Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

0
votes
1answer
44 views

Does $\frac{x}{n}$ converge uniformly on ℝ?

Does $x, \frac{x}{2}, \frac{x}{3}, \frac{x}{4}, \ldots$ converge uniformly on ℝ? I think that it does not since $\lim_{n\rightarrow+\infty} x/n = 0$. Then $|\frac{x}{n} - 0| = |\frac{x}{n}| < ...
1
vote
1answer
69 views

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? [duplicate]

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? So if it converges then $\lim_{b \to\infty}\int_{0}^{b}\sin(x^2)\;\mathrm dx$ exists and our integral converges to this ...
1
vote
1answer
22 views

Finding interval of convergence for complicated sum

I'm going through old exams for my Calc III course and came across a problem that I did not know how to do. The problem is: Find the interval of convergence of the series ...
1
vote
4answers
57 views

Prove that $\lim_{n\to \infty} n\cdot r^n=0$ where $(0\leq r <1)$ without using ratio test

$\lim_{n\to \infty} n\cdot r^n=0$, where $0\leq r <1$, can be obtained by vanishing condition (considering $\sum^{\infty}_{n=1}n\cdot r^n$, which converges, using ratio test). Is there a direct ...
0
votes
0answers
28 views

Random variables, indicator variables, conditional probability

Let $\{X_n \}$ be a sequence of independent identiacally distributed random variables. Let $A, B \in \mathcal{B}(\mathbb{R})$ be such that $P(X_1 \in B) \neq 0$. Let $S_n:= 1_{\{X_1 \in B \}} + ... + ...
1
vote
0answers
25 views

Binary system, number of $1$s, almost sure convergence

Could you check if my solution is correct? For $x \in [0,1]$ let $S_n$ be the number of times $1$ occurs in the first $n$ digits of $x$'s binary representation. Show that $\lim _{n \to \infty} ...
1
vote
1answer
55 views

Limit comparison test

Since the series $\sum_{n=1}^\infty a_n$ is convergent, so $\lim_{n\to\infty} a_n=0.$ Consider $$\lim_{n\to\infty} {\left(\dfrac{{a_n}^{1/2}}{n}\right)\over{\left(\dfrac{1}{n^2}\right)}} $$ which is ...
0
votes
1answer
30 views

Stuck on finding where $\sum_1^{\infty} (x+4)^{n}$ converges conditionally.

I have all the questions correct on my hw except for one: find where $\sum_1^{\infty} (x+4)^{n}$ converges conditionally. Radius of Convergence I got 1 for this, by using the root test and finding ...
1
vote
4answers
53 views

Proof that $0.33333… = \frac{1}{3}$ using $\epsilon-N$ method

This proof is quite prevalent on the web, yet I struggle using this particular method. Wikipedia (http://en.wikipedia.org/wiki/Limit_of_a_sequence) tells us: We call $x$ the limit of the sequence ...
1
vote
1answer
26 views

Uniform convergence and continous.

Let $(f_{n})$ be a sequence of functions. Is it possible that $(f_{n})$ converges uniformly where each functions (that is $f_{1},f_{2}, f_{3}\dots$) aren't necessarily continous?
0
votes
2answers
34 views

Finding an unbiased estimator for a parameter, dicrete variable

Let $X : \Omega \to \mathbb{N}$ be a random variable. Define $p_i = P(X=i), \ \ i \in \mathbb{N}$. Find an unbiased and consistent estimator for $p_1$. I need to find an estimator $\alpha_n(X_1 + ...
1
vote
1answer
14 views

Describing convergence/divergence of a complex sequence

Let (a$_n$)$_{n \in N}$ be a complex sequence and a $\in$ C. Show that the following statements are equivalent: $\forall$ $\varepsilon$ > 0 $\exists$ N $\in$ N $\forall$ n $\geq$ N : |a$_n$ - a| ...
3
votes
0answers
21 views

Convergence of Arnoldi method

I would like to compute the largest real eigenvalue of a matrix in the following form: $$\begin{bmatrix} 0 & I_n \\ P & Q \end{bmatrix},$$ where $I_n$ is the $n \times n$ identity matrix, $P$, ...
0
votes
1answer
57 views

Limit of a convergent serie

For a research project, after some manipulation I come up with a convergent serie that I have to prove its limit. The statement is the following: $ \lim_{n \rightarrow \infty } \displaystyle ...
0
votes
0answers
13 views

If $S_M$ denotes the measure of a submanifold, then $\frac 1{r^{n-1}n\omega_n}\int_{\partial B_r}u(x)\;dS_{\partial B_r}(x)\to u(y)$ for $r\to 0$

Let $S_M$ denote the "surface measure" of a submanifold $m$ $B_\varepsilon(y)$ denote the open ball around $y$ with radius $\varepsilon>0$ $\omega_n$ denote the volume of the $n$-dimensional unit ...
0
votes
0answers
49 views

Rate of Convergence of a Sequence Defined by a Function

In my notes, I have that if a sequence defined by a function, $x_{i+1} = f(x_i)$, converges to $c$ in the limit, i.e., $$\lim_{i\to\infty} x_i = c,$$ then the rate of convergence to this limit, ...
0
votes
1answer
14 views

How to show $[0, \frac{\pi}{2}) \cup (\frac{3\pi}{4}, 2\pi)$ is sequentially separated?

