Convergence of sequences and different modes of convergence.

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2
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2answers
50 views

Convergence of integrals but $\int_a^b|f_n(x)-f(x)|dx$ does not converge to $0$

Can I have an example of a sequence of continuous functions $(f_n)_n$ and a continuous function $f:[a,b]\to \mathbb{R}$ such that $$\int_a^bf_n(x)dx\to\int_a^bf(x)dx,$$ when $n\to+\infty$, but ...
1
vote
1answer
41 views

Interval of convergence

Find the interval of convergence of $\sum_{n=1}^{\infty }\frac {n^{2n}}{(2n)!}x^{n}$ I use ratio test and i found $\lim_{n\to\infty}|\frac{a_{n+1}}{a_{n}}|<1$ iff ...
3
votes
1answer
31 views

Newton Rhapson Algorithm Accuracy

I read somewhere that the NR algorithm in general (given an appropriate initial value) increases in accuracy by roughly two decimal places per iteration. Is this something that can be proven, or is ...
3
votes
0answers
46 views

extending the convergence of measures

Let $F$ the real vector space of all applications $\phi: X \times Y \rightarrow \mathbb{R}$ where $(X,\mathcal{B}_1, \mu)$, $(Y,\mathcal{B}_2 )$ measurable spaces with $X$ and $Y$ are compact metric ...
0
votes
2answers
22 views

Series Ratio Test Convergence

$\displaystyle\sum_{k=1}^\infty (2k)!/k!(k+1)!$ Let $a_k = (2k)!/k!(k+1)!$ $\lvert a_{k+1}/a_k\rvert \to 4$ as $k \to \infty$ Thus the series is divergent. Can someone double check ... my gut says ...
5
votes
3answers
72 views

Convergent Sequence, mean of previous three numbers

I am given three real numbers $x_0, x_1, x_2$ and the next number in the sequence is defined as the mean of the three real numbers. So: $$x_a=\tfrac{1}{3} \cdot (x_{a-3} + x_{a-2} + x_{a-1})$$ I am ...
1
vote
1answer
22 views

Bijective transformation of $L^2$-weak convergence sequence again weak converging?

Let $f_n$ converge weakly in $L^2(x)([0,1])$ to $f$, with $|f_n(x)|\leq C$ for almost all $x\in]0,1]$ and all $n$. Let $H:R\rightarrow R$ be strong monotone increasing and continuous with $H(0)=0$. ...
1
vote
1answer
45 views

Does weak convergence in $L^2$ implies convergence almost everywhere along subsequence?

If I know $\int_{[0,1]} f_{n}(x) g(x) dx \rightarrow \int_{[0,1]} f(x) g(x) dx$ as $n \rightarrow \infty$ for all $g \in L^2([0,1])$ (weak convergence in $L^2$) and $|f_n(x)|_{L^2} <C$ (uniformly ...
0
votes
0answers
31 views

A variant of the Riemann Integral

This question is related to this one. Let $S_k$, $k\in\mathbb{N}$ be a sequence of finite sets where $S_k\subset S_{k+1}\subset[0,1]$. Fixed $s$ in $S_k$, let $s'$ denote the predecessor of $s$ in ...
2
votes
0answers
31 views

Weakly closed convex set and norm convergent

Let $X$ be a normed space and let $\{x_n\}$ be a sequence in $X$ such that $x_n\to x$ weakly. Show that there is a sequence $\{y_n\}$ such that $y_n\in \operatorname{co}\{x_1,...,x_n\}$ and ...
0
votes
1answer
39 views

A counter-example to Dini's Theorem (after removing a hypothesis)

Recall Dini's Theorem: Let $K$ be a compact metric space. Let $f: K\to\mathbb{R}$ be a continuous function and $f_{n}: K\to\mathbb{R}$, $n\in\mathbb{N}$ be a sequence of continuous functions. ...
2
votes
3answers
109 views

