Stack Exchange Network

Stack Exchange network consists of 174 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.

0
votes
1answer
15 views

Superlinear convergence proof

We say that $p_n$ converges to $p$ $\underline{\it superlinearly}$ if $$ \lim_{n \to \infty} \frac{|p_{n+1}-p|}{|p_n - p|} = 0. $$ Show that if $p_n$ converges superlinearly that $\lim_{n \to \...
1
vote
1answer
36 views

Limit of some integrals.

Is this true: $$\frac{\int_{-\delta }^{\delta }(1-x^2)^ndx}{\int_{-1}^{1}(1-x^2)^ndx}\rightarrow 1$$ for a previously fixed $\delta\in(0,1)$?
0
votes
1answer
13 views

Convergence in mean of a series of random variables

I'm stuck on the following proof: Let $(X_1,.. X_n)$ random variables so that $E(Xi) = \mu$, $V(Xi) = \sigma^2$ final for all $1 \leq i \leq n$. It is also given that for all $i \neq j$, $Cov(Xi, Xj) ...
1
vote
1answer
12 views

Convergence of inverses

Suppose that $A,A_1,A_2,\ldots\in\mathbb R^{p\times p}$ are invertible square matrices such that $\|A-A_n\|=o(a_n)$ as $n\to\infty$, where $a_n\to0$ as $n\to\infty$. Is it true that $\|A^{-1}-A_n^{-...
0
votes
1answer
23 views

Is this sequence almost sure convergent?

Consider the sequence of independent random variables $\{X_n\}$ such that $$\begin{align} P(X_n = 1) &= 1/n \\P(X_n = 0) &= 1 - 1/n \end{align}$$ I saw this as an example of convergent in ...
1
vote
2answers
33 views

Proving that $y_n$ converges given $y_{2n}, y_{2n+1}$ converges

Suppose we have $\mathrm{lim}_{n \rightarrow \infty} y_{2n} = \mathrm{lim}_{n \rightarrow \infty} y_{2n+1} = M \in \mathbb R$. I'm trying to prove from this that $\mathrm{lim}_{n \rightarrow \infty} ...
2
votes
1answer
23 views

Apply the Monotone Convergence Theorem to show that λ is countably additive.

Let $ \lambda: \mathcal{M} \longrightarrow [0,+\infty]$ be defined by $$\lambda(E)= \int_{E} fd\mu \text{ for each } E \in \mathcal{M}. $$ Apply the Monotone Convergence Theorem to show that $...
1
vote
2answers
37 views

Find the closure of $C=\{f\in C([0,1]):f(0)=0\}$

Let $C([0,1])$ be the space of all real valued continuous functions $f:[0,1]\to \mathbb{R}$. Take the norm $$||f||=\left(\int_0^1 |f(x)|^2\right)^{1/2}$$ and the subspace $$C=\{f\in C([0,1]):f(...
16
votes
6answers
2k views

Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero?

If $a_n$ is a sequence such that $$a_1 \leq a_2 \leq a_3 \leq ...$$ and has the property that $\space$$a_{n+1}-a_n \longrightarrow 0$, Then can we conclude that $a_n$ is convergent? $$\space$$ I ...
2
votes
1answer
30 views

Convergence of Euler's definition of the Gamma function

I was reading the wikipedia article on the Gamma function and found out that the original definition of it was... quite clever actualy. Here's the article. Anyway the definition is $$ \Gamma(z) = \...
1
vote
0answers
16 views

Limit of an indicator martingale

Consider the Lebesgue measure on $[0,1]$. We define $I_k^n=[k2^{-n},(k+1)2^{-n})$, for $0\leq k\leq 2^n-1$, and $F_n=\sigma(\{I^n_k:k\})$. Finally $M_n=1_{I_0^n}2^n$. How do we show that $M_n$ is a ...
1
vote
5answers
57 views

$\sum_{\omega\in \mathbb{Z}(i)^*} |\omega|^{-2}$ does not converge.

