Questions tagged [convergence]

Convergence of sequences and different modes of convergence.

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197
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11k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\...
35
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643 views

Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$

Is there a way to assess the convergence of the following series? $$\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$$ From numerical estimations it seems to be convergent but I don't know how to prove it.
24
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411 views

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
18
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397 views

Does the sum of reciprocals of all prime-prefix-free numbers converge?

Call a positive integer $n$ prime-prefix-free if for all $k \ge 1$, $\lfloor \frac{n}{2^k} \rfloor$ is not an odd prime. (Odd because otherwise the property is trivial, as every integer greater than ...
10
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1answer
360 views

Pi series that converges arbitrarily fast.

The old series for $\pi$ is this alternating series: $$\pi = 4 \sum_{i=0}^{\infty}\frac{(-1)^i}{2i+1}$$ Now, as already noticed, the series is alternating: adding one term overshoots $\pi$ every ...
10
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0answers
366 views

Fréchet L-Spaces

NOTE: The question has now been posted on MathOverflow: Fréchet L-Spaces According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous ...
9
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0answers
167 views

Triple sum $\sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} \sum\limits_{c=1}^{\infty} \frac{\cos a \cos b \cos c}{a^2 + b^2 + c^2}$

We have poor water heating system in our countryside house (currently it takes 4 hours to warm up the water), and my father has decided to improve it; he bought a water tank and placed it up in the ...
9
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0answers
216 views

Convergence of a linear recurrence equation

Let $T \colon \mathbb{C}^n \to \mathbb{C}^n$ be a linear operator. Let $\{u_k\} \subset \mathbb{C}^n$ and $\{v_k\} \subset \mathbb{C}^n$ be two sequences of vectors. Suppose the spectral radius of $T$ ...
8
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0answers
268 views

does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?

I've recently discovered that modifying the standard Newton-Raphson iteration by "squashing" $\frac{f (t)}{\dot{f} (t)}$ with the hyperbolic tangent function so that the iteration function is $$N_f (...
8
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1answer
279 views

Consider the problem minimize $f(x)= x^4 −1. $

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
7
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1answer
126 views

Topological Algebraic Independence of power series

Let $p$ be a prime number, let $x$ be a variable, and consider two power series over the ring $\mathbb{Z}_p$ of $p$-adic integers: $a(x):=\underset{n\geq 1}{\sum}{\frac{p^n}{n!}x^n}=px+\frac{p^2}{2}x^...
7
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224 views

For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

Unless I'm making a mistake, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely then the ...
7
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1answer
63 views

$\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e.

Let $\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e. I can show the only if part by using the theorem ...
7
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0answers
608 views

Uniform convergence of Empirical Moment Generating Function

In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of $n$ variables $X_1,X_2, \dots, X_n$ as: $$ \begin{...
7
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0answers
562 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
7
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0answers
306 views

How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$?

How do I evaluate this sum :$$\displaystyle\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$$ Note : I used many criterions of convergence to show if it converges but i didn't up. Thank you for any ...
7
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1answer
336 views

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is $...
7
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1answer
568 views

Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a densely-...
7
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0answers
154 views

maximum renewal rate of a Markov chain

Consider a Markov chain $(X_t)$ on a state-space with a countably generated $\sigma$-algebra and assume this Markov chain allows for small sets of order one. This means there exist sets $\mathfrak{S}$ ...
6
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0answers
75 views

Prime number intercept

Suppose I arrange my (infinite) list of prime numbers in the following way: \begin{array}{c|c}x_i&2&5&11&17&23&31&\cdots\\\hline y_i&3&7&13&19&29&37&...
6
votes
1answer
123 views

convergent or divergent $\int_{-4}^{1} \frac{dz}{(z + 3)^3}$

Question Determine whether convergent or divergent. $$ \int_{-4}^{1} \frac{dz}{(z + 3)^3} $$ Thinking I'm not sure how best to go about this, whether I'm justified in my result. Basically I'm ...
6
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0answers
405 views

Define $\rho(f,g):=\int \frac{|f-g|}{1+|f+g|}d\mu.$. Show that $f_{n}\rightarrow f$ in measure $\Longrightarrow$ $\rho(f_{n},f)\rightarrow 0$.