Calc 2: convergent of divergent sequences

I would like to know if this sequence, $\sin\left(\frac{n \pi}{2}\right)$ ,is convergent or divergent? I have done this problem and I know that it is divergent through oscillation (I'm pretty sure). ...
2
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0answers
23 views

Preliminaries of the Martingale Representation Theorem

I cannot understand why we are taking a dense subset of $[0,T]$. Furthermore, I cannot see a result that would allow each such $g_n(B_{t_1},\ldots,B_{t_n})$ to be approximated in ...
0
votes
1answer
27 views

$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor\log_2n\rfloor}}{\lfloor\log_2n\rfloor+1}(x-x_0)^n$ convergence/divergence

I have a problem with determining whether these series are convergent/divergent at the endpoints of their radii of convergence. None of the tests or approaches I know seems to by applicable here... ...
1
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0answers
63 views

Backward from infinity?

The following question has been raised and answered lately: Problem 6 - IMO 1985 Please take a look at the Reverse method part of the answer given by this author. What's happening there is that we ...
1
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0answers
31 views

Recursive Least Square (RLS) algorithm for regression. [closed]

What is meant by convergence of recursive least square (RLS) in online regression ?
-1
votes
1answer
16 views

Sequence of Partial Sums for repeated decimal

I have been trying to figure out an explicit formula for the sequence of partial sums of a repeating decimal. Take 0.09 repeating for example. Using the fact that it is a geometric series with r < ...
0
votes
1answer
40 views

How to deal with discontinuous points when proving that step functions are dense in $PC[a,b]$

This question is a follow-up to my previous question: How does one prove that a space is dense in another under some norm? I figured out a way to solve (part of) the exercise. Given some function ...
3
votes
1answer
65 views

Fourier series that converges in $L^2$ but not pointwise

I've read this in my notes Let $f:\mathbb R\to \mathbb R$ be $2\pi$-periodic, and piecewise continuous with jump discontinuities such that $\displaystyle f(a)=\frac{1}{2}\frac{f(a^+)-f(a^-)}{2}$ . ...
11
votes
1answer
241 views

Interchanging pointwise limit and derivative of a sequence of C1 functions

Assume the following: $f_n$ is a sequence of $C^1$ functions on $[0,1]$ $f_n(x) \rightarrow 0$ pointwise. $f'_n(x) \rightarrow g(x)$ pointwise. Is it true that $g(x) = 0$ almost everywhere? I think ...
2
votes
1answer
25 views

Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
2
votes
2answers
36 views

Closed subspaces of $L^2(0,1)$

I would like to prove that the almost-everywhere constant functions, and the functions whose integral is 0 are closed subspaces of $L^2(0,1)$. It's readily seen that they are subspaces. I'm finding ...
14
votes
5answers
158 views

For $a_n,b_n\uparrow$ and $\sum \frac{1}{a_n}$, $\sum \frac{1}{b_n}$ divergent is the series $\sum \frac{1}{a_n+b_n}$ also divergent? [duplicate]

Let $a_n$ and $b_n$ are strictly increasing to $+\infty$ sequences such that the series $\sum \frac{1}{a_n}$ and $\sum \frac{1}{b_n}$ are divergent. Is it true that the series $\sum \frac{1}{a_n+b_n}$ ...
1
vote
2answers
111 views

Why does the sequence ${1/n}$ not converge in the positive reals?

I'm reading Baby Rudin at the moment and it claims something remarkable. Consider the sequence $$ x_n=\frac{1}{n}.$$ The book claims that this converges to zero in the reals: $$\lim_{n\to\infty} ...
0
votes
2answers
57 views

Convergence Proof Problem $\epsilon, n_0$ proof

I need help proving that for the following sequence it will converge to the given limit $p $using an $\epsilon$ $n_0$ argument (i.e given $\epsilon>0$, determine $n_0$ such that $|p_n - p| < ...
0
votes
1answer
28 views

Equivalence of weak-* convergence in Banach spaces.

