Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

2
votes
1answer
263 views

Conditional expectation and martingales

I have a few questions concerning martingales. Let $Y\in \mathcal{L}^1(\Omega,\mathcal{F},\mathbb{P})$ be given, and $(\mathcal{F}_n)$ a filtration, and define $X_n:=\mathbb{E}[Y|\mathcal{F}_n]$. We ...
1
vote
1answer
297 views

Convergence in probability implies convergence in mean under one additional condition

Prove that if random variables $X_n$ are dominated by an integrable random variable then $E[X_n] \to E[X]$ follows if $X_n$ converges to $X$ in probability. Hint: Use the following theorem : A ...
0
votes
2answers
98 views

Positive Sums: Relations

This is a lemma for: Trace Sum of a system of non-negative real numbers $\phi_\lambda$, $\lambda\in\Lambda$ can be defined as: ...
8
votes
1answer
992 views

Proof of a theorem of Cauchy's on the convergence of an infinite product

Well it is relatively well known that the condition for absolute convergence is given by the following theorem: In order that the infinite product $\prod _{n=1}^{\infty }\left( 1+a_{n}\right) $ may be ...
7
votes
1answer
296 views

convergence of the iterated cosine

it can be demonstrated by elementary means that the curves $y=\cos x$ and $y=x$ meet exactly once, at a value $x=\alpha$ satisfying: $$\cos \alpha = \alpha$$ it is also evident (empirically) that ...
7
votes
1answer
2k views

The strong and weak laws of large numbers: Why two?

The following questions are entirely based on the corresponding article from Wikipedia. The assumptions of both laws are the same, and the strong law has a more general claim than the one of the weak ...
7
votes
2answers
4k views

Proof that a sequence converges to a finite limit iff lim inf equals lim sup

This problem is purely for my own benefit, so I'd appreciate it if you offer help but don't spoil the proof for me. I've worked out the following solution, but I want to make sure that my reasoning ...
6
votes
1answer
236 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 ...
6
votes
4answers
381 views

Power Series with the coefficients $n!/(n^n)$

I'd be grateful if someone could tell me how to obtain the convergence radius of the aforementioned power series. Or, by Cauchy Hadamard, the limit of $(n!/(n^n))^{(1/n)}$ as n approaches infinity. ...
4
votes
3answers
103 views

Is the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$ necessarily $0$?

I have the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$. I know that every successive partial product will necessarily be smaller than the last, as we are multiplying always by a number smaller than ...
4
votes
2answers
198 views

How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?

I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
4
votes
1answer
154 views

$f_n(x_n)\to f(x) $ implies $f$ continuous - a question about the proof

My question refers to If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous. (the first answer) It's written there that the hypotheses imply $(f_n)$ converges to $f$ pointwise. ...
4
votes
3answers
207 views

$\sum_{n=2}^\infty \frac{1}{(\ln\, n)^2}$ c0nvergence

$$\sum_{n=2}^\infty \frac{1}{(\ln\, n)^2}$$ The series converge? Please verify my solution below
4
votes
2answers
531 views

Differences, geometric sequences and convergence - $ |x_{n+1} - x_n| \leq ac^n \implies \exists\ L : x_n \to L$

I've got a lingering question from a midterm in real analysis that I'd really like to have answered. The first time I answered the question, I received a 2/10 for absolutely mangling the definition of ...
4
votes
1answer
3k views

Bounded sequence and every convergent subsequence converges to L

Let $\{x_n\}$ be a bounded sequence such that every convergent subsequence converges to $L$. Prove that $$\lim_{n\to\infty}x_n = L.$$ The following is my proof. Please let me know what you think. ...
4
votes
2answers
490 views

Absolute and uniform convergence of $\sum _{n=1}^{n=\infty }2^{n}\sin \frac {1} {3^{n}z}$

I am trying to show $\sum _{n=1}^{n=\infty }2^{n}\sin \frac {1} {3^{n}z}$ converges absolutely for all values of $z$ $(z=0$ excepted$)$, but does not converge uniformly near $z=0$. I observed that ...
3
votes
1answer
73 views

Let $A=\{\sum_{i=1}^{\infty} \frac{a_i}{5^{i}}:a_i=0,1,2,3$ or $4 \} \subset \mathbb{R}$. Then which of the following are true??

Let $$A=\bigg\{\sum_{i=1}^{\infty} \frac{a_i}{5^{i}}\ :\ a_i\in\{0,1,2,3,4\} \bigg\} \subset \mathbb{R}.$$ Then which of the following are true: a. $A$ is a finite set. b. $A$ is countably ...
3
votes
4answers
88 views

Proving convergence of a series and then finding limit [duplicate]

I need to show that $\sqrt{2}$, $\sqrt{2+\sqrt{2}}$, $\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots$ converges and find the limit. I started by defining the sequence by $x_1=\sqrt{2}$ and then ...
3
votes
1answer
1k views

Weak convergence, together with convergence of norms, implies strong convergence in a Hilbert space.

