Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [convergence]

Convergence of sequences and different modes of convergence.

1
vote
1answer
12 views

Pointwise convergence implies convergence in the norm

Let $A$ and $B$ be normed spaces with norms $||\cdot||_A$ and $||\cdot||_B$ respectively, and let $\mathcal L(A;B)$ be the normed space of linear transformations from $A$ to $B$, with the norm $$||T||...
0
votes
1answer
25 views

How would the Dirichlet Test for convergence prove that $\sum_{n=1}^{\infty}\frac{\cos{n}}{n}$ does in fact converge?

I've been looking for a way to determine whether $\sum_{n=1}^{\infty}\frac{\cos{n}}{n}$ converges, and the test that I've most often seen recommended seen is the Dirichlet test for convergence. ...
0
votes
1answer
17 views

Integral test with $f$ negative and increasing?

The Integral Test states Assume $f$ is continuous, positive, and decreasing on [$1, \infty$). If $\int_1 ^{\infty}f(x)\,dx$ exists and is finite, then $\sum f(n)$ converges and vice versa. ...
0
votes
1answer
13 views

Shouldn't we check for conditionally convergent in ratio test done to see the intervals of convergence in power series?

(By A(n) I mean the power series)I understood that we use absolute value of A(n+1)/A(n) in ratio test because A(n) isn't neccessarily a positive value. We know when there is a limit of absolute value ...
0
votes
1answer
34 views

Show that limit of $n(1-(\frac{2n+1}{2n+2})^p)$ as $n\to\infty$ is $\frac{p}{2}$

For proving the convergence of the following series $$\frac{1}{2}^p+\frac{1\times3}{2\times4}^p+\frac{1\times3\times5}{2\times4\times6}^p+.... \text{ for } p>2$$ I try to use Rabe test to ...
-1
votes
1answer
23 views

interval of convergence of $\sum_{n=0}^{\infty} (nx)^{n}/n! $

Im not sure how to go about finding the interval of convergence of $\sum_{n=0}^{\infty} (nx)^{n}/n! $ I think i remember learning that you can either use the ratio test or cauchy root test to solve ...
0
votes
1answer
23 views

$f$ is holomorphic function on an annulus , show that $f(z)=0$ for all z in the annulus if some conditions are met

I guess the next problem should be addressed somehow with Laurent series: Let $0<r<s<\infty$, $A=A_{r,s}(0)$ and $f\in\mathcal{O}(A)$. Suppose that $\underset{n\rightarrow\infty}{lim}f(z_n)=...
1
vote
0answers
20 views

L2 metric entropy with bracketing

I am going through the proof of Theorem 3.1 in chen and white 1999. The authors use the upper bound of the $L_2$ metric entropy with bracketing $\mathcal{H}_{[]}(w, \Theta_n) = 2^kr_nB_n(1+d)\log(2^...
0
votes
1answer
21 views

example for theorem: pointwise but not uniform convergence

I'm reading through Rudin and was trying to understand this theorem- what is a good example to show that this theorem would not hold for pointwise but not uniform convergence? So, fn that converges ...
8
votes
0answers
106 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 ...
0
votes
1answer
50 views

Convergence of $\sum_{n=2}^{\infty}\left(\frac{\log n}{\log(\log n)}\right)^{-\log n/\log(\log n)}$

Initially, I need to prove, that $$\forall \lambda > 0 \ \forall\, \xi_i \sim \text{Pois}(\lambda), \xi_i\text{ are independent } \implies P\left(\limsup\limits_{n \rightarrow \infty}\frac{\...
1
vote
2answers
40 views

How to find $A_n$ such that $\sum_{n=1}^\infty A_n\sqrt{2} n \sin (nx)=1$

I meet a trouble to find $A_n$ such that the following equality holds. $$\sum_{n=1}^\infty A_n\sqrt{2} n \sin (nx)=1, \ \ \ \ 0<x<\pi$$ I am not sure if I can really find such $A_n$ since the ...
5
votes
2answers
135 views

How to remove this numerical artifact?