Let $X$ be a Banach space and $f,(f_n)_{n \in \mathbb N} \in X^*$. $f_n \xrightarrow{w^*} f$ if and only if a) $\sup_{n \in \mathbb N} \|f_n\| < \infty$ and b) $\exists A \subset X: ...
0
votes
0answers
34 views

Limit of measurable function is measurable

This question has been asked already here but I didn't get a satisfactory solution and didn't want to bring up an old question. Here is the question : Let $\{f_n\}$ be a sequence of measurable ...
1
vote
1answer
55 views

Extreme Value Theory - Show: Normal to Gumbel

The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have $$P(\max X_i \leq x) = ...
3
votes
1answer
30 views

How to prove the sequence $\{b_n\}$ with $0 \leq b_{n+1} \leq b_n + a_n$ converges, where $a_n \geq 0$ converges to $0$.

Let $\{a_n\}$ be a sequence of non-negative real numbers that converges to zero. Suppose the sequence $\{b_n\}$ of non-negative real numbers satisfies \begin{equation} 0 \leq b_{n+1} \leq b_n + a_n ...
1
vote
1answer
63 views

Show that convergence in the mean implies convergence of the means [closed]

Question: Let $X_n$, n = 1,... denote a sequence of real-valued random variables; $X_n$ is said to converge in mean if $\hspace{20mm}$$$\lim_{n\to\infty} E[|X_n-X|] = 0$$ Show that if $X_n$ ...
3
votes
2answers
251 views

Sequence problem

I have a calculus final two days from now and we have a test example. There's a sequence question I can't seem to solve and hope someone here will be able to help. With $a_1$ not given, what are the ...
11
votes
1answer
83 views

Convergence of $\sum_{n=1}^{+\infty}\frac{(-1)^{f(n)}}{n}$ where $f(n)$ is the number of prime divisors

Let $f(n)$ be the number of prime divisors of a number $n$ counted with their multiplicities. Show that the series $\sum_{n=1}^{+\infty}\frac{(-1)^{f(n)}}{n}$ converges and has sum $0$. Attempt ...
0
votes
1answer
14 views

Does this function go to zero faster than the norm of its argument?

Assume $f:\mathbb R^2\to\mathbb R$ is such that for all $\varepsilon>0$ exists $\delta>0$ such that, whenever $||x||<\delta$, also $||f(x)||<\varepsilon^2$. Can we see that $f$ is ...
0
votes
1answer
25 views

a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
1
vote
1answer
39 views

How to prove convergence

$a_n = n!/2^n$ Since $n!/2^n = 1/2 * 2/2 * 3/2 * ... * (n-1)/2 * n/2$ $1/2 <= n!/2^n$ I am having trouble finding an upper bound for the sequence. Any hints or help is much appreciated in ...
2
votes
1answer
34 views

Proof of Sequence Convergence

$a_n = \ln(n+1) - \ln(n)$ Consider the function $f(x) = \ln(1+ 1/x)$ As $n \to \infty$, $\ln(1+ 1/x) \to \ln(1) = 0$ Thus as $n\to \infty$, $a_n \to 0$ Hence the sequence is convergent. Does this ...
0
votes
1answer
40 views

Limit of recurrence relation $x_k= f(x_{k-1})$ for an increasing function $f:[0,1]\to[0,1]$

Is there a standard result out there that gives the following? It looks graphically like it must be true, but I'd like to appeal to a known result if possible. Let $f:[0,1]\to[0,1]$ be a continuous, ...
4
votes
4answers
152 views

Convergence of $\left(1+\frac{1}{n}\right)^n$, given that $\left(1+\frac{1}{2^n}\right)^{2^n}$ converges

