Convergence of sequences and different modes of convergence.

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1answer
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Rate of convergence of a solution to an equation with a free parameter

Suppose that $\epsilon>0$ is a solution to the following equation: $$\epsilon^2-a(m)\ln\epsilon-b(m)=0,$$ where $a(m)\to 0^+$ and $b(m)\to 0$ as $m\to\infty$. Suppose that a solution ...
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0answers
136 views

When does $x^{x^{x^{…}}}$ converge? [duplicate]

Provided that $x$ is positive, when does the following converge? $$x^{x^{x^{x^{...}}}}$$ So here is my work, but I don't really know if this is correct. Could anyone shed some light on this? I ...
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5answers
1k views

What happens if I repeatedly alternately normalize the rows and columns of a matrix?

Here is an algorithm: input matrix M (in-place) divide each row of M by its norm divide each column of M by its norm repeat What will M look like after this has been repeated many times? Can we ...
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3answers
51 views

Does $a_k$ diverges

I'm not sure, If I'm doing the convergence test correctly. From given, $\sum_{k=1}^{\infty} b_{k}$ diverges and $\sum_{k=1}^{\infty} \frac{a_{k}}{b_{k}}=2$ I have to check whether ...
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0answers
18 views

implication of continuity of function in proving convergence of sequence

I am reading a paper named "Asynchronous Broadcast-Based Convex Optimization Over a Network" by Nedic and I am confused about a part I attach as follows. I don't know the rational deducting from the ...
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1answer
16 views

Convergence of functionals in the dual space

Let $M\subseteq X$ be a subset of a normed space. I have been asked to show that the annihilator of $M$,$M^a$ is closed. To do this I assume that it isn't closed. I.e there exists some functional ...
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1answer
9 views

Need some help in understanding a root test with given answer

Trying to proof the convergent of the series in the box by root test, the answer is as shown. However two parts which I do not understand why does the equation became $(1+1/n)^n$ instead ...
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1answer
30 views

Is this proof of divergence of an alternating series correct?

Determine whether the series $\sum_{r=1}^\infty (-1)^{r-1}(\sqrt{r+1}-\sqrt{r})$ is convergent, absolutely convergent, or divergent. The way the textbook did it is that they let $b_r = ...
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1answer
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transformations divergent series

I need help, If $\forall n\in \mathbb{N}, x_n> 0 $ and $ \sum_{n=1}^{\infty} x_n$ is divergent then $ \sum_{n=1}^{\infty} \frac{x_n}{1+x_n}$ divergent? thanks for the help
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20 views

Almost sure limit of a sequence

I have a random variable $X$, which is uniformly distributed on $[0,1]$. The underlying probability space is as usual $(\Omega,\mathcal{F},P)$. For every $t$ with $0\leq t < 1$, I define a random ...
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0answers
14 views

arithmetic mean of a bounded sequence converges? [duplicate]

I have learnt that the means of a convergent sequence also converges. However, I am wondering whether can we say the means of a bounded sequence converge? or any other counterexample? Let ...
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2answers
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Convergence of series $\sum_{n = 1}^\infty \exp(-n^a)$, when $0 < a < 1$.

Consider the non-negative series $$\sum_{n = 1}^\infty e^{-n^a}, 0 < a < 1.$$ If $a = 0$, the series is divergent, and if $a \geq 1$, by root test, it is convergent. Root test doesn't give ...
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3answers
42 views

$1/z_n$ converges to 0 if and only if $z_n$ diverges. For complex numbers.

In real analysis there was an easy property that converted limits to infinity in limits at zero. More precisely, $1/z_n$ converges to 0 if and only if $z_n$ diverges (this is converges to infinity). ...
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1answer
26 views

Sequence of functions that fails certain conditions of Arzela-Ascoli theorem

For a closed, bounded interval $[a,b]$, let $\{ f_{n}\}$ be a sequence in $C[a,b]$. If $\{f_{n}\}$ is equicontinuous, does $\{f_{n}\}$ necessarily have a uniformly convergent subsequence? I would ...
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2answers
51 views

What does the series $\sum_{n=2}^\infty \frac{2}{n^3-n}$ converge to?

I know that the series converges. My questions is to what. I tried seeing if it was a telescoping series: $\sum_{n=2}^\infty \frac{2}{n^3-n} = 2\sum_{n=2}^\infty (\frac{1}{n^2-1}-\frac{1}{n})$ but it ...
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3answers
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How do I show the series $\sum{\frac{1}{\log(n)^{\log(n)}}}$ converges? [closed]

How do I show the series $\sum{\dfrac{1}{\log(n)^{\log(n)}}}$ converges?
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Uniqueness of solution to integral equation with “endogenous” kernel

Let $x,y\in C$ and consider the functional equation $T:y\mapsto x$ implicitly defined by the following integral equation: $$ x(t) = \int_{-\infty}^\infty K(x(t),t,s) y(s) \, ds, $$ with $K$ given. ...
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1answer
25 views

Radius of Convergence for a complex sin function

I encountered the following power series, and while I know a couple of ways to determine radius of convergence, I wasn't able to figure out how to evaluate the appropriate limit to get said radius. ...
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1answer
29 views

From known power series deduce the power series expansion of $ln(5-x)$ and infer the general term and radius of convergence

From known power series deduce the power series expansion of $ln(5-x)$ and infer the general term and radius of convergence. Now I said: ...
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1answer
33 views

Under what conditions will $f(x)^n$ converge pointwise and uniformly?

