Convergence of sequences and different modes of convergence.

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Given $\phi \in C^{1,b}(R)$, find $\phi_n$ countably piecewise affine functions whose derivatives converge to $\phi'$ uniformly where differentiable

Let $\phi \in C^{1}(\mathbb R)$ with bounded derivative. I am trying to build $\phi_n$ a sequence of countably piecewise affine functions, s.t. $\phi_n'$ converges uniformly to $\phi'$ on $N^c$, where ...
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1answer
18 views

Conditions for convergence of derivatives from pointwise convergence

Let $\{f_n\}_n$ be a sequence of functions $\mathbb R \rightarrow \mathbb R$ which converges pointwise to $f$, ie: $$f_n(x) \rightarrow f(x) \hspace{10pt}\hbox{for all $x$}.$$ What additional ...
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1answer
52 views

How is this trival?

I was reading an article today and on section 2 it is indicated that if we are given a Radon Measure $\mu$, and a real $p$ then fast convergence entails trivially almost sure convergence, where fast ...
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1answer
22 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 ...
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3answers
25 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 ...
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1answer
40 views

Convergence of an integration $t=\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy$

When I am reading Brian Hall's "Quantum Theory for Mathematicians", I came across an integration (frequently appeared in physics textbooks) $$t=\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy.$$ The ...
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0answers
33 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 ...
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1answer
47 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 ...
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20 views

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

$$f(x)=\frac{1}{x+a}$$ $$f:\mathbb{R}_{\le0}\rightarrow \mathbb{R}_{\le0},a<0$$ $$f:\mathbb{R}_{=0}\rightarrow \mathbb{R}_{=0},a=0$$ $$f:\mathbb{R}_{\ge0}\rightarrow \mathbb{R}_{\ge0},a>0$$ ...
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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 ...
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3answers
31 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.
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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}} ...
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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 ...
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0answers
18 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) ...
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2answers
67 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 ...
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1answer
40 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 $\mathbb{R}^n$ $$ ...
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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 ...
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1answer
37 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? ...
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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.
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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 ...
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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 ...
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0answers
34 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: ...
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0answers
36 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 ...
2
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1answer
43 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 ...
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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 ...
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73 views

When $\lim_{n\to\infty} \mathbb P(A_n) = \mathbb P(\lim_{n\to\infty} A_n)$? [closed]

When is the following statement true for a sequence $(A_n)_{n\in\mathbb N}$ of events: $$\lim_{n \to \infty} \mathbb{P} (A_n) = \mathbb{P} \left( \lim_{n \to \infty} A_n \right)?$$
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47 views

Positivity of an alternating series.

Greetings esteemed mathematicians. I've managed to prove that the following series \begin{equation} f_{\lambda}(\omega)= ...
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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 ...
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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 ...
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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, ...
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1answer
73 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 ...
2
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3answers
62 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 ...
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4answers
68 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 ...
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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
37 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 $\frac{1}{2}$
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1answer
27 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 $\lim (na_n)=0$ ?
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convergence in iteration solution [closed]

I use an iteration method to solve a spin system in different interaction, but in some values it can not converged at least after 1 day. I have 16 variables and it used values of K th step to compute ...
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1answer
20 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 ...
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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
51 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 ...
0
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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 ...
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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 ...
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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 ...
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5answers
790 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 ...
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2answers
33 views

locally uniform convergence vs pointwise convergence

I am finding lots on here about 'uniform convergence vs pointwise convergence' of a function but not the comparison for local uniform convergence. It somehow intuitively seems to me that pointwise ...
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3answers
60 views

Prove that $\sum\frac{(\log n)^2}{n^3}$ converges

This question is from Serge Lang's textbook, in a chapter that comes before the ratio and integral tests are introduced, so those can't be used. I've already proved that $\sum\frac{\log n}{n^3}$ ...
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2answers
51 views

Find $\sum_{n=1}^{\infty}a_nx^n$ given $a_0=3, \ 3na_n+3(n-1)a_{n-1}=2a_{n-1}$

Given $\ a_0=3$, $\,3na_n+3(n-1)a_{n-1}=2a_{n-1}$, find $\ f = \sum_{n=1}^{\infty}a_nx^n$. I have proved that when $\ \left\lvert x \right\rvert<1$, this exponential series function is convergent. ...
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1answer
55 views

To be or not to be Banach? That is the question.

On the set $H^1_0((0,2))$ we put the following norms. $$\|u\|_a^2= \int_{[0,2]}(u')^2.$$ $$\|u\|_b= \|u\|_\infty.$$ $$\|u\|_c= \|u\|_{L^2}.$$ Is $H^1_0((0,2))$ Banach with any of these norms?
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2answers
37 views

Counter Examples for Dominated Convergence Theorem and Fatou's Lemma

Is there an example to see why the dominated convergence theorem fails when there is no integrable function dominates the sequence $f_n(x)$? Also for Fatou's lemma, is there an example where the ...
0
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1answer
29 views

How to show that $L^p$ norm is monotone increasing?

I am trying to solve the following (very standard) exercise: Let $(X,\mathcal M,\mu)$ be a measure space and $f\in L^r\cap L^\infty$ for some $1\leqslant r<\infty$. Then $f\in L^p$ for ...