I am trying to solve a differential equation: $$\frac{d f}{d\theta} = \frac{1}{c}(\text{max}(\sin\theta, 0) - f^4)~,$$ subject to periodic boundary condition, whic would imply $f(0)=f(2\pi)$ and $f'(0)...
0
votes
0answers
27 views

Convergence in probability of running maximum

Suppose we have a sequence of integrable random variables $(X_n)$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that $n^{-1}X_n\to 0$ in probability as $n\to\infty$. Suppose further ...
1
vote
0answers
47 views

Does this explicit formula for the prime-counting function $\pi(x)$ converge?

This question is related to an answer I posted earlier at the following link. Explicit Formula for $\pi(x)$ A potential explicit formula for the fundamental prime-counting function $\pi(x)$ is ...
0
votes
3answers
44 views

Does $\sum_{i=2}^\infty \frac{1}{(\ln(n))^2}$ converge or diverge?

I've tried, the limit comparison test with several values and have tried finding some values for the direct comparison test but nothing really concrete has come out of it. $$\sum_{i=2}^\infty \frac{...
1
vote
0answers
17 views

Slightly alternative proof to the converse part of Cauchy's General Principle

I want to prove that: If $\forall \epsilon >0$, $\exists k \in \mathbb{N}$, such that $| u_{n+p}-u_n| <\epsilon $, whenever $n\geq k$, $p\in \mathbb{N}$, then $\{u_n\}$ is convergent. Proof: [...
2
votes
1answer
20 views

Strategies for proving continuity and differentiability of trigonometric series

Let $f$ be a function defined by a series $$f\left(x\right)=\sum c_n e^{inx}.$$ Sometimes, I can prove that the series converges pointwise (when it does), using the Dirichlet test. When the ...
2
votes
0answers
29 views

A criterion of convergence almost surely.

Suppose random variables $\{X_k\}_{k\in \Bbb{N}}$ are $i.i.d.$ and set $S_n=X_1+...+X_n$, show that if $S_n/n\rightarrow 0$ in probability and $$S_{2^n}/2^n\rightarrow 0 \ \ a.s.$$ then $S_n/n\...
0
votes
1answer
21 views

Convergence of series $a(n)$, where $a(n+1) = p + qa(n)$, $n \geq 1$

Suppose for $n \geq 1$, we have for some constants $p,q$ $a(n+1) = p + qa(n)$. Conditions on $p$ and $q$ for which the series converges? So, we have the series as $a(1)+a(2)+a(3)+\ldots $ $a(1) +...
0
votes
0answers
73 views

pointwise vs uniform convergence (Baby Rudin 7.4)

This is a basic question about the relationship between pointwise and uniform convergence. Suppose $f(x) = \Sigma_{n=1} ^ \infty \frac{1}{1+n^2x}.$ The question (Rudin 7.4) is what intervals does ...
0
votes
2answers
23 views

Finding Values for x for which $\sum_{n=1}^\infty \frac{x^n}{3^n}$ converges

My question is to find the values of x for which $\sum_{n=1}^\infty \frac{x^n}{3^n}$ converges and to also find the sum of the series for those values of x. I was going to use the ratio test, however ...
0
votes
3answers
57 views

Showing that $\sum_{n=2}^\infty \frac{2}{n^2-1}$ is convergent [duplicate]

I am trying to show that $$\sum_{n=2}^\infty \frac{2}{n^2-1}$$ is convergent by using telescoping sum. So far I have reduced $\frac{2}{n^2-1}$ via partial fractions to $\frac{1}{n-1} - \frac{1}{n+1}$....
1
vote
2answers
64 views

Using induction to prove a sequence $a_1=1$ and $a_{n+1}=3-\frac{1}{a_n}$ is increasing

I have a sequence defined by $a_1=1$ and $a_{n+1}=3-\frac{1}{a_n}$. I need to use induction to show that the sequence is increasing and $a_n<3$ for all $n$. Also to deduce that $a_n$ is ...
0
votes
1answer
28 views

Studying the convergence of integral alone (ratio: zero/zero)

I am looking for an examination of the convergence of the integral alone: $$\int_0^2 \frac {\sqrt{2-x}\,dx} {x^2-5x+6}$$ Any way to prove the convergence without calculation of the integral?
1
vote
2answers
65 views