Let $(X,\mathcal{M},\mu)$ be a measurable space, suppose $\mu(X)<\infty$. If $f$ and $g$ measurable functions on $X$, define $$\rho(f,g):=\int \frac{|f-g|}{1+|f-g|}d\mu.$$ Let $(f_{n})_{n\in\...
6
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0answers
267 views

Application of the Dominated convergence theorem for series

The following exercise is taken from a Calculus I course exam: Let $k\in \mathbb N$. Prove the existence of $$x = \lim_{k\to \infty}\sum_{n=1}^{\infty}\exp\left(-n+\frac{k}{n}e^{-\frac{k}{n}}\right)...
6
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0answers
142 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
6
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0answers
176 views

Absolute convergence in Banach space

Let $X$ be a Banach space. Show that, if $\sum_{n\ge 1}x_{n}$ is absolutely convergent then $\sum_{n\ge 1}x_{n}$ is convergent and $$\left\|\sum_{n\ge 1}x_{n}\right\|\le \sum_{n\ge 1}\|x_{n}\|$$ I ...
6
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0answers
235 views

Intuition for almost sure convergence = fast enough convergence in probability

I know the meaning of convergence in probability and almost convergence. From zero-one law, we can derive that if a sequence of random variables converges in probability fast enough, then it converges ...
6
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0answers
388 views

Prove that the Weierstrass-type function is nowhere differentiable

Given $0<\alpha\leq1$. Show that the function $$f(x)=\sum_{j=1}^\infty 2^{-j\alpha}\sin(2^jx)$$ is nowhere differentiable. I have solved the case $x=0$. Taking $t_l=2^{-l-1}\pi$, then $f(t_l)-f(0)=...
6
votes
1answer
233 views

Book on Convergence Concepts in Probability without Measure Theory

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
6
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0answers
382 views

Infinite Double Exponential Sum, with Functional Equation $g(x) = g(\sqrt{x})$

What is a closed form for $$ \lim_{n\to-\infty}\sum_{i=n}^{\infty}\frac{x^{2^i}(x^{2^i}-1)}{(x^{2^{i+2}}+1)} $$ The series has the form: $$... \frac{x^{\frac{1}{4}}(x^{\frac{1}{4}}-1)}{x+1} + \frac{...
6
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0answers
209 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. $\frac{\...
6
votes
1answer
119 views

For every space $X$, $C_p(X)$ is a topological group.

I try to show that for every space $X$, $(C_p(X), +)$ where $$+:C_p(X)\times C_p(X)\to C_p(X):(f,g)\mapsto f+g$$ and for every $x\in X$, $(f+g)(x)=f(x)+g(x)$ is a topological group. The family $$\{O(...
6
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0answers
263 views

I need help about some compactness arguments

I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
6
votes
1answer
406 views

For infinite series convergence/divergence: Why doesnt meeting the conditions of the Divergence test imply the Cauchy Convergence Critierion

Assume that the limit of the sequence is zero, $\lim_{n\to\infty}a_n=0$. So its not plainly obvious if the series $\sum a_n$ converges or diverges. I have wondered for some time. If $\lim_{n\to\...
6
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0answers
176 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 0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,\cdots\...
5
votes
3answers
82 views

Suppose $\{x_n\}$ is bounded. Prove $\frac{x_n}{n^k} \rightarrow 0$, as $n \rightarrow \infty$, for all $k \in \mathbb{N}$.

I've came up with the following proof for the question in the title. I'm looking for suggestions on presentation and on methodology as I'm relatively new to analysis. In particular, my book says a ...
5
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0answers
74 views

How to evaluate $\sum\limits_{n=0}^\infty\frac{1}{x^n+y^n}$ for $x,y>1$?

Is there an analytical expression for $\sum\limits_{n=0}^\infty\frac{1}{x^n+y^n}$ when $x,y>1$? If so, how do you solve it?
5
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0answers
107 views

Can you tell my proof of $\lim_{n\to\infty} (1 + \frac{1}{n})^{n} = \sum\limits_{n=0}^\infty \frac{1}{n!}$ is correct?