Let $(x_n)$ be a weakly convergent sequence in a Hilbert space $H$. If $\| x_n \| \to \| x \|$, show that $x_n$ converges strongly to $x$. Context This problem comes from a question in my exam ...
3
votes
3answers
122 views

Please more help me to find the convergence interval and the sum -by using residue theory- of the series.

The sum is that $$\sum_{n=0}^\infty \binom{3n}{2n} x^n$$ First of all, I need to check whether the sum converges or not and if it is convergent, which points? I am using ratio test. $$ ...
2
votes
3answers
118 views

Convergence of $\sum_n \frac{n!}{n^n}$

I'm working on a problem sheet and it ask to discuss the convergence of $$\sum \frac{n!}{{n}^{n}}$$ By D'Lembert's ratio test, $$\lim_{n->\infty}\frac{{a}_{n+1}}{{a}_{n}} = 1$$ and so, is ...
2
votes
3answers
104 views

$u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))$

how can i show this sequence $u_n$ is divergent: $$u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))\quad n\in \mathbb{N}^*;\quad \epsilon \in (0,1)$$ My attempts: \begin{align*} u_n&=\exp( n\log ...
2
votes
1answer
107 views

Convergence of multiple zeta function

The following term:$$\zeta(k_1,k_2,...,k_n)=\sum_{m_1>m_2>\cdots>m_n>0}\frac{1}{m_1^{k_1}m_2^{k_2}\cdots m_n^{k_n}}, m_i\in\mathbb{N}, k_i\in\mathbb{N}$$ is called the "multiple zeta ...
2
votes
3answers
62 views

$a_n$ is bounded and decreasing

my second question from [An inequality for the product $\prod_{k=2}^{n}\cos\frac{\pi }{2^{k}}$ Let $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and ...
2
votes
3answers
3k views

Proof that rational sequence converges to irrational number

Let $a>0$ be a real number and consider the sequence $x_{n+1}=(x_n^2+a)/2x_n$. I have already shown that this sequence is monotonic decreasing and thus convergent, now I have to show that $(\lim ...
2
votes
1answer
235 views

Uniform convergence of functions, Spring 2002

The question I have in mind is (see here, page 60, the solution is at page 297): Assume $f_{n}$ is a sequence of functions from a metric space $X$ to $Y$. Suppose $f_{n}\rightarrow f$ uniformly and ...
1
vote
1answer
92 views

What are conditions under which convergence in quadratic mean implies convergence in almost sure sense?

What are the conditions on the sequence on $\{X_n\}$ (apart from the degenerate random variable), under which it can be claim that $||X_n-X||_{L^2(\mathbb{R})}\rightarrow 0$ implies $X_n\rightarrow ...
1
vote
3answers
53 views

Dominated Convergence: Estimate

This is an application of: Spectral Measures: Domain Criterion I'm trying to check the estimate: $$\frac{1}{h}|e^{ixh}-1|\leq C\left(|ix|+1\right)\quad(h\in(-\varepsilon,\varepsilon))$$ for some ...
1
vote
0answers
75 views

Convergence in Probability of an estimator

Let $X_n$ be a Poisson process with mean $\lambda^*$. The following sequence estimates the parameter of the Poisson process: $ X_{n+1} = \hat{\lambda}_{n+1} + ...
1
vote
3answers
1k views

Weak convergence and strong convergence

Suppose $f_i$ is uniformly bounded in $W^{1,p}$ for some $+\infty>p>1$, then by passing to a sub-sequence, we can suppose $f_i$ is weakly convergent to $f$ in $W^{1,p}$. Assume furthermore that ...
1
vote
2answers
1k views

Is $\sum\limits_{n=3}^\infty\dfrac{1}{n\log n}$ absolutely convergent, conditionally convergent or divergent?

Classify $$\sum_{n=3}^\infty \frac{1}{n\log(n)}$$ as absolutely convergent, conditionally convergent or divergent. Is it, $$\sum_{n=3}^\infty \frac{1}n$$ is a divergent $p$-series as $p=1$, and ...
1
vote
3answers
210 views

How to calculate $\sum_{n=0}^\infty {(n+2)}x^{n}$

I want to calculate the sum of $$\sum_{n=0}^\infty {(n+2)}x^{n}$$ I have tried to look for a known taylor/maclaurin series to maybe integrate or differentiate...but I did not find it :| Thank you. ...
0
votes
1answer
70 views

Prove that $\int \limits _{0}^{\infty}\frac{e^{-2x}-e^{-ax}}{x}\text{d}x$ converges for any $a>0$

I'm doing this exercise: Prove that $F(a)=\displaystyle\int \limits _{0}^{\infty}\frac{e^{-2x}-e^{-ax}}{x}\text{d}x$ converges for every $a>0$. Calculate $F'(a)$ and deduce $F(a)$. I've ...
0
votes
1answer
27 views

A bounded net with a unique limit point must be convergent

Let $(x_i) _{i \in I}$ be a net in a compact Hausdorff space X, with the property that every convergent subnet has the same limit $x$. Does this imply that $(x_i) _{i \in I}$ converges? If I try to ...
0
votes
0answers
99 views

Semigroups: Entire Elements (II)

Problem Given a Banach space $E$. Consider a contraction C0-group: $$T:\mathbb{R}\to\mathcal{B}(E):\quad\|T(t)x\|\leq\|x\|$$ Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)\in E$$ ...
0
votes
1answer
147 views

A sequence converges if and only if every subsequence converges?