Definition of separated: $E$ is separated if $A, B \neq \varnothing$, $A \cup B = E$, and there are not convergent sequences in $A$ that have limit points in $B$, and vice versa. Definition of ...
4
votes
3answers
108 views

Which (convergent) series can one find the sum of?

I know about geometric series and how one can find the sum when they are convergent. I also have heard that one can prove that the $p$-series $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ has sum $\pi^2/6$ ...
1
vote
0answers
66 views

Does a Banach valued Cauchy series over arbitrary set converges?

Definitions: If $f$ is a function from a set $A$ (not necessarily countable) into a Banach space $V$, we say that the series of $f$ over $A$ converges if there exists an element $v \in V$ such that ...
1
vote
1answer
42 views

Newton Integral Convergence

please, I have a problem. I suppose it´s quite easy, however, I really don´t see what should I do with it. I should decide on convergence or divergence of this integral: $$\int_0^\infty ...
0
votes
1answer
18 views

A very simple question about multinomial distributions

Let's say you have a random vector $(x_1,\ldots,x_k)$ that has a multinomial distribution with parameters $n$ and $(p_1,\ldots,p_k)$. Suppose that we know $p_i>p_j$ for some $i,j$. Is it correct ...
2
votes
1answer
18 views

Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
1
vote
0answers
47 views

Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
1
vote
0answers
18 views

Multivariate Berry-Esseen/ Please help!

I've got a problem with understanding Berry-Esseen inequality for random vectors. You see, I keep coming across various forms of this theorem, all assuming a unit covariance matrix $I$, though it's ...
1
vote
1answer
24 views

$\int_{\Omega\setminus A_n}f\;d\mu\to\int_\Omega f\;d\mu$ for all measurable $A_n\downarrow\emptyset$

Let $(\Omega,\mathcal{A},\mu)$ be a measure space $(A_n)_{n\in\mathbb{N}}\subseteq\mathcal{A}$ such that $A_n\downarrow\emptyset$, i.e. $A_n\supseteq A_{n+1}$ and ...
2
votes
0answers
25 views

Formal Expansion of another Expansion

Given a function $f(x)=\sum_{n=1}^{\infty}\frac{c_n}{x^n n!}$, where $c_n$ are constants, we want to find the formal series expansion of the function $g(x)=\exp(f(x))$ in terms of $x$. I want to ...
4
votes
1answer
42 views

Convergence of the following sequence:

It could be exhaustion from the amount of work that I've done today, but I'd like to prove for myself that $$\lim_{n\to \infty} e^{-t\sqrt{n}}(1-\frac{t}{\sqrt{n}})^{-n}=e^{\frac{1}{2}t^2}$$ Here's ...
0
votes
2answers
49 views

Determine if the sequence- $a_1 = 2$ , $a_{n+1} = 72/(1+a_n)$ is convergent.

I've tried to prove that the sequence is convergent by using the monotonic sequence theorem but after computing the first few terms, I realized that the sequence is not monotonic thus, it isn't ...
0
votes
1answer
29 views

Determine if the sequence an = ln(n)/(n^(1/n)) is convergent.

I have some difficulty with showing that the sequence $$ a_n = \frac{\ln(n)}{n^{1/n}} $$ is divergent. Can anyone help me out with this? Thanks!
0
votes
0answers
34 views

If $f_n \to f$ pointwise a.e., $\int |f| < \infty$, and if $\int |f_n| \to A$, is $A=\int |f|$?

We work on some domain $\Omega$ which may or may not be bounded. If $f_n \to f$ pointwise a.e., if $\int |f| < \infty$, and if we know that $\int |f_n| \to A$ to some number $A$, is $$A=\int ...
0
votes
0answers
15 views

Where $|f| <\infty$ a.e. condition is used in Vitali Convergence Theorem

Vitali convergence theorem_Wiki Here above is a Wiki article about Vitali convergence theorem, which is referred to Rudin, Real and Complex Analysis. And I'm wondering where the fourth condition is ...
2
votes
2answers
27 views

Series Radius/interval of convergence

Help please! I have no idea how to do this question. I tried using the ratio test
1
vote
1answer
27 views

Prove that the sequence $(b_n)$ converges

Prove that if $(a_n)$ converges and $|a_n - nb_n| < 2$ for all $n \in \mathbb N^+$ then $(b_n)$ converges. Is the following proof valid? Proof Since $(a_n)$ converges, $(a_n)$ must be bounded, ...
0
votes
1answer
60 views

Proving that if the sequence $X_n$ converges to $x$, then ${X_n}^a$, where $a$ is a positive rational, converges to $x^a$.

