Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

0
votes
1answer
34 views

arithmetic of converge series question [duplicate]

If $\sum _{n=1}^{\infty \:}a_{n\:}$ converges and $\sum _{n=1}^{\infty \:}b_{n\:}$ converges. how to proof that $\sum _{n=1}^{\infty \:}a_{n\:}-b_n$ also converges?
-1
votes
3answers
78 views

Convergence of $\sum_{i=1}^\infty \sin^2(\frac{1}{i})$ and $\sum_{i=1}^\infty\cos^2(\frac{1}{i}).$

I need to check convergence of these sums: $$\sum\limits_{i=1}^\infty \sin^2\left(\frac{1}{i}\right)\qquad\sum\limits_{i=1}^\infty\cos^2\left(\frac{1}{i}\right).$$ Does comparing these sums to ...
3
votes
1answer
75 views

What is the proper way to handle the limit with little-$o$?

I was hoping to show that $$\left(1-\frac{x}{n}+o\left(\frac{2x}{n}\right)\right)^n \xrightarrow{n\to\infty} e^{-x}$$ which would be just fine without the little-$o$. Trying binomial formula: ...
2
votes
0answers
51 views

How do I prove the special case of the central limit theorem?

Let $(X_n)$ be an i.i.d. sequence such that $\mathbb P(X_1=1)=\frac{1}{2}+\varepsilon$ and $\mathbb P(X_1=-1)=\frac{1}{2}-\varepsilon$ for some $\varepsilon\in(0, \frac{1}{2})$. I'd like to show that ...
3
votes
1answer
203 views

Convergence of marginal distribtution

Here I have a question which looks a little bit weird: $(q_n)_n$ is sequence of probability density functions of the couple $(x,y) \in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. ...
0
votes
2answers
60 views

Integral Test for Convergence - Log of a Log

I need to investigate the convergence of the series $\sum_{n=3}^{\infty}\dfrac{1}{n\ln n}$ So, doing the integral test, I end up with (shortcutted because integration is boring): ...
1
vote
1answer
40 views

Convergence of random variable

I've been facing the following problem: Let $(X_k, Y_k)_k$ be a sequence of $2$-dimensional, independent random variables, each with uniform distribution over $ B(0,k) $ Verify if the following ...
0
votes
1answer
37 views

Show that $\mu(f_n^+) \rightarrow \mu(f^+) $ and $\mu(f_n^-) \rightarrow \mu(f^-) $, using Fatou's Lemma.

I'm starting learning about Fatou's lemma. How would you apply it to solve the following problem: Let $g^+ = max (g,0)$ and $g^- = max (-g,0)$. Let $f_n$ be integrable on measure space with measure ...
0
votes
1answer
215 views

Proof for multivariate Newton-Raphson method

How can the proof for Newton's method for a single variable be extended to the multivariate version? Forgive me if this is trivial, but I don't seem to get it. Any links or proofs would be greatly ...
1
vote
1answer
63 views

How to analyze convergence or divergence of the integral $\int_1^{\infty}(t^2+\ln^2t)^{-1}dt$

Analyze convergence or divergence of the integral $\displaystyle\int_1^{\infty}(t^2+\ln^2t)^{-1}dt$ since $\displaystyle\int f(y)^{-1}dy=yf(y)^{-1}-F(f(y)^{-1})+C$ ...
0
votes
1answer
39 views

tightness of sequence of degenerate probabilities

If $\delta_x$ denotes for $x\in \mathscr{R} $, the degenerate distribution at $x$, prove that the sequence $\delta_{x_n}$ of probabilities on $(\mathscr{R,B})$ is tight iff $x_n$ is bounded. This is ...
1
vote
0answers
38 views

Can someone please check if I have solved this convergence question correctly?

I got this problem today. Can someone please check my proof and confirm if it is correct or point out the place where it is wrong? I think it is correct, but it is so hard to see what is going on, so ...
0
votes
1answer
29 views

Existence of expected value with complex power

Suppose that $X$ be a random variable taking values on $(0,+\infty)$ with density function $f(x)$ and we have $\mathbb{E}(X^2)<\infty$. Can we conclude that $$\mathbb{E}(X^{-2-it}) ...
1
vote
0answers
107 views

How to show a sequence of functions does not converge uniformly

Let $f$ be a continuous function on $[0, \infty)$ such that $0\leq f \leq Cx^{-1-\rho}$, where $C$ and $\rho$ are positive constants. Let $f_k(x)=kf(kx)$. $\textbf{Question}$: Show that $f_k$ does ...
1
vote
1answer
237 views

Simpson's composite rule rate of convergence.

Hello I have wriiten a program in Matlab that determines an Integral using Simpsons rule and it also determines the rate of convergence. I tried my program on the following examples: $f(x)=\sin{ x}$ ...
1
vote
1answer
60 views

Is this function familiar to anyone?