$Lip_\alpha$ is not closed in $C[0,1]$

As the title says I am trying to show that $Lip_\alpha$ is not closed in $C[0,1]$. $Lip_\alpha$ is the class of functions on [0,1] that belong to $Lip_\alpha([0,1];K)$ where $f \in Lip_\alpha([0,1];K)$...
0
votes
2answers
42 views

Does this sequence converge? Alternating and exponential

$$\sum_{k=1}^{\infty}\left(-1\right)^k\frac{\left(k+1\right)^{k+1}}{k^{k+2}}$$ I started to use Dirichlet's test. However, the latter half does not decrease to 0. I am unsure of what to do.
2
votes
1answer
69 views

Showing $h_n$ does not uniformly converge

$$f_n(x)=x(1+1/n) \text{ if } x \in \mathbb{R}$$ $$g_n(x) = \begin{cases} (1/n)& x = 0, \text{ or } x \in \mathbb{I} \\ b+1/n& \text{if } x \in \mathbb{Q} \text{ with } ...
0
votes
3answers
40 views

Intuitively, $\sqrt{n}$ is not convergent. However, $|\sqrt{n+p}-\sqrt{n}|<\epsilon, \forall \epsilon>0, p\geq 1$

$|\sqrt{n+p}-\sqrt{n}|<\epsilon$ Clearly, $|\sqrt{n+p}-\sqrt{n}|=\frac{p}{\sqrt{n+p}+\sqrt{n}} \leq p/\sqrt{n} \rightarrow 0$. But by definition of a cauchy sequence, if we can choose $\exists N: ...
0
votes
2answers
33 views

Every third term alternates and I need to find if it converges.

$$\frac{1}{\sqrt 1} + \frac{2}{\sqrt 2} - \frac{3}{\sqrt 3} + \frac{1}{\sqrt 4} + \frac{2}{\sqrt 5} - \frac{3}{\sqrt 6} + \frac{1}{\sqrt 7} + \frac{2}{\sqrt 8} - \frac{3}{\sqrt 9} + \dots $$ By what ...
3
votes
2answers
34 views

Does Fourier imply Laplace?

Can we find a function $f(t)$ for which $$\int_{-\infty}^{+\infty}f(t)e^{-j\omega t}dt,$$ converges but $$\int_{-\infty}^{+\infty}f(t)e^{-st}dt,$$ does not ? Here, $j^2=-1$, $\omega$ is a real number ...
0
votes
0answers
22 views

Convergence of Random Variable (convergence in probability)

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
-1
votes
0answers
21 views

Convergence of random variable ! (Probability) [on hold]

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
0
votes
1answer
19 views

Convergence in measure is the equivalent with convergence in metric

We work in the framwork of a measurable sapce $X$ with a complete and finite measure $ \mu $. We say that a sequence of functions $\left( f_{n}\right) $, defined on a measurable space, converge in ...
0
votes
0answers
9 views

Region of convergence of transfer function from differential equation

I learned in my signal processing class that an LTI system can be defined using a linear constant coefficient differential equation. Whenever we have 'initial rest' condition, the LTI system is causal....
0
votes
0answers
23 views

Prove $O(\frac{1}{T})$ convergence rate

Suppose we have the following first-order non-homogeneous recurrence relation $$z_{t+1} \leq \frac{1}{(1+b_1c_t)^2}\big(\left(1+b_2c_t^2 \right)z_t + b_3c_t^2\big) $$ where $t$ is an integer which ...
0
votes
0answers
33 views

Prove $φ_n =2^{-1+2n}(-1+2^n)\to f$ pointwise

(Particular case of the general theorem and proof) Let $f:X\to[0,\infty]$ be measurable function defined on the interval $(k2^{-n},(k+1)2^{-n}]$ and $φ_n:X\to [0,\infty)$ monotone increasing ...
-1
votes
1answer
33 views

Prove that the sequence $( \frac{1}{n} , \frac{1}{n}) \subseteq \mathbb{R}^2$ converges to $(0, 0)$ with respect to the norm $\|\cdot\|_2$. [on hold]

Prove that the sequence $( \frac{1}{n} , \frac{1}{n}) \subseteq \mathbb{R}^2$ converges to $(0, 0)$ with respect to the norm $\|\cdot\|_2$.
0
votes
2answers
69 views

