks why $\pi=4$ is wrong. And some answers(see the answer of user TCL, for example) stated that this has to do with functions and their derivatives. ...
3
votes
2answers
73 views

Investigate the convergence of $\sum a_n$, where $a_n =\int_{0}^{1} \frac{x^n \sin(\pi x)}{1-x} dx$.

Problem: Investigate the convergence of $\sum a_n$, where $a_n =\int_{0}^{1} \frac{x^n \sin(\pi x)}{1-x} \, \mathrm{d}x$. I'm thinking about changing $\frac{1}{1-x}$ to $\sum x^k$ and then ...
2
votes
3answers
336 views

Why does this sum converge?

I know that the following sum converges to 2 via WolframAlpha, but I am not sure why. $$\sum_{k=1}^\infty k \left[\frac{2}{k} - \frac{4}{k+1} + \frac{2}{k+2}\right] = 2$$ WolframAlpha gives the ...
1
vote
0answers
27 views

Convergence of a hypergeometric function

The hypergeometric function, ${}_{2}F_1(a,b,c;z)$ can be written in terms of a power series in $z$ as follows, $${}_{2}F_1(a,b,c;z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} ...
4
votes
2answers
115 views

Average $lcm(a,b)$, $ 1\le a \le b \le n$, and asymptotic behavior

What is the average value for $\mathrm{lcm}(a,b)$, with $ 1\le a \le b \le n$, for a given $n$, and what is the asymptotic behavior? The $\mathrm{lcm}$ is the least common multiple. I have ...
0
votes
0answers
21 views

Monte Carlo (or other) Approximation to Infinite Summations

This question is sort of paired with a similar post I wrote in Stack Overflow, but not a repeat because I am asking about using another mathematical method. StackOverflow I am trying to approximate a ...
1
vote
1answer
60 views

Dominated convergence theorem for complex-valued functions?

Suppose there is a sequence $\{f_n(x)\}$ such that $\lim_{n\rightarrow\infty}f_n(x)=f(x)$. I've previously used the dominated convergence theorem for interchanging the limit and the integral in ...