# Tagged Questions

36 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 ...
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 ...
43 views

### What is $f$? Finding where a function converges pointwise?

I have a question. Let $f_n(x) = x^{4n} + \frac1{n^2}$. AS $n \to \infty$, $f_n$ converges pointwise to a function $f$ on $[0,1]$ What is $f$? Now if I am understanding correctly, couldn't ...
40 views

### Does $L^2$ strong convergence and bounded $L^\infty$ imply convergence in $L^\infty$ along subsequences?

Take a sequence of functions $f_n \in L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$ (convergence in $L^2$ norm) and $|f_n|_{L^\infty},|f|_{L^\infty} <R$ (uniformly bounded in ...
23 views

### Convergence and uniform convergence of a sequence of functions

Suppose that ($f_n$;$n \geq 1$) is a sequence of functions defined on an interval [a,b].We will say $f_n$ tends to $f$ uniformly on $[a,b]$ as n tends to infinity if for every $\epsilon>0$ there ...
20 views

### Proving that a function is a contraction map

I have a function defined by: $F(X) = a \left( \frac{X-A}{\|X-A\|_2} - \frac{B-X}{\|B-X\|_2} \right)$ with $X,A,B \in \mathbb{R}^3$ $a\in \mathbb{R}_+$. Is this a contraction map? If yes I want to ...
46 views

### Do one-sided limits exist for this functi0n?

Given a function $f$ with $f(x) := \begin{cases} \\ 0, & \text{if }x\in\mathbb{R}\setminus\mathbb{Q}\\ 1, & \text{if }x\in\mathbb{Q}\\ \end{cases}$ , what is $\lim\limits_{x\to0^+}f(x)$ and ...
51 views

### Prove $\frac{x}{1+n^2x^2}$ is uniformly convergent

I need to prove that $$f_n(x)=\frac{x}{1+n^2x^2}$$ Converges uniformly to $0$. I've tried a solution: (Scratchwork) Want ...
41 views

### Where does $f_n(x)=\frac{x^n}{1+x^{2n}}$ converge uniformly?

I'm given the function $$f_n(x)=\frac{x^n}{1+x^{2n}},$$ where I can assume $f_n:[0,\infty)\to \mathbb{R}$. I need to determine and show the sets over which the ...
66 views

### Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that convergence pointwise implies convergence in norm $\Vert .\Vert$ ? I think not because if there would be the natural ...
44 views

### Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
23 views

### Searching for a function

I'm searching for a funcion with this values: f(0)=0 f(1)=1 f(2)=3/2 f(3)=7/4 ... lim from x to infinity: f(x) = 2 I don't want the recursive way to define f.
47 views

### Need to know how: $L^{p}$ convergence for function.

I feel terrible at the moment as I have exhausted everything possible to understand this and I am still stuck. I have looked everywhere for some sort of resource to the concept at hand, and yet, they ...
88 views

### How to read $\bigcap\limits_{r=1}^{\infty}\bigcup\limits_{k=1}^{\infty}\bigcap\limits_{n,m\ge k}^{\infty}\{x:|f_n-f_m|<\frac{1}{r}\}$

Can somebody explain me step by step why does this $\bigcap\limits_{r=1}^{\infty}\bigcup\limits_{k=1}^{\infty}\bigcap\limits_{n,m\ge k}^{\infty}\{x:|f_n-f_m|<\frac{1}{r}\}$ represents the set of ...
33 views

### Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
35 views

### Convergent integral of divergent function

On one of my calculus lectures i was told that exist convergent inmproper integrals (in infinity) of divergent function. I was searching for an example in the internet, but I didn't find any. Has any ...
43 views

### Convergence rate of exponential function

If I have two exponential function, say $f_1(t)=4e^{-3t}+6e^{-7t}$ and $f_2(t)=\frac{2e^{-3t}+5e^{-7t}}{e^{-3t}+9e^{-7t}} - 2$ who are all converge to $0$. Then, the convergence rate of $f_1(t)$ can ...
86 views

### Prove that the succession $x_{n}$ where $x_{1}=1$ and $x_{n+1} = \sqrt{3x_{n}}$ is convergent

Well, ive been having a weee bit of problem solving this homework, can anyone give me a hand? Prove that the sequence $x_{n}$ where $x_{1}=1$ and $x_{n+1} = \sqrt{3x_{n}}$ is convergent and calculate ...
65 views

### What function, f(x), converges to linear function when x -> inf as well as when x -> -inf

I know that $\sqrt{1+x^2}$ does, but it is not good enough, because I need it to converge to $a+b x$ when $x\to\infty$ and $c+d x$ when $x\to-\infty$, with $a,b,c,d\in\mathbb{R}$. How to find the ...
18 views

