Questions tagged [sequence-of-function]

Use this tag only when your query is about sequences of functions. Don't use this tag for any other sequence such as sequences of real numbers or sequences of complex numbers etc.

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22 views

Equivalence of sets about sequence of real valued function

Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of real-valued functions defined on $\mathbb{R}$ and let $f : \mathbb{R} \to \mathbb{R} .$ Let $\epsilon > 0.$ Define $E_n( \epsilon ) = \{x \in R : |f_{...
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34 views

A term that is not bounded by 1.

$f_{n}(x)= \frac{n^3 x^{3/2}}{ 1 + n^4 x^2}\}$ I think when $0 < x <1$ this expression is bounded by 1, am I correct? if so how can I prove it? My trials: At $x = 1,$ the following fraction $\{...
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1answer
22 views

Proving that the pointwise limit of a sequence of function is 0. [on hold]

$$f_{n}(x)=\frac{n^3 x^{3/2}}{ 1 + n^4 x^2}.$$ I know that the pointwise limit is zero but I do not know how to prove it, could anyone help me in this please?
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18 views

Finding a direct function for $f_n = f_{n-1} + (x-a_{n-1})^2$ where $a_{n-1}$ is root of $f_{n-1}$ closest to $\alpha$ and $f_1 = x-a$.

Problem Suppose $f_1 = (x-a)$ and $f_n = f_{n-1} + (x-a_{n-1})^2$ where $a_{n-1}$ is a root of $f_{n-1}$ closest to $0$. Is there a function for $f_n$ that only depends on $x$? Examples As a first ...
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12 views

How to find all accumulation points of a sequence

First of all, I had I apology for snipping the equation out from my worksheet instead of writing it down. I'm new here. sequence I need to find all accumulation points I tried to solve it by hand, ...
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1answer
29 views

Show that a sequence converges in the $d_\infty$ metric.

Recall the $d_\infty$ metric on the set of continuous functions $C([0,1],R)$: $d_\infty(f,g) = \max\{|f(x)−g(x)|:x\in[0,1]\}$ Consider the sequence in $C([0,1],R)$ defined by $$f_n(x) = \frac{1}{(n^...
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1answer
55 views

Topology of set-theoretic limits

The convergence of a sequence of subsets of a fixed set $\Omega$ is defined in-terms of set-theoretic limit which does not use any metric or topology on $2^{\Omega}.$ Now I wonder: What is the ...
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85 views

Is there a short form of a polynomial function applied to itself $i$-times?

Given a polynomial $f$ of degree $m$: $$f(x)=\sum_{j=0}^{m} a_jx^j$$ Now this polynomial is applied to itself $f(f(f(f...(f(x))))$ for $i$($-1$) times $->f^i(x)$. The resulting function is a ...
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21 views

Convergence of the sequence of derivatives of a smooth function [closed]

Let $f$ be an infinitely differentiable function (real or complex valued). We can then define a sequence of functions $f_n := f^{(n)}$. What do we know about the convergence of this sequence? If the ...
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25 views

Given a sequence $n_{i+1} = f(n_i)$ with $f$ a polynomial of degree m. Any way to get the index $i$ of a sequence element $v$?

A sequence $$n_{i+1} = f_m(n_i)$$ with $f_m$ a polynomial of degree $m$. With this also $n_i$ is a polynomial: $$n_i=\sum_{j=0}^{m\cdot i} a_jx^j$$ If $n_1$ has polynomial degree $m$ then the ...
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3answers
35 views

About convergence in distribution

Suppose that a sequence of real valued random variables $(X_n)_{n\geq 1}$ converges in distribution to a random variable $X$. Does the couple $(X_n, X)_{n\geq 1}$ converge in distribution to $(X, X)$?
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1answer
18 views

Decreasing sequence of Lebesgue measurable functions that converge $0$ and that don't converge in mean

Does there exist a decreasing sequence of Lebesgue-measurable non-negative functions $(f_n)_{n}$ such that $f_n \to 0$ pointwise on $\mathbb{R}$, but $f_n$ does not converge to $0$ in mean, by which I ...
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39 views

Computing the limit $\lim_{n\to\infty} (n^3x^{3/4})/(1+n^4x^2)$

Find $$\lim_{n \rightarrow \infty} \frac{n^3 x^{3/4}}{ 1 + n^4 x^2}.$$ The overall goal is to find the uniform limit of a sequence of functions, or show that the sequence does not converge uniformly....
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39 views

Sequence of continuous functions on [0,1] converging to unbounded function.