Consider $$f(z)=\sum_{w\in C}\frac{1}{z-w}$$ Where $C$ is the set of complex integers. What I would like to know is where can I find any information about this function (name perhaps). For instance, ...
19
votes
1answer
301 views

Does the series $\sum\frac{a_n}{a_{n+1}}\frac{1}{n}$ always diverge if $0<a_n<1$?

Does the series $$\sum_{n=1}^{\infty}\frac{a_n}{a_{n+1}}\frac{1}{n}$$ diverge for any $a_n$, satisfying $0<a_n<1$, $n=1, 2, 3\dots$ ?
1
vote
1answer
105 views

Rate of convergence of an exponential function

If I have a function $$f = \exp(\sqrt{n} \cdot \frac{\sqrt{\log{n}}}{\sqrt{n}-\sqrt{\log n}}),$$ I can notice, that $$\lim_{n \to \infty} f = \infty,$$ but also I can notice that it goes very slowly ...
1
vote
1answer
40 views

Proving convergence/divergence for $p$-series

I have an exam in Calc 2 coming up. As such, I am doing previous exams given by our current professor. However, the exams lack a solution set, so I will post the question, and the answer I wrote down ...
1
vote
1answer
50 views

Can someone look at my proof about the convergence of $e^{-tA}$

Hi I am trying to prove that if A is a symmetric positive definite matrix then $e^{-tA}\rightarrow 0$ as $t\rightarrow\infty$. So I have attempted an answer but I'm not sure it is correct. ...
1
vote
0answers
71 views

Does $ \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\left ( \sum_{m=1}^{n}\frac{1}{m} \right )\right) $ converge? [duplicate]

I am trying to determine whether the $$ \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\left ( \sum_{m=1}^{n}\frac{1}{m} \right ) \right) $$ converges or not. I have tried the popular tests, but all ...
1
vote
1answer
33 views

Convergence of the sequence of operators on a Banach space.

Let $T$ be a bounded linear operator on a Banach space $E$, and let $(T_n)$ be a sequence of bounded linear operators on $E$ such that $$\| (T - T_n)x \| \to 0 \ \ \text{ as } n \to \infty$$ for all ...
1
vote
0answers
42 views

Weak convergence of a sequence of cdfs

Suppose $F_n \to F $ weakly, $ x \in c(F)$ and $ x_n $ is a real sequence converging to x. Prove that $F_n(x_n) \to F(x) $. Here $F_n$, $F$ are cdfs and $c(F)$-set of continuity points of $F$. I ...
1
vote
1answer
89 views

Quick question on convergence

Shouldn't the radius of convergence be defined as: $$\frac{1}{R} = \lim_{n\rightarrow \infty} \left(a_n \right)^{\frac{1}{n}}$$ Not sure what they are doing above..
1
vote
1answer
190 views

Radius of Convergence and the Frobenius Method

Consider the equation $$4xy'' + 2y'+ y = 0$$ I know that $x=4$ is a regular singular point, and in the notation that my uni uses, we say that: $$(x-x_0)^2 y'' + (x-x_0)p(x)y' + q(x)y = 0$$ where ...
2
votes
0answers
48 views

problem on almost sure convergence

Let {$X_i$} be iid with finite second moment. Let $Y_n = \frac {2} {n(n+1)} \sum_{i=0}^n i*X_i $, n$\ge$1 Show that $Y_n \to E(X_1) $ I tried to define $Z_i = \frac {2} {(n+1)} i*X_i $ Then $Y_n = ...
1
vote
1answer
37 views

This is a ratio test question.

How do you use the ratio test to show whether converges or diverges? Using symbolab gave me something with "series root test", however it is not covered in my course. Would it be possible to ...
0
votes
0answers
23 views

real sequence and convergence in probability

$X_n$ is a sequence of random variables.$X_n \equiv a_n$, $a_n $ is a real sequence. Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n \to \lim_{n\to\infty} a_n$ in ...
1
vote
1answer
46 views

Alternating Series Proof

I should be able to figure this out, but it has me a bit confused conceptually. I'm really just not sure how to approach it in a rigorous fashion. Any help? If $a_0, a_1, a_2, . . .$ is a decreasing ...
0
votes
1answer
86 views

convergence in probability of function of random variables

Suppose that $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d random variables. If we have $E(|X_1|^k) <\infty$ for some $k>0$ and $f(x)$ is a bounded continuous function on $\mathbb{R}$. Is the ...
3
votes
3answers
80 views

Show that $f_n\to f$ uniformly

Let $(f_n)$ be a sequence of continuous functions on a compact set $K$ with pointwise and continuous limit $f$. Show that $(f_n)$ converges uniformly to $f$. My professor gave me a proof, ...
1
vote
0answers
31 views

Does from this follow that the sequence $(f_n)$ converges uniformly to $f$?

