Convergence of sequences and different modes of convergence.

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2
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1answer
26 views

Example of a pointwise convergent functional sequence that is not compactly convergent.

I'm looking for an example of a pointwisely convergent functional sequence $\{f_n\}_{n \geq 0}$ (where $f_n:\mathbb{R}\to\mathbb{R}$) that is not compactly convergent. I'm not sure if it is even ...
0
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2answers
69 views

Is the Taylor series of the $f: \mathbb{R} \to \mathbb{R}$ evaluated at $0$ converges pointwise (in the whole $\mathbb{R}$)?

I would like to solve this problem: Consider the $f: \mathbb{R} \to \mathbb{R}$ function which is $n$-times differentiable for any $n \in \mathbb{N}$. Is it true that the Taylor series of ...
1
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3answers
28 views

Is this a counter example for a comparison test for sequences?

I’ve recently started learning about sequences and convergence and divergence, and I came across the comparison test for sequences. What I have is that: What if $a_n$ is defined as a periodic ...
3
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0answers
24 views

Limit of $\frac{1}{2^n}\sum_{n=0}^{2^n-1}f(e^\frac{2k\pi}{2^n}i)$ for a complex-analytic function

Let consider a Laurent series $\displaystyle{ \sum_{k\in\mathbb Z}a_kz^k }$ with complex coefficients and converging inside the annulus $A=\{\ z\in\mathbb C\ |\ r<|z|<R\ \}$, with ...
3
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2answers
62 views

How is the Radius of Convergence of a Series determined?

Consider $$\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{(n+1)^2}$$ which by the ratio test the ratio of two consecutive terms converges to $|x|$ as $n\rightarrow \infty$ and has a radius of convergence equal ...
3
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0answers
39 views

Coconvergent topology basis?

Consider the space $X=\{\frac1n:n\in\mathbb N_+\}$, with the "coconvergent topology": $$\mathcal O=\{A:(A=\varnothing)\lor(\sum_{x\notin A}x<\infty)\}$$ That is, a nonempty set is open iff its ...
1
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1answer
50 views

Convergence of $\sum_{n=1}^{\infty}(-1)^n\frac{x^n+1}{n}$

I have trouble with the following sum: $$S=\sum_{n=1}^{\infty}(-1)^n\frac{x^n+1}{n}$$ Since $a_n=\frac{2}{n}$ decreases monotonically and tends to $0$, it converges by Liebniz criterion. Then, by ...
0
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0answers
21 views

Convergence of a sequence using Banach contraction principle: $x_{n+1}=\frac{1}{x_n+a},x_1>0,a \in \mathbb{R},n=1,2,…$

$$f(x)=\frac{1}{x+a}$$ $$f:\mathbb{R}_{\le0}\rightarrow \mathbb{R}_{\le0},a<0$$ $$f:\mathbb{R}_{=0}\rightarrow \mathbb{R}_{=0},a=0$$ $$f:\mathbb{R}_{\ge0}\rightarrow \mathbb{R}_{\ge0},a>0$$ ...
1
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3answers
33 views

Prove that $\frac{2^{x+1}+(x+1)^2}{2^x+x^2}\rightarrow 2$ as $x \rightarrow \infty$

Could someone please show me the proof that $\frac{2^{x+1}+(x+1)^2}{2^x+x^2}\rightarrow 2$ as $x \rightarrow \infty$ I have no idea where to begin with this one. Thanks.
0
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3answers
34 views

Show that $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \frac{1}{2}$ as $x \rightarrow \infty$

Could someone please show me the algebraic steps in showing that $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \frac{1}{2}$ as $x \rightarrow \infty$? As the way I see it $\frac{1}{2}x^{\frac{2}{x}} ...
1
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0answers
34 views

Proving Convergence of a Derivative Free Method

Hopefully this question is on topic for the mathematics community (rather than the statistics) since the optimization method I am using relies on a statistical model. Well I am building an algorithm ...
1
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0answers
20 views

pick a sequence of parameters to preserve original convergence

Suppose for any fixed $\delta >0$, we have $$X_n(\delta) \overset{\mathbb{P}}\to 0 \quad \text{as}\ \ n \to \infty.$$ Does there exist a sequence $\delta_n \to 0$ such that $$X_n(\delta_n) ...
1
vote
1answer
41 views

Difficulties of convergence of the partial sums of the Fourier inversion formula

Define the Fourier inversion formula over $\mathbb{R}^n$ by $$ f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\hat{f}d\xi $$ where $\hat{f}$ is the Fourier transform over $\mathbb{R}^n$ $$ ...
2
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1answer
2k 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. ...
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2answers
51 views

Finding the radius of convergence for the given series.

How to find out the radius of convergence for $\sum x^{n!}$? I tried the ratio and the root test. But no luck. Kindly help !
1
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1answer
41 views

Examine the uniform convergence of the series $\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$ if $x \in [0, \infty]$

Examine the uniform convergence of the series $$\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$$ if $x \in [0, \infty)$ Which series should I choose in Weierstrass M-test to show that is divergent? ...
0
votes
1answer
39 views

For which values of $p$ does the serie $\sum\limits_{n=2}^\infty\frac{1}{n^p\ln(n)}$ converge?