I know how to prove that the sequence $$\left(1+\frac{1}{2^n}\right)^{2^n}$$ has a limit. Can I use this knowledge to quickly get the fact that $$\left(1+\frac{1}{n}\right)^n$$ also has a limit?
0
votes
2answers
46 views

if $\dot{x}<0$ and $\dot{|x|}<0$, what can we say about the convergence of $x$?

we have $\dot{x}=\frac{dx}{dt}<0$ and $\dot{|x|}=\frac{d|x|}{dt}<0$, that is: the derivative of $x$ is negative, the derivative of the absolute value of $x$ is negative, and $t$ is time. what ...
0
votes
1answer
40 views

Let $Y=\sup_n|X_n|$, Show that $Y<\infty$ a.s.

Suppose $\lim_{n\to\infty}X_n=X$ a.s. and $|X|<\infty$ a.s. Let $Y=\sup_n|X_n|$, Show that $Y<\infty$ a.s. If $\lim_{n\to\infty}X_n=X$ a.s. then $S_1:=\{w:\lim_{n\to\infty}X_n=X\}$ has ...
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vote
2answers
57 views

sequential continuity vs. continuity

A short and hopefully simple question for someone with more experience in topology: If a topology is induced by a mode of convergence and in fact nothing more is known apriori, wether if this topology ...
6
votes
1answer
104 views

The sequences $x_n$ and $y_n$ converges

Let $\quad2{x}_{n+1}=1+{y}^{2}_{n},\quad 2{y}_{n+1}=2{x}_{n}-{x}^2_{n},\quad n\in\mathbb{N};\quad 0\leq {y}_{0}\leq \frac{1}{2}\leq {x}_{0}\leq 2.$ Prove that the sequences $ \begin{Bmatrix} ...
0
votes
1answer
20 views

Convergence in probability of a random variable

I need to prove that $(X_n^2 -X)^2\to 0$ in probability $\Rightarrow X_n^2\to X$ in probability. I tried solving it with the triangle inequality, but it didn't get me anywhere. Is there another ...
2
votes
5answers
173 views

Is integral convergent?

I have a problem with following integral: $\int_1^\infty \frac{\sqrt{x}}{1+x} \sin(2x)dx$ I was trying to prove convergence (or divergence) of this integral, however without any success. My best ...
5
votes
0answers
110 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
0
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0answers
31 views

Tensor Product of Hilbert Spaces: incomplete?

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{H}\otimes_0\mathcal{H}$ its algebraic twofold tensor product. Define a scalar product on it as ...
2
votes
4answers
61 views

Prove convergence of sequence defined recursively $a_{n+1} = \sqrt{6+a_n}$ [duplicate]

We have a sequence: $$a_1=\sqrt{6}$$ $$a_{n+1} = \sqrt{6+a_n}$$ The problem is to check convergence and then find the limit. We know that sequence converges when it is monotonic and bounded. With a ...
0
votes
1answer
26 views

Finding initial value to get a convergent sequence

Assume that a sequence of real number $\begin{Bmatrix} {S}_{n}\end{Bmatrix}$ satisfes: ${S}_{1}=b;\quad {S}_{n+1}={S}^{2}_{n}+\left(1-2a \right){S}_{n}+{a}^{2}(n\in \mathbb{N});\quad a,b\in ...
2
votes
1answer
38 views

Relations between convergence in nets and topologies.

I want to prove that given a net $S$ and topologies $T$ and $T'$, then $T\subset T'\iff$ when $S$ is convergent in $T'$ is also convergent in $T$. I'm proceeding this way: First I'd like to show that ...
3
votes
1answer
52 views

Will Fourier Series converge even if you only use Prime Integer frequencies?

So there is a Fourier Series for a function $f(x)$ with period $P$: $$ f(x) = \frac{A_0}{2} + \sum_{n=1}^{N} A_n \cdot \cos \left(\frac{n 2 \pi x}{P} + \phi_n \right) $$ Let $\frac{2 \pi x}{P} = t$ ...