I'm trying to prove that the series $$ \sum_{\omega\in \mathbb{Z}(i)^*} |\omega|^{-2} $$ The problem can be viewed as the sum above the fundamental region $\Omega^* = \{m\omega_1 + n\omega_2 : m,n\...
-4
votes
0answers
24 views

Discussing the convergence of a sequence $x_n$ [on hold]

If $k, x_1$ are positive, and $x_{n+1}$ = $\sqrt {k + x_n},$ discuss the convergence of the sequence $x_n$, according to whether $x_1$ is less than or greater than $\alpha$, the positive root of $x^2 =...
1
vote
1answer
27 views

How to show that the space of probability measures on $\mathbb{R}$ is separable under Lévy metric

The Lévy metric between distribution functions $F$ and $G$ is given by: $$\rho(F,G) = \inf\{\epsilon : F(x-\epsilon)-\epsilon\leq G(x)\leq F(x+\epsilon)+\epsilon\}.$$ Another way to write this is: ...
1
vote
3answers
68 views

Convergence of the series for $a \in \mathbb R$ $\sum_{n=1}^\infty\sin\left(\pi \sqrt{n^2+a^2} \right)$

Convergence of the series for $a \in \mathbb R$ $$\sum_{n=1}^\infty\sin\left(\pi \sqrt{n^2+a^2} \right)$$ I saw this problem in a calculus book and it gave a hint that says HINT First show that $$\...
2
votes
0answers
15 views

Finding an interval over which a polynomial interpolant $P_n$ converges to a function $f$ as the degree $n$ increases

I understand that due to Runge's phenomenon, increasing the degree, $n$, of a polynomial interpolant can actually increase the error between the interpolant, $P_n$, and the function, $f$, you are ...
0
votes
0answers
35 views

Discuss the convergence of a sequence [duplicate]

If $k, x_1$ are positive, and $x_{n+1}$ = $\sqrt {k + x_n},$ discuss the convergence of the sequence $x_n$, according to whether $x_1$ is less than or greater than $\alpha$, the positive root of $x^2 =...
1
vote
1answer
26 views

Convergence of the sum of iid scaled by $n^\alpha$

I am interested in the convergence of the sequence $\mathbb{P}(|X_1+...+X_n|/n^\alpha<z)$ where $z>0$, $\{X_n\}_n$ is an i.i.d. sequence with mean zero and finite variance. I can easily prove ...
1
vote
1answer
34 views

Help with convergence tests for series

I have a few questions to ask about series and convergence tests. I have been struggling to study everything fully and if someone can give me advices I will be really thankful.This is what I know so ...
2
votes
1answer
22 views

Contradiction in radii of convergence? Where is my error?

I'm working through Baby Rudin and I came across what seems to me to be a contradiction, but it could be an error on my part. It has to do with radii of convergence of power series. First, let $\{...
0
votes
0answers
11 views

Prove convergence of a sum of random variables

I am trying to grab on to some intuition about the area where random variables start looking a bit more like calculus. I've learned about random variables and the weak law of large numbers, but seem ...
0
votes
4answers
90 views

Is there any way to show that $\sum_{n=2}^{\infty}\frac{1}{n^2-1}$ converges without integral test?

In my real analysis class, we are trying to show that $$\left(\frac{2 \times 2}{1 \times 3}\right)\left(\frac{4 \times 4}{3 \times 5}\right) \left(\frac{6 \times 6}{5 \times 7}\right)...$$ converges. ...
7
votes
5answers
663 views

Stuck at proving whether the sequence is convergent or not

I have been trying to determine whether the following sequence is convergent or not. This is what I got: Exercise 1: Find the $\min,\max,\sup,\inf, \liminf,\limsup$ and determine whether the ...
0
votes
0answers
32 views

Studying convergence of an integral at the vary of alpha

I need to understand this exercise from my math book: study, at the vary of alpha belonging to R, the convergence of the following integral and then calculate the value for $a = 1$. $$ \int_0^{4} \...
4
votes
1answer
53 views

Convergence of $\sum\limits_{n=0}^{\infty} (1-|a_n|)$

So I must prove that if $(a_n)$ is a sequence of points in $\mathbb{C}$ with $0< |a_n| < 1 \; \forall n \in \mathbb{N}$ and verifying that $|b| \leq \prod\limits_{n=1}^{\infty} |a_n|$ with $0&...
3
votes
1answer
19 views