I've been stuck on this problem for a while. I splitted a into $p/q$, so it would be $({X_n}^p)^{1/q}$, and I got the convergence of ${X_n}^p$ to be $x^p$ since it is just induction using the product ...
1
vote
0answers
57 views

Prove the uniform convergence

Let $a_n$ be a monotonic sequence convergent to a. Let f : R $\to$ R be a continous and monotonic function. Then we define a series of functions as follows : $$f_n(x) := f(x+a_n)$$ Prove that the ...
0
votes
1answer
39 views

Euler method uniform convergence

I have a question for you guys. Given a differential equation $$\dot{x}=f(x)\qquad x\in\mathbb{R}^n$$ on a compact interval $[0,T]$. If one considers for every $k\in\mathbb{N}$, the Euler's ...
1
vote
1answer
26 views

Almost sure convergence, arithmetic mean, variance

I am stuck proving that this sequence $$\sigma^2_n:= \frac{1}{n-1} \sum_{i=1}^n (X_i - \frac{X_1+...+X_n}{n})^2$$ is converegent almost surely to $D^2X_i = \sigma^2$. We assume that $X_1, X_2, X_3, ...
0
votes
0answers
25 views

Interchanging infinite double sum and expectation

Let $\xi_i$ be a sequence of independent and identically distributed standard normal random variables and consider sequences $\{b_i\}$ and $\{c_j\}$ such that $\sum_i b_i<\infty$ and $\sum_j ...
0
votes
1answer
35 views

Puzzled at this alternating series problem.

I have rechecked this problem so many times, and even my tutor got stuck on this. Can someone tell me what I did wrong? My homework says I got at least one question wrong. And my tutor was confused ...
1
vote
1answer
28 views

Determine the radius of convergence of $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ (by the ratio test if possible)

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ Applying the ratio test gives $\frac{({n+1})^{({n+1})^{1/3}}}{n^{n^{1/3}}}z<1$. So ...
2
votes
1answer
38 views

Divergent series of independent RV

I'm trying to prove that if $\{X_n\}_{n=1}^{\infty}$ is a sequence of independent random variables with the same distribution and $P(X_1 \neq 0)>0$, then the series $\sum_{n=1}^{\infty} X_n$ is ...
1
vote
1answer
36 views

Convergence of infinite series of function with factorial and power

Determine whether the series is convergent or divergent: $$\sum_{n=0}^\infty \frac{(3n)!+4^{n+1}}{(3n+2)!}$$ I guess we have to use comparison test for this question, but I am not sure what to use ...
2
votes
1answer
134 views

Continuous time Uniform Integrability and convergence in L1 for (super)martingale

I need to prove the following theorem by reasoning as in discrete-time, since we have proven this theorem in discrete time. Theorem Let $M$ be a right-continuous supermartingale that is bounded in ...
1
vote
0answers
22 views

if a sequence converges in measure in $L^p$, then converging for weak topology.

Given a finite measure space $(A,\Sigma,\mu)$, for $p \in (1,\infty)$, if {$f_n$} is a bounded sequence in $L^p(A)$ converging in measure to $f \in L^p (A)$, then {$f_n$} converges to $f$ for the ...
-1
votes
4answers
62 views

Root test for $\sum_{n=8}^\infty \left(1+\frac{3}{n}\right)^{n^2}$

I got that it would be infinity and diverge, but my answer seems to be incorrect. What did I do wrong? $$\lim_{n\to\infty}\left(1+\frac3n\right)^{n^2}= \lim_{n\to\infty} ...
0
votes
1answer
32 views

Root Test for Convergence or Divergence (ln problem)

I'm stuck at this problem. How would I approach this (I have to use the root test for this one)? The ln and e and my power is throwing me off. This is how far I have gotten.
0
votes
1answer
15 views

Interval of convergence for a series

I am currently trying to determine the interval of convergence, but I keep getting 0 for all my questions. I have attached one of the questions that I am unable to solve completely and I would really ...
0
votes
1answer
14 views

Asymptotic convergence of the total length of a graph

I encoded the following algorithm: suppose we're in (0,1)x(0,1) and I randomly create a "village" one at a time. At each step, I link a newly randomly created village to the closest village already ...
0
votes
2answers
60 views

radius of convergence of $\sum_{n=1}^\infty n!^2x^{n^2}$ [closed]

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n!^2x^{n^2}$
0
votes
1answer
31 views

Almost uniform convergence implies a.e. pointwise convergence proof

I've just read a proof of the statement "On a finite measurable space, $(f_n)_{n \geq 1}$ and $f$ measurable and finite a.e. functions, if $(f_n)_{n \geq 1}$ converges almost uniformly to $f$, then it ...