I am currently studying analysis with Rudin's PMA myself without looking at proofs for theorems stated in the book. I'm now at the stage where I should prove $e = \lim_{n\to\infty} (1 + \frac{1}{n})^{...
5
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0answers
201 views

Convergence of series alternating at varying “rates”

Motivation: We all know the alternating harmonic series $$\sum (-1)^{n+1} \frac 1n = 1 - \frac 12 + \frac 13 - \frac 14 \cdots$$ is convergent. This is a basic consequence of the alternating ...
5
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0answers
142 views

Examples for Ermakoff's convergence test.

There are several convergence tests (Ratio, Root, Direct comparison, Integral, etc). When one test is inconclusive, another more powerful one (or a combination of tests) is used. For example, the ...
5
votes
1answer
68 views

$(a_n)$ is a sequence that converges to a>0. Prove that $\exists \delta \gt 0$ and $M \in \mathbb N$ such that $\forall n \ge M, a_n \ge \delta$.

$(a_n)$ is a sequence and $a_n \rightarrow a$, $a \gt 0$. Prove that $\exists \delta \gt 0$ and $M \in \mathbb N$ such that $\forall n \ge M, a_n \ge \delta$. Hello, everyone. This is my first post ...
5
votes
2answers
173 views

What's the implication of l'Hospital's rule on rate of convergence?

Consider $h(x)=f(x)/g(x)$, if l'Hospital's rule is applicable, then $$\lim h(x)=\lim\frac{f'(x)}{g'(x)}.$$ Does this fact implies $h(x)$ and $f'(x)/g'(x)$ converge at the same speed? E.g. if $f'(x)/g'(...
5
votes
1answer
246 views

Concept of “eventually almost surely” as an artefact of measure-theoretic axioms?

This is a serious question despite provocative title. Ever since I found out about Cox's theorem, I got quite enthusiastic about an alternative approach to formalising probability theory and started ...
5
votes
2answers
254 views

Sum $\sum\limits_{n,m=1}^\infty \frac{1}{(n+m)!},$

I am looking at: $$\sum_{n,m=1}^\infty \dfrac{1}{(n+m)!},$$ my task is to show that it is absolutely convergent and to find its sum. I have found the sum doing the following: $$\sum_{m,n=1}^\...
5
votes
0answers
103 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
5
votes
2answers
167 views

Show that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$.

Suppose that $\Sigma_{k=1}^\infty a_k$ converges. Prove that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$. Attemtp: ...
5
votes
0answers
125 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq L^{2}\...
5
votes
0answers
322 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ d\mu_{\frac{1}{\sqrt{n}}M_n}\stackrel{P}\rightarrow\int_{\mathbb{...
5
votes
0answers
82 views

Prove that $||\sum_{n=0}^{+\infty}{x_n}||\le\sum_{n=0}^{+\infty}||{x_n}||$ when series $\sum_{n=0}^{+\infty}{x_n}$ are absolutely converge?

I think it should be proved that: Since $$||\sum_{n=0}^{N}{x_n}||\le\sum_{n=0}^{N}||{x_n}||$$ so $$\lim_{N\to+\infty}||\sum_{n=0}^{N}{x_n}||\le\lim_{N\to+\infty}\sum_{n=0}^{N}||{x_n}||$$ so $$|...
5
votes
0answers
223 views

Can one use $e^n$ instead of $2^n$ in Cauchy condensation test?

Cauchy condensation test is useful for testing the convergence of infinite series. The test is stated here as follows: for a positive non-increasing sequence $f(n)$, the sum $\sum_{n=1}^\infty f(n)$ ...
5
votes
1answer
260 views

Necessary or sufficient conditions for rationality of a limit of a sequence of rational numbers

Consider a convergent sequence of rational numbers $a_n$ with a limit $\lim_{n\rightarrow\infty} a_n = b$. Does there exists some kind of necessary condition for $b$ to be rational that only uses the ...