Proof Verification for Uniform Convergence on Sequence of Functions

just looking for a verification on a proof. Thanks in Advance Let $f_n$ be a sequence of functions such that $f_n=\frac{x^{2n}}{1+x^{2n}}$ defined on $[-2,2]$. Prove or Disprove Uniform Convergence ...
0
votes
1answer
37 views

Examples of use of integrals to show convergence of a limit

Recently I got the problem where I had to show the existence and finiteness of the limit of $a_n$ where $a_n$ is bounded and such that both limits \begin{align} \lim_{n\to\infty} \cos(a_nt) \ \ \ \...
0
votes
0answers
15 views

Coordinate-wise gradient descent converges to least-squares solution

Does somebody know a reference (or maybe short proof/argument) for the following claim: Coordinate-wise gradient descent converges to a least-squares solution. Coordinate-wise gradient descent: ...
1
vote
1answer
36 views

Convergence in C[0,1] under different norms

I was wondering whether there exists a sequence of functions $\{x_n(t)\}$ in the space of continuous functions in the closed interval $C[0,1]$ such that $\|x_n(t)\|_{\infty}=1 \; \forall n \in \mathbb{...
1
vote
0answers
40 views

Convergence (in distribution) of Markov process

For each natural number $N \geq 1$, let $X^N = (X^N_k)_{k = 1}^{\infty}$ be an homogeneous Markov process taking values in $R_+$, with transition kernel \begin{equation} p_N(A|X^{N}_{k-1}) = P[X^{N}_{...
0
votes
1answer
42 views

Modification of the law of large numbers for Binomial random variables. [duplicate]

Let $(p_n)_{n \geq 1}$ be a sequence of numbers in $[0,1]$ such that $p_n \to p$. Let $(X_n)_{n \geq 1}$ be a sequence of independent random variables where $X_n \sim Bin(n,p_n)$. Is it true that $X_n/...
1
vote
1answer
46 views

$ \sum _{n=1}^{\infty }\sin\left(\frac{\left(-1\right)^n}{n}\right) $ Does this sum converge or diverge?

$$ \sum _{n=1}^{\infty }\sin\left(\frac{\left(-1\right)^n}{n}\right) $$ Does this sum converge or diverge? I tried this: $$ \sum _{n=1}^{\infty }\sin\left(\frac{\left(-1\right)^n}{n}\right) = \...
1
vote
0answers
17 views

Question on Bochner spaces: Convergence of the weak time derivative

Let $U$ be a $C^2-$compact manifold and consider two non negative sequences $f_n,g_n$ such that $f_n \overset * \to f$ weakly in $L^{\infty}(0,T;\mathcal M_{*}(U))$ $g_n \overset * \to g$ ...
2
votes
1answer
53 views

Bounded Sequence in $L^\infty$ and Interpolation in $L^p$

a) Let $1\leq p_1\leq p\leq p_2\leq \infty$ and for $\alpha \in [0,1]$ $\frac {1}{p}=\frac {\alpha}{p_1}+\frac {1- \alpha}{p_2}$ Prove that if $f\in L^{p_1}\cap L^{p_2}$, then $f\in L^p$ and we have ...
1
vote
1answer
15 views

How to prove the convergence region for matrix Laurent expansion?

I know how to do it for scalar Laurent series. However, consider $$\mathbf{F}(z) = \sum_{k=0}^{\infty} C A^k B z^{-k}, $$ where $F, A,B,C$ are matrix with proper dimension. $z \in \mathbb{C}$. I ...
0
votes
0answers
18 views

Finding the limit of a sequence having epsilon [duplicate]

If $a_n$ → a, $b_n$ → b, then $c_n$ = ($a_1$$b_n$+$a_2$$b_{n−1}$+···+$a_n$$b_1$)\n → ab. Hint: Write $a_n$ = a + $\epsilon$$_n$, where $\epsilon$ → 0. Then: $c_n$ = ($a_1$$b_n$+$a_2$$b_{n−1}$+···+$...
0
votes
0answers
45 views

finding upper and lower bounds of finite sums

Find upper and lower bound for the following finite sum $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ My attempt: $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ = $\sum_{i=1}^n ...
-1
votes
0answers
12 views

Weakly star convergence on $L^{\infty}$.

i would like hints for the convergence in the yellow LINE, please.
0
votes
0answers
45 views

