Questions tagged [convergence-divergence]

Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.

3,244 questions with no upvoted or accepted answers
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242
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13k 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(\...
86
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2k 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.
29
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483 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. ...
20
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418 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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67 views

Recursive integrals a la $\int_{\int_y^xg(t)dt}^{\int_x^yg(t)dt}g(t)dt$

Inspired by this question I was wondering whether "recursive" integrals have been studied or if they appear anywhere in applications. What I mean is the following: Let $I(x, y) = \int_x^y g(t) dt$ and ...
10
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197 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 ...
10
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0answers
266 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 $T$ ...
10
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422 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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139 views

Are the unconditionally convergent series, with terms in a Banach algebra, closed under the Cauchy product?

We have a Banach algebra $\mathbb L$, and two sequences $(A_0,A_1,A_2,\cdots),\;(B_0,B_1,B_2,\cdots)\in\mathbb L^{\mathbb N}$, for which the sums $\sum_{n\in\mathbb N}A_n,\;\sum_{n\in\mathbb N}B_n$ ...
8
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261 views

Generalized limits

Cross-posted to Mathoverflow. $\DeclareMathOperator{\Lim}{Lim}$ $\DeclareMathOperator{\dom}{dom}$ $\DeclareMathOperator{\shift}{\sigma}$ $\DeclareMathOperator{\cesaro}{C}$ After reading Terry Tao's ...
8
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0answers
365 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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724 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 ...
8
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318 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 ...
8
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1answer
308 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
133 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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238 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
72 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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717 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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1answer
351 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
606 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
169 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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1answer
147 views

Is $\pi (x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})$ correct at all?

It should be the case that, in some appropriate sense $$\pi (x)\sim \operatorname{Ri}(x)-\sum_{\rho}\operatorname{Ri}(x^{\rho}) \tag*{(4)}$$ with $\operatorname{Ri}$ denoting the Riemann function ...
6
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0answers
49 views

On the convergence of the series $\sum \frac{(-1)^{\lfloor ne\rfloor}}{n}$

The question is pretty simple: It the series $\sum \frac{(-1)^{\lfloor ne\rfloor}}{n}$ convergent or not ? As usual in this situation we let $S_n:=\sum_{k=0}^n(-1)^{\lfloor ne\rfloor}$ and ...
6
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0answers
92 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
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1answer
127 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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564 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
393 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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143 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
226 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
271 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
408 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
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1answer
258 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
396 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
219 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
127 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
270 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
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0answers
182 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
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0answers
52 views

Almost surely convergence for independent random variables and bounded real sequence

Stumbled upon an exercise in my probability course which was given a couple of years ago as an assignment problem and was trying to solve it. However, I am struggling a bit with probability theory (...
5
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0answers
166 views

Proof that $(1+\frac{1}{n})^n$ can't converge to a rational number

One of my colleagues challenged me (and his students) with the following: "Assume you don't know that $\lim_{n\to +\infty}(1+\frac{1}{n})^n=e$. Prove the sequence $u_n=(1+\frac{1}{n})^n$ ...
5
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0answers
166 views

Relationship between $p$-adic numbers and analytic continuation of $1+x+x^2+x^3+…$

The infinite sums $1 + 2 + 4 + 8 + ...$ $1 + 3 + 9 + 27 + ...$ $1 + 5 + 25 + 125 + ...$ $1 + 7 + 49 + 343 + ...$ ... of powers of primes do not converge in the usual sense. However, by analytically ...
5
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0answers
76 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
115 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
205 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
votes
1answer
69 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
1answer
295 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
0answers
110 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
175 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
149 views

Approximation of $L^2$ function by smooth functions on a manifold

Let $M$ be a $C^2$ compact Riemannian manifold with boundary. Suppose $f \in L^2(M)$ is such that $0 \leq f \leq 1$. Is it possible to find $f_n \in C^\infty(M)$ such that $0 \leq f_n \leq 1$ for ...
5
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0answers
159 views

Methods of constructing rapidly convergent series

It's fairly easy to see that the series $$1-\tfrac{1}{3}+\tfrac{1}{5}-\cdots=\tfrac{1}{4}\pi$$ is : 1. Convergent to the value given, and - 2. Very slowly converging, which can be seen just by ...
5
votes
0answers
331 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{...

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