Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [convergence-divergence]

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

This tag is for questions about Convergent Sequences and Series and their existence, and by extension also for divergent ones. Also for convergence/divergence of improper integrals.

Formally, a sequence $S_n$ converges to the limit $S.$

$$\lim_{n\to \infty}S_n=S $$ if, for any $\epsilon>0$, there exists an $N>0$ such that $|S_n-S|<\epsilon$ for $n>N$.

A sequence diverges if it does not converge.

For more specialized questions about divergent series, such as summation methods, consider the tag .

21990 questions
1
vote
1 answer

Convergence of r.vs

I'm working on the question below and I appreciate if you can guide me on how I can solve it. Here is the question: Consider $X_j$'s j = 1, 2, ... as independent bernulli random variables. These random variables, although are independent, but are…
user52144
1
vote
1 answer

General formula for the partial sums

I m having trouble figuring out how to find the general formula for partial sums of the following two series. $\sum_{i=2}^n \frac{1}{i^2-1} = \frac{3}{4}-\frac{1}{2n}-\frac{1}{2(n+1)}$ $\lim_{n\to\infty} \frac{4n^2 -n^3}{10+2n^3} = -\frac{1}{2}…
user3352074
  • 39
1
vote
1 answer

Convergence in Probability!

I'm working on a problem and after couple of hours, I'm not still sure how I should approach it. Could you give me some hints? Consider $X_1, X_2, \ldots$ as independent random variable where: $\Pr(X_n = k) = (1-p_n)p_n^k$ for $k = 0, 1, 2, \ldots$…
user48405
1
vote
0 answers

Consistent estimator of theta

Let Y1,....,Yn be a random sample from a uniform distribution on the interval (0, theta), theta >1. Let Xn= max (Y1,...,Yn). Find the consistent estimator of: log (3 ((theta)^5) + theta) Usually I have dealt with a given estimator and showing if…
L.mak
  • 201
  • 1
  • 3
  • 8
1
vote
1 answer

Conditional convergence of $\sum \frac{(-1)^{[\sqrt{n}]}}{n^{p}}$

I need to prove that the following series: $$ \sum \frac{(-1)^{[\sqrt{n}]}}{n^{p}} $$ is conditionally convergent when p=1. Any hints would be welcome.
syntagma
  • 1,003
1
vote
2 answers

Not sure about a convergent series

I solved this series $$\sum_{n=2}^\infty \frac 1 {n^2\ln n}$$ wit Condensation Test and I got now $$\sum_{n=2}^\infty \frac{1}{2^nnln(ln(2))}$$ Can I use now a geometric series like $\sum_{n=0}^\infty \frac 1 {2^n}$ with comparasion test for say…
arcilli
  • 69
1
vote
1 answer

Can we Conclude Strong Convergence?

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain, $p\in (1,\infty)$. Suppose that $u_n\in L^p(\Omega)$. By using the fact that $L^p(\Omega)$ is uniformly convex, we know that if $u_n\rightharpoonup u$ and $\|u_n\|_p\rightarrow \|u\|_p$, then…
Tomás
  • 22,559
1
vote
1 answer

Convergence in $L^p(\Omega)$ Norm

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Let $u,v,v_n\in L^p(\Omega)$ and suppose that $$\|u+v_n\|_p\rightarrow\|u+v\|_p$$ Is true that $$\|v_n\|_p\rightarrow\|v\|_p$$ Thanks
Tomás
  • 22,559
1
vote
1 answer

First order convergence explanation understanding

I was wondering if i am understanding the reasoning wrong or if the source has a mistake? Source: To my understanding it should read: $$\frac{x_2-x_1}{x_1-x_0} =…
Sonny Da Silva-Peters
  • 316
1
vote
1 answer

Proving an estimator is consistent

For the set of i.i.d. $X_{1},\ldots,X_{n} \sim \mathrm{Uniform}(0,\theta)$, the method of moments estimator is $$\hat{\theta} = \frac{2}{n} \sum_{k=1}^{n}X_k.$$ now to show that $\hat{\theta}$ is a consistent estimator of $\theta$, I want to…
Twenty-six colours
  • 1,881
1
vote
1 answer

How prove that $\int_2^{\infty}\left|\frac{\cos\sqrt{x}}{x^{\alpha}\ln x}\right|dx$ diverges with $\frac{1}{2}\leq \alpha \leq 1$

How would one prove that $\int_2^{\infty}\left|\frac{\cos\sqrt{x}}{x^{\alpha}\ln x}\right|dx$ diverges with $\frac{1}{2}\leq \alpha \leq 1$? It converges for $\alpha > 1$ as $\int_2^{\infty}\left|\frac{\cos\sqrt{x}}{x^{\alpha}\ln x}\right|dx \leq…
Eugene Korotkov
  • 409
1
vote
1 answer

convergence of $\int_{0}^{1}\frac{1}{\sqrt[3]{(1-x^2)^5}}\, dx$

I would like to prove that this integral is divergent. Thanks for any suggestions. $$\int_{0}^{1}\frac{1}{\sqrt[3]{(1-x^2)^5}}\, dx$$
Juan Carlos
  • 169
  • 7
1
vote
4 answers

How do you find the sum of this alternating series?

$$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)(n+1)!}.$$ I found out from my fellow peers at stack exchange see here, that this series converges from the alternating series test. But how do you find the sum? I know if you use wolfram alpha you get:…
Dank memer
  • 79
1
vote
2 answers

Prove that the sequence is convergent

How can we show that the sequence $$a_n=\sqrt[3]{n^3+n^2}-\sqrt[3]{n^3-n^2}$$ is convergent?
Nasibabuba
  • 546
1
vote
1 answer

pointwise convergence contested

From the joint image, how does $f_n \rightarrow 0$ pointwise on $\mathbb{R}$? For instance, I can't find an integer $A$ such that $n>A \Longrightarrow |f_n(x)|\le 1/2, \forall x\in \mathbb{R}$ ?
Smilia
  • 2,024
1 2 3
…
35 36