Proving that a sequence is convergent and finding its limit

If $x$ and $A$ are positive and $x_1 = \frac{1}{2}(x + A/x)$,$x_2 = \frac{1}{2}(x_1 + A/x_1)$, and so forth, prove that the sequence $x_n$ is convergent and determine its limit. Use your result to ...
2
votes
2answers
76 views

Uniform convergence of $\sum\limits_{n=1}^{+\infty} \left(\sin{{1} \over {n}}\right) x^n$

$f_n(x)=\left(\sin{{1} \over {n}}\right) x^n$ pointwise convergence: $|f_n(x)|=\left(\sin{{1} \over {n}}\right) |x|^n \sim {{|x|^n}\over{n}}$ for $n \rightarrow +\infty$ $\sum\limits_{n=1}^{+\infty}{...
0
votes
1answer
15 views

Nested interestection forms neighbourhood basis

Let $X$ be a topological space and $x \in X$. Suppose that there exists a countable collection $(U_n)_{n \geq 1}$ of open sets such that $U_{n + 1} \subseteq U_n$ and $\bigcap\limits_{n \geq 1} U_n = \...
2
votes
2answers
28 views

Chose $a_n$ so that $\frac{M_n}{a_n}$ converges to 0 in distribution

Let $X_1, ... , X_n$ be i.i.d. random variables with distribution Function $F$ and $M_n = \max\left\{X_1, \dots , X_n\right\}$ . Now i need to find $ a_n \in (0, \infty )$ so that $\frac{M_n}{a_n}$ ...
0
votes
0answers
20 views

What is the limit of the series (summation) of the q-Pochhammer symbol or the ~q-Pochhammer symbol?

I am interested in knowing if the following series converges or not: \begin{equation} \sum_{n=1}^{\infty} \prod_{i=1}^n \left(1-e^{-\sqrt{i}} \right) \qquad Expression \; 1 \end{equation} If that is ...
2
votes
2answers
36 views

Is a completely regular space whose convergent sequences are eventually constant discrete?

If $X$ is a metrizable topological space where the only convergent sequences are eventually constant sequences, then $X$ must be a discrete space. But I'm interested in whether something stronger is ...
-1
votes
0answers
56 views

Any counter example for the below claim?

Let $f(t)$ be a real valued function and continuous such that it lies in $(-1,1)$ and $\operatorname{erfc}(t)$ be a complementary error function. Many calculations I have run using inverse Symbolic ...
0
votes
1answer
38 views

Convergence in $L^p$ implies convergent $\mu$ a.e

I have this question: Given a $\sigma$-finite measure space, and $\forall n$ $0\leq f_n\in L^1(\mu)$ such that $\int_Xf_nd\mu=1$, show that $\frac{f_n}{n^2}\rightarrow0$ a.e. Any ideas?
3
votes
2answers
88 views

Studying the convergence of the series $\sqrt{2}+\sqrt{2+\sqrt{2}}+\sqrt{2+\sqrt{2+\sqrt{2}}}+ \cdots$

Studying the convergence of the series $$\sqrt{2}+\sqrt{2+\sqrt{2}}+\sqrt{2+\sqrt{2+\sqrt{2}}}+ \cdots$$ I saw this problem and I tried to do it my own way but I don't know what I'm doing wrong ...
1
vote
0answers
20 views

Iterating a sequence and verifying its convergence

I am given a sequence $(f_n)_n$ where $n\in N$. $f_n : \Re \rightarrow \Re: x \mapsto 1$ $f_1:\Re \rightarrow \Re$ is defined as follows $$f_1 (x) = 1 + \int_0^x f_0 (t) dt$$ One sees that the ...
0
votes
3answers
41 views

Computing terms of a sequence and proving it's convergent

Let $c_n$ be the sequence defined by $$c_n =1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} -2\sqrt n$$ a) Compute $c_1$, $c_2$, and $c_3$. b)Prove that $c_n$ is a ...
0
votes
1answer
26 views

Discussing the convergence of a sequence

If $k, x_1$ are positive, and $x_{n+1}$ = $\sqrt {k + x_n},$ discuss the convergence of the sequence $x_n$, according to whether $x_1$ is less than or greater than $\alpha$, the positive root of $x^2 =...