Proving $\displaystyle{\lim _{n\to\infty}}\sqrt[n]{n} = 1$ given that $\inf \{\sqrt[n]{n}|n\in \mathbb{N}\} = 1$ [duplicate]

Here's my attempt at trying to prove it by definition: $$\exists \mathcal{E} > 0 | \exists N\in \mathbb{N}|\forall n > N:$$ $$|\sqrt[n]{n}-1|<\mathcal{E}$$ But from this point on, I'm not ...
0
votes
0answers
38 views

Determine that if $a_n \longrightarrow L$ then $\sqrt[n]{a_1\cdot…a_n}\longrightarrow L$ [duplicate]

I managed to prove that when $a_n \longrightarrow L$ then $\frac{n}{\frac{1}{a_1}+...+\frac{1}{a_n}} \longrightarrow L$, and $\frac{a_1+...+a_n}{n}\longrightarrow L$, and all these things scream the ...
0
votes
0answers
11 views

Convergence of KL-divergence along a convergent sequence of measures

My question is about Lemma 12 and 13 (page 6) of of this paper https://arxiv.org/abs/1802.09583. The Lemma 13 in particular proves, ``Let $\log(g)$ be bounded. If $P_n \rightarrow P$, then $KL((P_n)_g ...
-3
votes
0answers
28 views

Series - convergence - logarithm [closed]

I need help for: $$\sum_{n=2}^{\infty}{\frac{ 1}{(n\ln(n)\sqrt{\ln(\ln(n))}}}$$ Does it converges?
2
votes
2answers
36 views

Example of a sequence needed that satisfies certain conditions

Can you please provide examples of sequence $\{a_1,a_2,\cdots\}$ such that $\sum_{i=1}^{\infty}a_i\to \infty$ while $\sum_{i=1}^{\infty}(a_i)^2<\infty$ with each $a_i\in[0,1)$. Thank you.
0
votes
1answer
58 views

How can we prove that this integral converges?

If we have integral in the form $\int_0^\infty \frac{1}{(r+a \exp (-xt) )]\sqrt{(1+(a \exp (-xt))^2)}} \ dt $ and if we take difference of such two integrals with the same $r>0$ and different (or ...
1
vote
4answers
127 views

Calculating $\displaystyle {\lim _{n\to\infty}}\frac{1+2+3+…+n-1}{n^2}$

My first attempt was using limit arithmetic, but it fails because one of the operands is infinite, so that didn't work. I then tried using the squeeze theorem: $$b_n = \frac{1+2+3+...+n-1}{n^2}$$ $$...
0
votes
0answers
10 views

Convergence of Frechet Derivative Functional related to Palais-Smale Sequence

Let $\phi : H_{0}^{1}(\Omega)\to\mathbb{R}$ be an energy functional for bounded domain $\Omega$ such that $\phi'$ be its Frechet derivative, and there exists a bounded sequence $\{u_{n}\}_{n\in\mathbb{...
0
votes
1answer
75 views

Does the Integral $\int_{-\infty}^{\infty} e^{-iwx}\cos(kx) \ dx$ converge?

I am trying to determine whether or not the integral $$\int_{-\infty}^{\infty} e^{-iwx}\cos(kx) \ dx$$ converges, to determine if $\cos(kx)$ has a Fourier transform. Intuition tells me it does not ...
1
vote
1answer
63 views

Any convergent sequence is bounded. Don’t we need to use the absolute value in this proof?

We have the following elementary result on real sequences. Any convergent sequence is bounded. This is basically the proof given in my notes: Suppose that $a_n \to a \in \mathbb{R}$. Now choose $\...
-1
votes
1answer
22 views

question about convergence and divergence of $\sum (n-2)^3\,e^{-n(x+2)}$

I have the problem that I could not see my fails in the my calculations and why does exists just one right way of showing the convergence or divergence of that formula. the first version is the ...
5
votes
3answers
82 views

Does a sequence $a_n$ converge if $|a_n-\frac{1}{n} \sum_{i=1}^n a_i| \to 0$, as $n \to \infty$?

Let $\{a_n\}$ be a real sequence. If $\lim\limits_{n\to \infty} \left|a_n - \frac{1}{n}\sum\limits_{i=1}^n a_i \right|= 0$, do we have $\lim\limits_{n\to \infty} a_n$ convergent? This is somehow an ...