Convergence of sequences and different modes of convergence.

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Series convergence $x+\frac{2^2x^2}{2!}+\frac{3^2x^3}{3!}+\frac{4^4x^4}{4!}+\cdots$

Choose the right option. The series $x+\dfrac{2^2x^2}{2!}+\dfrac{3^2x^3}{3!}+\dfrac{4^4x^4}{4!}+\cdots$ is convergent if a. $0<x<1/e$ b. $x>1/e$ c. $2/e<x<3/e$ d. $3/e<x<4/e$ ...
3
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1answer
57 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 ...
11
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2answers
292 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
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0answers
36 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
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3answers
51 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 ...
7
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1answer
124 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 ...
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2answers
69 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 ...
1
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1answer
28 views

Uniform convergence and continuous.

Let $(f_{n})$ be a sequence of functions. Is it possible that $(f_{n})$ converges uniformly where each functions (that is $f_{1},f_{2}, f_{3}\dots$) aren't necessarily continuous?
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2answers
575 views

Convergence of Monotone sequences? example

An example of an unbounded increasing sequence that satisfies the assumptions of the convergence of monotone sequences...? According to the convergence of monotone sequences if a sequences is ...
0
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1answer
42 views

Convergence of expected values and integrability

I'm trying to prove a result for a homework assignment, and I got to a point that if the following result is true, then the result follows. Let $X_n$ be a sequence of positive random variables and ...
3
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1answer
36 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 ...
0
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1answer
42 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 $0<r<1$ ...
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0answers
26 views

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 ...
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1answer
57 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 ...
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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 ...
2
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1answer
685 views

Is $\sum_{n=1}^{\infty} \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$convergent or divergent

I have worked on this my answer is L= Div and B Consider the series $\displaystyle \sum_{n=1}^{\infty} a_n$ where $$a_n = \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$$ In this problem you must ...
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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 ...
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1answer
115 views

Prove series converges absolutely

Dot product is defined $\sum_{n=1}^\infty x_ny_n$. Let V consist of all sequences for which $\sum_{n=1}^\infty x_n^2$ converges. Prove that this series converges absolutely. I know I need to use the ...
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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 ...
0
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2answers
77 views

If $f_n\to f$ a.e. and are bounded in $L^p$ norm, then $\int f_n g\to \int fg$ for any $g\in L^q$

Suppose $p>1$ and $q$ is its conjugate exponent. Suppose $f_n \rightarrow f~a.e.$ and $\sup_n\|f_n\|_p < \infty $. prove that if $g \in L^q,$ then $\lim_{n \rightarrow \infty} \int f_ng=\int ...
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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 ...
2
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1answer
43 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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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 ...
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1answer
29 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 ...
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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 $f_n$ ...
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4answers
85 views

Convergence of $\sum\limits_{n=2}^\infty \ln\left(\dfrac{n^2}{n^2-1}\right)$

I made sure it passed the nth term test. Next I thought the easiest way, given that it's wrapped in ln, would be to use log rules to make it $\ln(n^2)-\ln(n^2-1)$ and then compare it to $\dfrac ...
36
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0answers
2k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
2
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4answers
44 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.
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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 ...
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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 $, ...
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1answer
24 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 ...
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. ...
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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} + ...
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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 ...
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 ...
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6answers
153 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 ...
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3answers
101 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.
4
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2answers
127 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 ...
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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 ...
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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 ...
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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: $$-x^{-3/2}\cos x -\int \frac 12{x^{-1/2}}\cos{x} dx $$ ...
6
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2answers
70 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 ...
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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 ...
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2answers
58 views

Question about the Riemann-Lebesgue Lemma proof

Ok, so one of the formulations of the Riemann-Lebesgue Lemma says: $$ f\in L^1(\mathbb{R}) \implies \hat{f}(\omega)\to 0\;\mbox{ when } \;|\omega|\to\infty.$$ I get all the steps of the proof, except ...
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1answer
45 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 ...
2
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1answer
304 views

Uniform convergence of $f_n = (n^a x^2)/(n^2 +x^3)$

My question is, if you have the sequence $$f_n = \frac{n^\alpha x^2}{n^2 +x^3}$$ on $[0, \infty)$, for values of a for $0<\alpha<2$ does the sequence uniformly converge? I guess another way to ...
0
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0answers
38 views

what does this mean $n\delta$ where $n\rightarrow \infty$ and $\delta \rightarrow 0$

$\lim_\limits{n \rightarrow \infty} \sup\limits_{\delta \rightarrow 0} \left(n \delta\right)^{-1} \left|T_n(\theta)_{ij}\right| < \infty$ a.s, $1 \leq i \leq p$, $1\leq j \leq p$ what does the ...
6
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0answers
166 views

Does $\displaystyle \lim_{m \to +\infty}f_{2,m}(x)$ converge?

This is related to a previous question where, as stated there, $f_{2}(n)$ gives the greatest power of $2$ that divides $n$. Specifically the sequence $\lbrace ...