Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

0
votes
0answers
79 views

Integration by part for Lebesgue integral

I tried to prove one theorem in convergence of random variables and found myself in a little bit of trouble when doing integration by part. The reason being it involves a Lebesgue integral which I am ...
0
votes
1answer
80 views

Finding the value of distribution function of a converging random variable

There is this example in a note that I think this is supposed to be a simple problem, but I still find it not as straightforward. Consider a sequence of random variables $X_n\equiv1/n,X\equiv0$. Then ...
4
votes
1answer
48 views

Question about uniqueness function series

Please pardon me if this is elementary, but I've looked hard for an answer to this and am very surprised I have yet to find a good one. My question is simple: under what condition(s) does ...
1
vote
2answers
44 views

Find the radius of convergence of two series

I have to tell the convergence-radius of the following series: $\sum_{n=0}^\infty n^5x^n$ Here I thought about using Euler and I get $\lim \frac{1}{(1+\frac{1}{n})^5} = 1$ So the convergence-radius ...
1
vote
0answers
92 views

Convergence in distribution and convergence of expectation.

Suppose that $\sup_n \mathbb{E}[|X_n|^{r+\epsilon}]<\infty$ and $\sup_n \mathbb{E}[|Y_n|^{r+\epsilon}]<\infty$ for some $\epsilon>0$ and $r\geq 1$. Denote by $F_n(x)$ and $G_n(x)$ the ...
3
votes
1answer
78 views

Infinite sum convergence $ \sum_{i\geq 1}\frac{1}{x^i-y^i}$

For certain values of x and y, the sum $$\sum_{i=1}^{\infty}{\frac{1}{x^i-y^i}}$$ converges...is there a way to get the exact value, given x and y?
0
votes
1answer
46 views

Order of convergence of a sum

Let $(X_t)_{t\geq 0},\;X_0=0$, be a positive stochastic process such that \begin{align*} \mathbb{E}\left[\sum_{n=1}^{\infty}X_t^n\right]=\sum_{n=1}^{\infty}\mathbb{E}[X_t^n]<\infty. \end{align*} ...
2
votes
1answer
81 views

Example of sequence converging in $d_{l^\infty}$ but not in $d_{l^1}$.

I'll denote by $X$ the space of real sequences $(a_n)$ such that $\sum |a_n|$ converges. Let $d_{l^1}$ be the metric $$ d_{l^1}((a_n),(b_n))=\sum|a_n-b_n| $$ and $d_{l^\infty}$ be the metric $$ ...
2
votes
0answers
52 views

measure theory and convergence

1) Let $\Omega=[0,1]$, $F = B([0,1])$, $P$ be Lebesgue measure on $[0,1]$ ($P([a,b])=b-a$). Set $$A_n^i:=\left[{\frac{i-1}{n},\frac{i}{n}}\right]$$ and $$X_n^i(\omega):=\chi_{A_n^i}(\omega)$$ ...
2
votes
2answers
554 views

Find an interval of convergence and an explicit formula for $f(x)$

Let $f(x) = 1 + 2x + x^2 + 2x^3 +x^4+...$ If $c_{2n} = 1$ and $c_{2n+1} = 2$ $\forall n \ge 0$ find the interval of convergence and an explicit formula for $f(x)$. The answers are $I = (-1,1)$ and ...
5
votes
1answer
62 views

$\int_\Omega F(u_n)\to0$ implies $\int_\Omega F(au_n)\to 0$ for $a\in [0,\infty)$?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $F:[0,\infty)\to [0,\infty)$ be a convex, strictly increasing and continuous function satisfying $F(0)=0$. Suppose that $u_n\in ...
4
votes
2answers
238 views

Definition of convergence in $C^\infty(\Omega)$

I am not convinced or may be don't understand, the way they define convergence and then topology as a consequence of convergence. $\Omega$ is open subset of $\Bbb R^n.$Define standard topology on ...
7
votes
1answer
124 views

convergence of series with $k!$

check if the following series converges: $\sum\limits_{k=1}^{\infty} (-1)^k \dfrac{(k-1)!!}{k!!}$ where $k!!=k(k-2)(k-4)(k-6)...$ I came across this exercise while going trough some old exams. I'm ...
1
vote
1answer
567 views

Show that Y is a closed subspace of l2

This might be a straight forward problem but I wouldn't ask if I knew how to continue. Apologies in advance, I am not sure how to use the mathematical formatting. We are currently busy with inner ...
4
votes
3answers
163 views

Convergence of $\sum \frac{a_n}{(a_1+\ldots+a_n)^2}$

Assume that $0 < a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that $$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} < \infty $$ ? I think it is but I can't prove it. Of course if $a_n ...
2
votes
1answer
64 views

Intuition behind closed subsets of a metric space?

Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space. Consider a metric space $$(X,d)$$ Then consider a subset of this space$$F$$ What the book ...
-1
votes
1answer
68 views

Alternating functional Series Convergence SOS…

Does the following series converge? $\sum_{k=0}^\infty \frac{(-1)^k x^k \sqrt{k}}{k!}$ what is the radius of convergence?!!
2
votes
1answer
82 views

Multiplication in $\mathcal D'(R)$.

I was reading in my text book that multiplication of elements of $\mathcal D'(R)$ is not a continuous function although it is defined and can be seen from the convergence of $\sin(nx)$ to $0$ as $n$ ...
1
vote
1answer
79 views

Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.

I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
1
vote
1answer
85 views

Convergence in distribution of a random variable/vector

I'm trying to prove a version of Slutsky theorem and I appreciate if you can guide me on that. Suppose $X_n \rightarrow^{d} X > 0$ (i.e. $X_n$ convergence in distribution to X which is positive) ...
0
votes
1answer
90 views

Does convergence in $C(\Bbb R)$ imply convergence in $\mathcal D'(\Bbb R)$

Sequence of continuous functions $(f_k)$ which converge to $f$ in $C(\Bbb R)$. Does it imply that $f_k$ will converge to $f$ in $\mathcal D'(\Bbb R)$ as well? Or we need something more to make this ...
1
vote
4answers
66 views

Is $(a_n)_n$ with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ a Cauchy sequence?

Let $0 < q < 1$ and $(a_n)_n$ be a squence with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ for all $n ∈ ℕ$. I need to show that this is a Cauchy sequence. I'm not sure how to start this one, as we ...
0
votes
1answer
98 views

Show convergence for this sequence only by using the definition

I need to prove convergence for $(b_n)_{n ∈ ℕ}=\left(\frac{(-1)^nn}{2n+1}\right)_{n∈ℕ}$ and also show the limit. I may only use the following definition: $∀ɛ > 0∃n_0∈ℕ∀n≥n_0:|a_n-a|< ɛ$. So far ...
0
votes
3answers
44 views

Evaluating $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$

I am trying to evaluate $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$. Factoring and using partial fraction decomposition, I have found that the indefinite integral is: $$\ln{|x-5|}-\ln{|x-3|} + C$$ But ...
1
vote
2answers
57 views

Prove the convergence of the sequence.

Prove the convergence of the following sequence: $$x_1 = \sqrt{a}$$ $$x_{n+1} = \sqrt{a + x_n}$$
3
votes
1answer
58 views

Evaluating order of convergence

I think this is quite a simple question, I just want to make sure I understood all correctly. Here's the problem: I have a numerical method, which is in some way dependent on its spacing $h$ (like ...
5
votes
4answers
313 views

Evaluating $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$

Could you help me evaluate $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$? I've calculated that the recurrence relation for this integral is: $\int_0 ^{1} x^n e^x dx = x^ne^x | ^{1} ...
1
vote
1answer
47 views

series convergence

i ran into this question: prove or show false: if $\sum_{n=1}^{\infty}a_{n}$ is a converging series, but the series $\sum_{n=1}^{\infty}a_{n}^2$ diverges, then $\sum_{n=1}^{\infty}a_{n}$ is ...
7
votes
1answer
173 views

methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$

As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which ...
2
votes
0answers
14 views

Polynomials, integrals convergence

Let $P_n(x)= \frac{x^n(bx -a)^n}{n!}, \ \ \ a,b,n \in \mathbb{N}$. Prove that $\int_0 ^{\pi}P_n(x) \sin xdx \rightarrow 0 \ \ \ \ $ and $ \ \ \ \ \int_0 ^r P_n(x)e^xdx \rightarrow 0$ $ \ \ \ \ \ \ (n ...
1
vote
1answer
105 views

A problem on almost sure convergence

Consider a sequence of random variables defined on the standard unit interval probability space : $$X_n = \begin{cases} 2^n & \text{when} \quad \frac{1}{2^n} \leq \omega \leq ...
1
vote
2answers
88 views

Checking $\displaystyle \int_{0}^{\infty} {\frac{\sin^2x}{\sqrt[3]{x^7 + 1}} dx}$ for convergence

Given $\displaystyle\int_{0}^{\infty} {\frac{\sin^2x}{\sqrt[3]{x^7 + 1}} dx}$, prove that it converges. So of course, I said: We have to calculate $\displaystyle \lim_{b \to \infty} ...
1
vote
2answers
28 views

Develop the next function:$f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0)$

Develop the next function:$\displaystyle f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0).$ For the first part: $\displaystyle\frac ...
1
vote
2answers
108 views

How to show that these integrals converge?