I want to prove this and intuitively it makes sense. But I'm having a hard time coming up with a proof. So if a sequence converges, then we have a natural number for which the distance between all ...
0
votes
1answer
98 views

Prove that the sequence converges to zero [closed]

$$a_n=\sqrt{n+3} - \sqrt n$$ Can someone please give me a detailed way of how to prove that this sequence converges to zero?
0
votes
3answers
96 views

How to show that $\sum {2^j + j \over 3^j - j}$ converges

I'm not quite sure how to go about growing the numerator or shrinking the denominator to perform a tricky comparison test, or hashing out the ratio test, so any help would be much appreciated!
0
votes
7answers
109 views

Induction and convergence of an inequality

Problem statement: Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty $. , $n\in \mathbb{N}$ My progress LHS is ...
-2
votes
0answers
190 views

Fibonacci Numbers - Complex Analysis [duplicate]

Possible Duplicate: Complex Analysis - Integral over a circle of radius R Hey guys~ Does anyone know where to find the solutions to this problem set on page 106 involving the fibonacci ...
7
votes
3answers
365 views

Convergence of a sequence of non-negative real numbers $x_n$ given that $x_{n+1} \leq x_n + 1/n^2$.

Let $x_n$ be a sequence of the type described above. It is not monotonic in general, so boundedness won't help. So, it seems as if I should show it's Cauchy. A wrong way to do this would be as follows ...
6
votes
3answers
737 views

vercongent sequences

Definition- We say a sequence $(x_n)$ verconges to $x$ if there exist an $\epsilon>0$ such that for all $N\in \Bbb{N}$, $n\ge N \implies |x_n-x|<\epsilon$. Loosely speaking, by convergent ...
6
votes
1answer
70 views

Suppose $\sum\limits_{n=1}^\infty b_n$ diverges and $b_n\gt 0$, show that the series $\sum\limits_{n=1}^\infty \frac{b_n}{1+b_n}$ also diverges

As the title says, given a series $b_n > 0$, where $\sum_{n=1}^\infty b_n$ is divergent: Show that the series $$\sum_{n=1}^\infty \frac{b_n}{1+b_n}$$ is also divergent. So I've defined the series ...
6
votes
4answers
354 views

Please help me to prove the convergence of $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}-\ln n$

Please help me to prove that $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{n}-\ln n$ converge and it's limit $\gamma \in [0,1]$. Then, using $\gamma_{n}$ find the sum: ...
5
votes
2answers
198 views

Prove $ \sum \frac{\cos n} { \sqrt n}$ converges

How to Prove $ \sum \frac{\cos n} { \sqrt n}$ converges Using Abel’s theorem ? I think it can be done using $\cos n = Re(e^{in})$ { Real Part of Complex Number } How to proceed ?
4
votes
1answer
49 views

Splitting an infinite unordered sum (both directions)

This question is a follow-up to this. Let $A, B_1, B_2, \dots$ be countable sets such that $\bigcup_{i \in \mathbb{N}}B_i = A$ and the $B_i$'s are disjoint. Let $f : A \to \mathbb{R}$ take elements ...
4
votes
5answers
89 views

Proving the convergence of $\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$

How to prove that the series $$\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$$ is convergent? What about finding the sum? My attempt: $$\ln (1-1/n^2)= \ln(n-1) -2\ln n + \ln(n+1)$$ The term in the ...
4
votes
1answer
203 views

Cauchy Sequences, not converging to zero

True or False? If $\{x_n\}$ and $\{y_n\}$ are Cauchy and $x_n + y_n > 0$, for all $n\in\mathbb{N}$, then $\left\{\frac{1}{(x_n + y_n)}\right\}$ cannot converge to zero. I believe the claim to be ...
4
votes
2answers
567 views

Using the Banach Fixed Point Theorem to prove convergence of a sequence

Use the Banach fixed point theorem to show that the following sequence converges. What is the limit of this sequence? $$\left(\frac{1}{3}, \frac{1}{3+\frac{1}{3}}, ...
3
votes
1answer
43 views

Convergence of $\int f dP_n$ to $\int f dP$ for all Lipschitz functions $f$ implies uniform integrability

I would like to prove or give a counterexample for the following statement: Let $(S,d)$ be a complete and separable space. We define: $$ \mathcal{P}^1(S) := \{P: \mathcal{B}_S \rightarrow [0,1] \mid ...