### When does the argument of a function converge given that the image converges?

Suppose $x_n \to x$ and $f(v_n)=x_n,f(v)=x$. Under what (minimal) conditions on $f$ does $v_n \to v$?
156 views

### Mathematical Analysis help

Let $f$ be differentiable on $(0,\infty)$ and suppose $\lim_{x\rightarrow\ \infty}(f(x)+f'(x))=L$. Then, $\lim_{x\rightarrow\ \infty}f(x)= L$ and $\lim_{x\rightarrow\ \infty}f'(x)= 0$ My approach: ...
90 views

### Does convergence in $C(\Bbb R)$ imply convergence in $\mathcal D'(\Bbb R)$

Sequence of continuous functions $(f_k)$ which converge to $f$ in $C(\Bbb R)$. Does it imply that $f_k$ will converge to $f$ in $\mathcal D'(\Bbb R)$ as well? Or we need something more to make this ...
50 views

### Approximating Lipschitz funtion by $C^1$ function.

Let $G:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function and $L$ its Lipschitz constant. Suppose that $G(0)=0$. It is known that $G$ is almost everywhere differentiable and $G'\in L^\infty$ with ...
80 views

### Show uniform convergence of indefinite function series

How can i show uniform convergence on function series like this one: $\sum\limits_{k=1}^{\infty} (\sqrt{1-x^{n}}-1)$ ? I have a given interval of [0 / 0.5] I thought about using the Weierstrass ...
241 views

### Sequence of continuous functions which converges to a continuous limit [duplicate]

Any help with this: construct a sequence of continuous functions defined on $[0,1]$ which converges pointwise but not uniformly to a continuous limit ? Thank you.
92 views

### Check convergence of $f_{n}(x)=x^{n}-x^{2n}=x^{n}(1-x^{n})$

Check convergence of $f_{n}(x)=x^{n}-x^{2n}$ where $x\in(0,1)$ Please verify my answer, I'm not sure I'm doing it correctly. Thanks in advance!
194 views

### Can a sequence of functions converge to different functions pointwise and on average?

Is it possible for a sequence of functions in $\mathcal{C}\left[0,1\right]$ to converge to one function pointwise (not necessarily to a continuous function) and to a different function in average ...
146 views

### Scaling function and dilation equation in MRA

In the book I am reading, the author explains how we can use the dilation equation to obtain the scaling function through the process of iteration. In particular, we use the dilation equation, which ...
119 views

### Does this series approach an actual value?

Does this series converge to an actual number for any value of $x$? $$\sum_ {k = 1}^{\infty} \frac{\ln(k)\sin( 2\pi kx) }{ k}.$$ I tried summing the series for $x=2/3$ on wolfram alpha, and it seems ...
50 views

### Study of a series of functions

I've to study this series: $$\sum_{n=1}^\infty e^{\sqrt n\,x}$$ My teacher wrote that with the asymptotic comparison with this series: $$\sum_{n=1}^\infty\frac{1}{n^2}$$ My series converges ...
115 views

### uniformly convergent

Define functions $f_n\colon [0,1]\to\Bbb R$ by $f_n(x)=n^px\exp(-n^qx)$ where $p$, $q>0$ and $f_n\to 0$ pointwise on $[0,1]$ as $n\to \infty$ and $\sup|f_n(x)|=(n^{p-q})/e$. Assume that ...
107 views

### limit of convergent series

What is the limit of $U_{n+1} = \dfrac{2U_n + 3}{U_n + 2}$ and $U_0 = 1$? I need the detail, and another way than using the solution of $f(x)=x$, as $f(x) = \frac{2x+3}{x+2}$ because I can't show ...
688 views

### Meaning of functions that vanish at a point

I realize this may be a very thick question, but I have been wondering for some time. Sometimes I am asked to prove or read proofs involving "functions that vanish at a point" or "every point" or ...
98 views

### Problem with uniform convergence

Let $f_n$ ($n=1,2,\dots$) be a sequence of functions $f_n\colon \mathbb R\to \mathbb R$ of class $C^1$ such that $f_n \rightrightarrows 0$, $f_n' \rightrightarrows 0$. Assume moreover that functions ...
187 views

### Convergence in measure and pointwise convergence in continuity points

Hi can you help me with the following: $\{f_n\}$ a sequence of increasing functions with $f_n\to f$ in measure on $[a,b]$. Show that $f_n(x)\to f(x)$ at every $x$ where $f(x)$ is continuous. ...
178 views

### About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$

On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
102 views

192 views

### Limit of function of a set of intervals labeled i_n in [0,1]

Suppose we divide the the interval $[0,1]$ into $t$ equal intervals labeled $i_1$ upto $i_t$, then we make a function $f(t,x)$ that returns $1$ if $x$ is in $i_n$ and $n$ is odd, and $0$ if $n$ is ...