Convergence of sequences and different modes of convergence.

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6
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2answers
824 views

Do weak convergence and convergence of norms imply convergence in $L^2$?

Let $(f_n)_n\subseteq L^2(0,1)$ s.t. $$ f_n \rightharpoonup f, \qquad\qquad \Vert f_n\Vert_2 \to \Vert f\Vert_2 $$ where $\rightharpoonup$ means weak convergence. Is it true that $f_n \to f$ ...
4
votes
2answers
559 views

Test for convergence of the series $\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$

Could I have a hint for testing the convergence of the following series please? $$\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$$ Edit The integral test does not work because $\int_1^n\frac{1}{(\ln ...
4
votes
4answers
3k views

Show that $\sqrt{n^2+1}-n$ converges to 0

I want to use the definition of the limit to show that $\sqrt{n^2+1}-n$ converges to 0. The definition is as follows: if $\sqrt{n^2+1}-n$ converges to 0, then $\forall \epsilon>0$, there exists an ...
4
votes
6answers
816 views

Does $\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2+2}$ converge?

I'm trying to find out whether $$\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2+2}$$ is convergent or divergent?
3
votes
2answers
83 views

Convergence to $N(0,1)$ in distribution

Question is as following: $X\sim Po(\lambda)$ $$\frac{X-\lambda}{\sqrt{\lambda}} \,{\buildrel d \over \rightarrow}\, N(0,1)$$ as $\lambda \rightarrow \infty$. Obs. One is asked not ...
2
votes
1answer
62 views

Number of roots of a sequence of a uniformly convergent holomorphic functions implies an upper bound for the number of roots of their limit

Let $G$ be an open, simply connected region in $\mathbb{C}$. We define a sequence of holomorphic functions $(f_n)_{n \in \mathbb{N}}, f_n: G \to \mathbb{C}$ as almost uniformly convergent iff $(f_n)$ ...
2
votes
2answers
123 views

For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
2
votes
1answer
270 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
2answers
133 views

If $\{X_n\}$ converges in probability to $1$, where does $\{1/X_n\}$ converge to?

Without using the continuous mapping theorem, I want to show that, given $\{X_n\}$ is a sequence of random variables converging in probability to $1$, $\{1/X_n\}$ converges in probability to $1$. The ...
1
vote
5answers
160 views

Show that $\int^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$ converge.

Show that $$\int\limits^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$$ converge. I have utterly no clue on this integral. Please give me some hints. Thanks you.
10
votes
1answer
182 views

$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.

Let $(X, M, m)$ be an arbitrary measure space. Let $f_n, f \in L^1_m(X)$. Assume that $$\int_X |f_n - f| \, dm \leq \frac{1}{n^2} \text{ for all }n \geq 1. $$ Then I want to show that $f_n ...
7
votes
3answers
908 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
240 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
1answer
796 views

$C(X)$ with the pointwise convergence topology is not metrizable

I need to show that if $X$ is an uncountable Tychonoff space, then $C(X)$ is not metrizable. All I've been able to show so far is that that $F(X)$, the space of all functions with pointwise topology, ...
6
votes
4answers
386 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. ...
5
votes
1answer
319 views

Subsequence convergence in $L^p$

I recall a fact that for functions $f_1,f_2,\ldots\in L^1$ such that $\|f_n-f\|_1\rightarrow 0$ as $n\rightarrow\infty$, there exists a subsequence $f_{n_i}$ that converges to $f$ almost everywhere. ...
4
votes
3answers
112 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
1answer
158 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
2answers
557 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. ...
3
votes
4answers
93 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
1answer
139 views

Need a unique convergence (UC) space's Alexandrov extension be a UC space?

Background Say a topological space $X$ is a unique convergence (UC) space iff every sequence of points of $X$ converges to at most one point of $X$; a unique convergent clustering (UCC) space iff ...
3
votes
1answer
412 views

Prove that if $\sum a_k z_1^k$ converges, then $\sum a_k z^k$ also converges, for $|z|<|z_1|$

Show that if a power-series converges for any value of $z_{0}$ of $z$, it will be absolutely convergent for all values of $z$ whose representation points are within a circle which passes through ...
2
votes
2answers
40 views

For what values does this method converge on the Lambert W function?

Someone from another question had noted that the following statement $$W(x)=\ln\left(\frac x{\ln\left(\frac{x}{\ln\left(\frac x{\ln(\dots)}\right)}\right)}\right)$$ Can be found from the identity ...
2
votes
3answers
128 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
1answer
111 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
1answer
400 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 ...
2
votes
1answer
1k views

The series $\sum a_n$ is conditionally convergent. Prove that the series $\sum n^2 a_n$ is divergent.

Ratio and root tests won't help. And I can't use the comparison test because $ |a_n| $ is not necessarily smaller than $ n^2a_n $. Can I use limits? We know: $\lim\limits_{n \to \infty} a_n = 0 $ ...
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 ...
1
vote
1answer
109 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
1answer
335 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 ...
1
vote
0answers
76 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
2k 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
212 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
2answers
109 views

Positive Sums: Product

Lemma for: Trace Positive Sum Given the TAS $\overline{\mathbb{R}}_+$. For product sums: $$\omega:I\times J\to\overline{\mathbb{R}}_+:\quad{\sum}_{I\times J}\omega={\sum}_J{\sum}_I\omega$$ ...
0
votes
1answer
100 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
7answers
113 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 ...
9
votes
1answer
1k 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
353 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
3answers
395 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
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
356 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
215 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 ?
5
votes
1answer
188 views

Divergence of $ \sum_{n = 2}^{\infty} \frac{1}{n \ln n}$ through the comparison test?

I have shown that it diverges through the integral test, but I am curious about how this would be shown using the comparison test. I can't use harmonic series because this is lesser than it. I had one ...
5
votes
2answers
10k views

How to find the partial sum of a given series?

On my last exam there was the question if the series $\sum_{n=2}^{\infty}\frac{1}{(n-1)n(n+1)}$ converges and which limit it has. During the exam and until now, I am not able to solve it. I tried ...
4
votes
2answers
193 views

Almost Everywhere Convergence versus Convergence in Measure

I am having some conceptual difficulties with almost everywhere (a.e.) convergence versus convergence in measure. Let $f_{n} : X \to Y$. In my mind, a sequence of measurable functions $\{ f_{n} \}$ ...
4
votes
1answer
62 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 ...