For which values of $p$ does the serie $\sum\limits_{n=2}^\infty\frac{1}{n^p\ln(p)}$ converge? I'm trying to use the ratio test but I can't get a simple term in which use limit easily enough.
9
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2answers
623 views

Stone-Čech compactifications and limits of sequences

I've been working on some old prelims from my university when they used to just be on point-set topology. We don't cover a couple of the topics so I've been teaching myself some of the material, one ...
1
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1answer
16 views

existence of a sequence of continuous functions with two conditions

$\displaystyle \int_0^1 \lim_{n\to\infty} f_n(x)\,dx = \lim_{n\to\infty}\int_0^1f_n(x)\,dx $ There is no function $\,g:\left[0,1\right]\to \mathbb R\,$ lebesgue integrable such that $\,\left\lvert ...
1
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2answers
295 views

Convergent subsequences of $x_n = \sin n$ and $y_n = \cos n$…

As in title, $x_n = \sin n$ and $y_n = \cos n$. Can we find some index sequence $\{n_k\}$ such that both $\{x_{nk}\}$ and $\{y_{nk}\}$ converge? (Not necessarily to the same limit) I'm fairly ...
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0answers
33 views

Show that $S_n/n$ converges almost surely if $S_{2^n}/2^n$ converges almost surely

Let $X_n$ be independent random variables such that $\dfrac{S_n}{n}\to0$ in probability and $\dfrac{S_{2^n}}{2^n}\to0$ almost surely. Show that, $\dfrac{S_n}{n}\to0$ almost surely. Here ...
2
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0answers
40 views

Weak convergent without completeness implies strong convergence

I want to know if the following holds without completeness: In a normed linear space $H$, $x_n$ is weak convergent to $x$, and $\lim_{n\to\infty} \|x_n\| = \|x\|$ then: ...
2
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1answer
45 views

Convergence of random variables taking integer values

Let $X_n$ be random variables taking integer values, and let $X_n\to X$ in distribution. Show $X$ also takes only integer values. $P(X_n=j)\to P(X=j)$ for each integer $j$. $\displaystyle ...
2
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3answers
64 views

Convergence and Limit of a Sequence

I am looking at the sequence and I'm trying to see what happens as $n\to\infty$ $$ a_n = \frac{\prod_{1}^{n}(2n-1)}{(2n)^n} = \frac{1\cdot3\cdot5\cdot...\cdot(2n-1)}{(2n)^n} $$ By inspection, I can ...
1
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0answers
37 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
0
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1answer
18 views

Show that $S_n$ converges almost surely to $\infty$

Suppose $X_n$ are independent random variables with $P(X_n=-n^2)=\dfrac{1}{n^2}$, $P(X_n=-n^3)=\dfrac{1}{n^3}$ and $P(X_n=2)=1-\dfrac{1}{n^2}-\dfrac{1}{n^3}$ for all $n\geq2$. Let ...
1
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1answer
53 views

Proof of Product Rule for Sequences using definition of infinitesimal and properties of infinitesimal sequences.

I have been trying to understand this proof for the product rule of sequences, where the author makes use of some properties for infinitesimals, to prove this theorem. This is quite a long question, ...
3
votes
3answers
91 views

Convergence in $L_1$ and Convergence of the Integrals

Am I right with the following argument? (I am a bit confused by all those types of convergence.) Let $f, f_n \in L_1(a,b)$ with $f_n$ converging to $f$ in $L_1$, meaning $$\lVert f_n-f \rVert_1 = ...
0
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1answer
77 views

Almost sure convergence of nonrandom sample

This is a question about almost sure convergence. Consider the following set-up: There are $B$ banks. Each has size $S_{b}$, which follows a size distribution $f_{S}$ with mean E[S]. $f_{S}$ is ...
0
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1answer
54 views

How to prove a set is norm-closed?

I have to prove that the given space is 'norm-closed convex.' I proved the 'convex' part. But I don't know how to prove a set is 'norm-closed' I think I have to do the followings. Let X be a ...
2
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0answers
47 views

Positivity of an alternating series.

Greetings esteemed mathematicians. I've managed to prove that the following series \begin{equation} f_{\lambda}(\omega)= ...
0
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0answers
31 views

trouble with convergent and divergent series

Determine whether $$z_n=\left(e^{n^{2(i-1)}}\right)^{1/n}$$ is convergent or divergent? I have having trouble with this. How to proceed with this?
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0answers
108 views

Is $1+2+3+4+\cdots=-\frac 1{12}$ true? [duplicate]

Hello (it's my first post here!), I have a strange question. I heard that (under certain conditions): $$ 1+2+3+4+\ldots=\sum_{k=1}^{\infty}k=-\frac{1}{12} $$ Is it REALLY true? And - if yes - how to ...
2
votes
4answers
71 views

Uniqueness of a Limit epsilon divided by 2?