What test do I use to show that the following integral converges? If you could provide me with the process that leads to the answer that would really help. $\displaystyle ...
5
votes
2answers
138 views

“Nearly” Harmonic Series

It's well known that $$ \sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0. $$ What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$? ...
2
votes
1answer
61 views

Convergence of sequence

Does the following: $$ \begin{align} x_0 & = a \\ x_1 & = x_0 + \frac{1}{1}(x_0 + x_0(c - 1)) \\ x_2 & = x_1 + \frac{1}{2}(b + x_1(c - 1)) \\ x_3 & = x_2 + \frac{1}{3}(b + x_2(c - 1)) ...
0
votes
1answer
47 views

Convergence of random variable to a negative constant

Let $X_n$ be the sequence of R.Vs and $X_n\overset{P}{\rightarrow}A$ (or $X_n\rightarrow A$ almost surely) where $A<0$ I want to prove that $Pr[X_n < 0] \rightarrow 1$ (or $X_n < A$ almost ...
0
votes
0answers
173 views

Central Limit Theorem for Dependent Non-Identical Random Variables.

If $X_{(1)}, X_{(2)},\ldots$ are mutually dependent as in the case of ordered statistics and we need to find the sum $S_N$ of all $X_{(i)}$ like $\sum_{i=1}^{N\to \infty} X_{(i)}$. How do we apply ...
3
votes
7answers
135 views

How to prove that $1/n!$ is less than $1/n^2$?

I want to prove $$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test. How to prove ...
0
votes
2answers
82 views

An equivalent expression of Cauchy Criterion?

For a sequence $\{a_n\}$, if $$ \forall \epsilon>0 \ \exists N>0, \forall k \in \mathbf{N}, \ |a_{N+k}-a_N|<\epsilon \ $$ Then $\{a_n\}$ converges and hence is a Cauchy sequence. Now how ...
0
votes
1answer
105 views

Radius of convergence - ratio test for power series/real numbers

Having trouble with when to apply the ratio test for power series and when to apply the ratio test for real numbers. For example, find radius of convergence of these.... $\sum_{n=0}^{\infty}(-1)^n ...
1
vote
0answers
28 views

Is $\phi^T_tP_t^{-1}\phi_t\to 0$ when $P_{t+1}=\sum_{k=0}^t\phi_k\phi_k^T+P_0$?

Let $\phi_t\in\mathbb{R}^n$, $\forall t\geq0$, and $\sup_t\|\phi_t\|_2^2\leq M<\infty$(euclidean norm). Define $n\times n$ positive definitive matrices as follow, ...
2
votes
1answer
30 views

Convergence by using Cauchy Criterion

this is the sequence: $(a_n)=\frac{1}{n+1}+\frac{1}{n+2}+\cdot\cdot\cdot+\frac{1}{2n}$ And this is what I tried to do so far: $|a_{n+1} - a_{n} | = \frac{1}{2n+1}+\frac{1}{2n+2}-\frac{1}{n+1} = ...
1
vote
3answers
148 views

Comparison test for series $\sum_{n=1}^{\infty}\frac{n}{n^3 - 2n + 1}$

I am trying to prove the convergence of the series $\sum_{n=1}^{\infty}\frac{n}{n^3 -2n +1}$ with the simple comparison test. I know it can be done with other tests but this question came up in my ...
4
votes
2answers
228 views

Convergence of $\int_0^\infty \sin(t)/t^\gamma \mathrm{d}t$

For what values of $\gamma\geq 0$ does the improper integral $$\int_0^\infty \frac{\sin(t)}{t^\gamma} \mathrm{d}t$$ converge? In order to avoid two "critical points" $0$ and $+\infty$ I've ...
1
vote
1answer
47 views

Convergence of $\int_0^1 \sqrt[3]{\ln(1/x)} \mathrm{d}x $

Does $$\int_0^1 \sqrt[3]{\ln\left(\frac{1}{x}\right)} \mathrm{d}x$$ converge? WA says it is equal to $\Gamma(4/3)$, however calculating the antiderivative seems approachless to me and can't compare ...
6
votes
1answer
180 views

Existence of a power series converging non-uniformly to a continuous function

I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that: $f$ converges and is continuous on the closed unit disk $D$ and the series $\sum_n a_n z^n$ does not converge ...
3
votes
1answer
112 views

Show that sequence approaches fixed point of a function

Problem Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
3
votes
1answer
77 views

convergence of an integral ( with an inner integral)

I need to figure out for which values of $p \in R $ does the following integral converge? $$\int_0^{1} \frac{x^p}{\int_0^{x}\ln(1 + \sin(t) + t)dt} {dx} $$ Please note that I don't have to ...
3
votes
1answer
302 views

Borel-Cantelli Lemma

I have some difficulties understanding the following: Let $(X_n)$ be a sequence of independent random variables s.t. $P[X_n=1]=1−P[X_n=0]=\frac{1}{n}$ After using the Borell Cantelli lemma, I ...