Let $f(x)$ be a continuous function on $[0,1]$. Under what conditions on $f$ will the sequence $f_n(x)$ = $(f(x))^n$ converge uniformly and pointwise?
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convergence, bounded sequences, and limits

Just want to get this straight in my head, so when doing proofs, proving that a sequence is convergent and proving if it is bounded is pretty much doing the same thing. They both use the same ...
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1answer
40 views

Determining convergence of $\frac{n}{e^n}$

I am trying to determine if $\sum_{i=1}^\infty \frac{n}{e^n}$ is converging This is what I have so far $\sum_{i=1}^\infty \frac{n}{e^n}$ converges by the geometric series test since $\frac{1}{e}$ ...
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2answers
63 views

Prove that $\lim_{n \rightarrow \infty}\sqrt[n]{n}=1$ [duplicate]

How to prove $$\lim_{n \rightarrow \infty}\sqrt[n]{n}=1.$$ I have problem in proving this statement at the beginning my textbook says: Suppose $f_{n}=\sqrt[n]{n}=1+h_{n}$ where does this ...
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Understanding convergence of fixed point iteration

I was reading some slides explaining the convergence of the fixed point iteration, but honestly I'm not seeing or having an intuitive idea of how fixed-point iteration methods converge. Assuming ...
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0answers
26 views

Proving particular case of Abel's series convergence criterion

I need the following to proof a particular case of the Abel's general convergence criterion which goes as follows: Let $\Omega$ be a non-empty subset of $\mathbb{C}$ and A a non-empty subset of ...
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0answers
26 views

When does $f_n(\beta_n)\rightarrow^p f(\beta)$ and

Let's suppose that $\beta_n\rightarrow^p \beta$ $f_n(x)\rightarrow f(x)$. When can I be sure that $f_n(\beta_n)\rightarrow^p f(\beta)$ ? I'm also looking for assumptions/theorems valid for a ...
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0answers
24 views

Optimising 2D acceleration to intercept a moving target

Related to this question: Accelerating one moving body to intercept another body in 2d I have a spaceship that needs to intercept a moving target in 2D. They have a relative velocity and I need to ...
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2answers
27 views

Possible limits from the Bolzano-Weierstrass Theorem

The Bolzano-Weierstrass Theorem states that Every bounded sequence has a convergent subsequence. For reference, I'll call the original bounded sequence $S_n$ and any subsequences $S'_k$. I have ...
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2answers
21 views

Convergence of functions in norm and pointwise

I understand that $L^2$ convergence does not imply pointwise convergence and vice versa. But I think that $L^2$ convergence must imply pointwise convergence a. e. $x$? So, since Schwartz functions are ...
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Is this limit almost sure, in distribution, or other?

Suppose I have a sequence of random variables $X_i$ and a pair of scalars $a,b$ with the property that $$\lim_{i\to\infty} \Pr(a\leq X_i \leq b)\to1$$. What is the name for this kind of limiting ...
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1answer
16 views

Asymptotic behavior

I very much dislike the "Big Oh" notation. It just doesn't stick in my mind. Suppose $f$ is a continuous function and $f \in \text{O}( 1/|x|^{1+\epsilon})$ when $|x| \rightarrow \infty$ and for $0< ...
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2answers
58 views

How can I show that this integral converges?

So here it is: $\int\limits_2^{+\infty}\left(\cos\frac{2}{x}-1\right)dx$. I've tried to use Cauchy, Dirichlet and Abel's tests, but can't seem to figure this out. Mathematica says in converges, but ...
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2answers
30 views

When is $\sum_{N=1}^{\infty}\exp\left(\ln\left(\frac{\sqrt{N}\ln(N)}{g(N)}\right)-\frac{(g(N))^2}{\ln(N)}\right)<\infty$?

Let \begin{align} \sum_{N=1}^{\infty}\exp\left(\ln \left(\frac{\sqrt{N}\ln(N)}{g(N)}\right)-\frac{(g(N))^2}{\ln(N)}\right) \end{align} Question: Let $g(N)=a(\ln(N))^{t}$ where $a \geq 0$ is some ...
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What's about $ \sum_{n=1}^{\infty} \frac{ \mu\left( \sigma (n)\right)}{n^3} ,$ where $\mu(n)$ is Möbius function and $\sigma(n)=\sum_{d\mid n}d$?