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions that are uniformly continious on a compact set $K$ and that converges pointwise to a function $f$ that is uniformly continious on $K$, ...
1
vote
0answers
56 views

Extension of a Pseudodifferential Operator

Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual ...
0
votes
3answers
95 views

Does $L^2$ strong convergence and bounded $L^\infty$ imply convergence in $L^\infty$ along subsequences?

Take a sequence of functions $f_n \in L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$ (convergence in $L^2$ norm) and $|f_n|_{L^\infty},|f|_{L^\infty} <R$ (uniformly bounded in ...
1
vote
1answer
382 views

Counterexamples for L1-convergence not implying L2-convergence and vice versa

I am trying to find counterexamples for the following statements. Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap ...
13
votes
1answer
523 views

Does the sequence $\{\sin^n(n)\}$ converge?

Does the sequence $\{\sin^n(n)\}$ converge? Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
0
votes
1answer
36 views

General approaches to test convergence of $\sum a_n z^{n^p}$ when z is a complex number of unit length 1

Suppose $|z| = 1 $ for some complex $z$, I know $\displaystyle \sum z^{n}$ diverges since by summing the geometric sequence, but what can we say about things like: $\displaystyle \sum z^{n^p}$ ...
1
vote
1answer
44 views

Rate of convergence of mean in a central limit theorem setting

I recently asked a question here that was the following: If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = P(Z_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that ...
1
vote
1answer
33 views

Is strong convergence?

Let $F_n(f) = f(\frac{1}{n}) - f(-\frac{1}{n}) \in \big ( C^{(1)} [-1, 1] \big ) ^*$. For every $f \in C^{(1)} [-1, 1]$ we have pointwise limit: $$ \lim_{n \to \infty} F_n (f) = \lim_{n \to \infty} ...
3
votes
1answer
168 views

question about uniform integrability

Am I correct with usage of this generalised Dominated Convergence lemma: a sequence $(f_n)$ in $L^1$ on a bounded domain is strongly convergent if and only if $(f_n)$ is uniformly integrable and ...
0
votes
1answer
76 views

What does it mean for a series to be convergent?

I have the definition: Let $(a_n)$ be a sequence of real numbers. Let $s_n=a_1+a_2+...+a_n$. We say the series $a_1+a_2+...$ is convergent if the sequence of partial sums $(s_n)$ is convergent. The ...
2
votes
1answer
96 views

limit of the integrations of a sequence of integrable functions

Let $(f_n)^\infty_{n=1}$ be a sequence of Lebesgue integrable functions on $[0,1]$ such that $f_n$ converges to $f$ almost everywhere in $[0,1]$. Suppose further (a). ...
1
vote
2answers
32 views

Convergence of $ \sum_{n=1}^\infty \frac{1}{n^q Log[n]^p}$ when $q$ is smaller than $1$

So for $$\sum_{n=1}^\infty \frac{1}{n^q \log(n)^p}$$ If $q$ is greater or equal to $1$, the case is okay, I can perform comparison test or integral test respectively. But what if $q$ is smaller ...
0
votes
1answer
24 views

Series and concavity

If $u(x)$ is strictly concave, can I say: $$ \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^{n+1}\cdot u(n) < \infty. $$ I am having trouble finding counterexamples. Thanks.
0
votes
0answers
75 views

Uniform convergence on compact sets

If I have a sequence of functions $(f_n)$ that converges to a function $f$ that is continuous (and therefore uniformly continuous on compact sets) can I then say that $(f_n)$ converges uniformly to ...
1
vote
2answers
87 views

Continuous approximation of upper semicontinuous indicator function

Let us define the upper semicontinuous indicator function as $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x<0. \end{cases}$$ I need to find some (at least one) ...
1
vote
1answer
106 views

Function relating Euler's constant and the golden ratio

Okay, I was messing around on Excel with some coefficients and I stumbled onto this. Not sure if it converges but it gets pretty damn close around the 1024th term mark. Was wondering if somebody could ...
2
votes
1answer
25 views

Denseness: $\overline{\mathcal{l}^2_0}=\mathcal{l}^2$

How to prove that the finite sequences are indeed dense within the space?
1
vote
3answers
71 views

Convergence test of a series? $\sum_{n=0}^\infty \frac{(-\pi)^n}{2^{2n+1}}$

What is the best way to test the convergence of the following series? My first guess is to use the Leibniz rule, but the exercise also asks to calculate it's sum, that makes me think that this is a ...
3
votes
2answers
97 views

What is $L^p$-convergence useful for?

Why do people care about $L^p$-convergence $f_n \rightarrow f$? Are there any interesting application of $L^p$-convergence? For example, if $p=\infty$, then the limit $f$ of the sequence $f_n$ of ...