I have been reading about this theorem in a book called 'Calculus: Basic Concepts for High-schools', it is a very good book (so far) and I can highly recommend it. Well the author goes on to prove ...
2
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0answers
53 views

Prove that this infinite product converges uniformly,

Let $(a_n)_{n=1}^\infty$ be a sequence of complex numbers such that (i) $0<|a_n|<1$, (ii) $\sum_{n=1}^\infty(1-|a_n|)<\infty$ Prove that the infinite product $$\prod_1^\infty \frac{(a_n ...
1
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1answer
49 views

Check the convergence (& absolutely) of parametric integral [closed]

$$\int\limits_{-1}^{1} \left(\frac{1+x}{1-x}\right)^{\alpha} \ln(2+x)dx$$ Don't know where to start..
1
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3answers
37 views

Is convergence in probability sometimes equivalent to almost sure convergence?

I was reading on sufficient and necessary conditions for the strong law of large numbers on this encyclopedia of math page, and I came across the following curious passage: The existence of such ...
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0answers
39 views

Convergence and limit of a sequence $x_n=1+\frac{x^2_{n-1}}{2},n\ge2,x_1=\frac{3}{8}$

$x_{n+1}-x_n=\frac{x^2_n-2x_n+2}{2}>0$ sequence is increasing. I don't know how to prove that it is bounded. Limit should be $\frac{1}{2}$
0
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1answer
29 views

A question on convergent series of positive terms

Let $\sum a_n$ be a convergent series of positive terms ; then we know $\lim \inf (na_n)=0$ ; can we derive from here that if $\{a_n\}$ is decreasing , then $\lim (na_n)=0$ ?
0
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1answer
22 views

Completeness of a system: For all $n$ and within an interval?

Why is the system $\sin((2n-1)x)$ for $n=1,2,\cdots$ complete in $L^2[0,\frac\pi2]$? This means that the Euclidean norm converges for $n=1,2,\cdots$ and for all $x\in[0,\frac\pi2]$ How does one prove ...
0
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1answer
130 views

Fractional-Recursive Sequence

Here from the fraction set we have a really hard question to be answered...suppose that a sequence is defined as $a_{n} = a_{n-1} - \dfrac 1{a_{n-1}}$, where $a_0$ is given. ...you already know what ...
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2answers
25 views

Convergent series? The pairwise product of the harmonic series with a nonnegative, diminishing to zero, divergent series

Suppose we have a sequence $\{\alpha_n\}$ such that $\alpha_n \ge 0$, $\sum_n \alpha_n = \infty$ and $\alpha_n \rightarrow 0$. Is it true that $\sum_n n^{-1} \, \alpha_n$ converges?
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0answers
52 views

How to prove that this series converges. [duplicate]

I want to prove that this series converges: \begin{equation} 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+... \end{equation} I normally use this one as a standard series to test other series for their ...
13
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8answers
1k views

Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges

I need to prove that the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges. I do not have to find the limit. I have tried to prove it by proving that the sequence is monotone ...
0
votes
1answer
31 views

Example of each $\{f_n\}$ Riemann integrable such that $\sum f_n$ converges point-wise to $f$ which is not Riemann-integrable

I am looking for an example of a sequence $\{f_n\}$ of real valued Riemann integrable functions on a closed bounded interval such that $\sum f_n$ converges point-wise to a function $f$ which is not ...
2
votes
2answers
69 views

Relation between convergences in $L^{p}$ for probability spaces.

I have read that for a probability space $(\Omega,\Sigma,P)$ it is true that $f \in L^{p}(\Omega,\Sigma,P)$ implies $f \in L^{q}(\Omega,\Sigma,P)$ if $p>q$, and hence $L^{2} \subset L^{1}$. I'm ...
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5answers
801 views

Why does L'Hôpital's rule work for sequences?

Say, for the classic example, $\frac{\log(n)}{n}$, this sequence converges to zero, from applying L'Hôpital's rule. Why does it work in the discrete setting, when the rule is about differentiable ...
4
votes
2answers
142 views

Why doesn't this work for Rudin Exercise 3.8

The problem is 3.8 exercise in baby Rudin: If $ \sum{a_n} $ converges and $\{b_n\}$ is bounded and monotonic, prove that $\sum{a_nb_n}$ converges. Why can't I just do this?: Let $M$ be an ...
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5answers
49 views

Help with proof that $\sum_{n \in \Bbb{N}} \frac{1}{an + b}$ also diverges?

We know that $\sum_{n \in \Bbb{N}} \frac{1}{n}$ diverges. So it seems likely that $\sum_{n \in \Bbb{N}} \frac{1}{a n + b}$ will for any real $a, b$. I'm having trouble proving it just for the ...
4
votes
5answers
102 views

Prove that $\sum\frac{n+1}{(n+2)n!}$ converges

Show that $\displaystyle\sum\limits_{n=1}^{\infty}\frac{n+1}{(n+2)n!}$ converges, using the integral test. I noticed that $\displaystyle\sum\frac{n+1}{(n+2)n!} = \sum\frac{(n+1)^2}{(n+2)!}$, but ...