Let $ \mu (n)$ the Möbius function and $ \sigma (n)$ the sum of divisors function, then the arithmetical function $g(n)= \frac{ \mu\left( \sigma (n)\right)}{n^3} $ isn't multiplicative since ...
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2answers
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Proving recursive sequence converges to $\sqrt{2}$ [duplicate]

Consider the sequence $x_{n+1} = \frac{1}{2} (x_n + \frac{2}{x_n})$, $x_1 = 2$. Prove that it converges to $\sqrt{2}$. I want to show that all of $x_n$ is bounded below by $\sqrt{2}$ using induction. ...
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2answers
37 views

Does this sequence of functions converge uniformly to $0$?

Suppose that $f$ is continuous and $|f(x)|<1$ for all $x$ in $[a,b]$. Does $[f(t)]^n$ converge uniformly to $0$? Is it a no? Because if I have $x^n$ then at $x$ very close to $1$ it takes very ...
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2answers
35 views

A sequence of vectors $\{a_i\}$ in $\mathbb{R}^n$ is a Cauchy sequence . . . Prove that Cauchy sequences are convergent.

A sequence of vectors $\{a_i\}$ in $\mathbb{R}^n$ is a Cauchy sequence if for each $\epsilon> 0$ there exists $I \in \mathbb{N}$ such that $\|a_i -a_j\| < \epsilon$ for all $i, j \geq I$. ...
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1answer
31 views

Confusion when finding convergences using divergence and integral test?

I am having a bit of confusion doing the divergence and integral tests, specifically when I am trying to visualize the functions to get a better idea of why the methods work. For example, take the two ...
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0answers
15 views

Bounds on the size of Voronoi cells

I am working on an algorithm for which bounds on the size of voronoi cells will come in handy. Suppose that the domain $D$ is partitioned according to the Voronoi cells $D_1,\dots,D_n$ with Voronoi ...
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2answers
31 views

How to prove the Fourier coefficient of a convergent sequence converges as well

So the question is, $\,f_1, f_2,...,f:T\to C$ are integrable functions with $ \,f_n\to f$ in $\parallel \cdot \parallel_1 $ as $n\to \infty$. Let $k\in Z$. Prove that, the Fourier coefficient, $ ...
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2answers
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power series of interval of convergence of [-1,1]

Is there a power series whose interval of convergence is [-1, 1] and which is conditionally convergent both at $-1$ and $1$? .
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1answer
42 views

An example of a sequence of functions that is not pointwise convergent

I've been searching for an example of a sequence $f_n(x)$ of functions that is not pointwise convergent, i.e.: $$\lim_{n\rightarrow \infty}\left | f_n(x) - f(x) \right | = 0$$ but I cannot find one. ...
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1answer
30 views

Convergence of $\sum\frac{(n-a)^2}{(n-b)^3}$ with $a,b$ complex numbers.

Find the complex constant $a, b$ for which $\sum\frac{(n-a)^2}{(n-b)^3}$ converges and diverges.
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1answer
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why does this function converge pointwise

https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch9.pdf Example 9.4: $f_n(x)=\begin{cases}2n^2x, & \text{if 0 ≤ x ≤ 1/(2n)} \\ 2n^2(1/n-x), & \text{if $1/2n$ < x < 1/n }\\ 0, ...
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1answer
7 views

Convergence of a Branching process

Consider the Branching process: $\{ \xi_i^n , n \ge 1, i \ge 1\}$ are i.i.d. taking values $0, 1, \ldots$ and $Z_0 := 1, \; Z_{n+1} := \sum\limits_{i=1}^{Z_n} \xi_i^{n+1}$. Assume $\mu := ...
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1answer
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Convergence of a Sequence of functions Uniformly Mean Pointwise [duplicate]

For each of the following, give an example of a sequence of functions $f_n(x)$ that converges to f A. uniformly but not in the mean square sense. B. in the mean square sense but pointwise nowhere. ...
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0answers
17 views

every sequence has a convergent subsequence in a space? [duplicate]

Let $X$ is a compact and Hausdorff topological space, does every sequence in $X$ have a convergent subsequence?
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1answer
18 views

(Resolved) Does the sum of a subset of the Harmonic sequence converge iff its density approaches 0?

Update: This question has been resolved. I have made some mistakes in this post. I will leave my post here for readers to find out my mistakes. I have noticed that the post is a bit too long. So if ...
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0answers
10 views

Can you describe two graphs as convergent over an interval?

When describing two graphs that seem to follow the same path (literally, not parallel as there is an intersect) is it mathematically correct to say that they seem to converge over an interval? Or to ...
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1answer
48 views

Pointwise and uniform convergence of a series of functions

Define a sequence of functions on $[0,\infty)$ such that $\forall n\in\mathbb{N}$, $$ f_n(x)\triangleq \begin{cases} 1 & x\in[n,n+\frac{1}{n}]\\ 0 & \text{otherwise} \end{cases} $$ Does the ...