I'm working on problem 3 from Exercises 2E of Axler's "Measure, Integration and Real Analysis" which says: I thought of something like $f_n (x) = \frac{1}{x + \frac{1}{n}}$ , but then this converges ...
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52 views

Determine if a sequence of functions is uniformly integrable

Let {$f_n$} be a sequence of continuous functions on $[0,1]$, and $|f_n|\le1$ on $[0,1]$. Is {$f_n$} uniformly integrable over $[0,1]$? For the same sequence, if we assume {$f_n$} is integrable and $\...
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21 views

Why the convergence is pointwise in this question and why the translation is continuous?

I was reading this question here Proving that $f'$ is measurable on $\mathbb R$ if$f$ is differentiable on $\mathbb R$. But I did not understand the following points in the solution: 1- why the ...
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139 views

Evaluate the limit $\lim\limits_{n\to0}\frac{(x)+(2x)+\cdots (nx)}{n^2}$

Find the limit of $\lim_{n\rightarrow ~0}\frac{(x)+(2x)+\cdots (nx)}{n^2}$, where, $(x)=x-[x]$ and $[x] $ is the greatest integer function(the fractional part function). I feel, as $n \rightarrow 0$ ...
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37 views

If $f_n \rightarrow f$ pointwise and $f_n \rightarrow f'$ uniformly, then does f = f'?

I'm motivated to ask this question because of the following question. Consider the sequence $\{f_n\}$ defined on $[0, \pi]$ by $f_n(x) = \sin^n(x)$. Show that $\{f_n\}$ converges pointwise. Use the ...
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22 views

A bounded sequence in $W^{1,1}((0,1))$ has no converging subsequence in $L^\infty ((0,1))$

I am doing the exercise 8.2 from the brezis' book of Functional Analysis and PDEs, And I'm having trouble with the second question, which says: Construct a bounded sequence $(u_n)$ in $W^{1,1}((0,...
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1answer
26 views

Any (convergent) sequence of functions in $\mathcal L^p$ that are bounded by a function $w\in\mathcal L^p$ converge in $\mathcal L^p$?

I know it's true that if $(u_j)_{j\in\mathbb{N}}$ is a sequence of integrable functions in $\mathcal L^1$, $u_j\rightarrow u$ and $|u_j|\leq w, \forall j\in\mathbb N$ for some $w\in \mathcal L^1$ then ...
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1answer
19 views

General Formula for a sequence with indices increasing in the order $(2 * i)$, where i = 0,1,2,3…

I have generated a sequence which looks really simple : $f(n) = 0, 1, 1, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 7, 7, 15 ....n$ The indices are having the same values in the order of $(2*i)$ where $i = 0,1,2,...
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1answer
26 views

Proving a sequence of functions converging pointwise doesn't converge uniformly on $(0,\infty)$ but does in $[a,\infty)$ where $a>0$

Let $f_n : (0,\infty) →\mathbb{R} , \,f_n(x) = \frac{1}{nx}$: i) Prove that the $\lim_{n\rightarrow \infty} f_n(x)$ exists $\forall x > 0$. ii )Prove that the convergence is not uniform in $(0,\...
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1answer
43 views

Turning a continuous everywhere differentiable nowhere function into a smooth function by infinitely many times definite integration?

Let $W(x)$ be a real-vlued function defined on a (possibly infinite) interval $\text{T}\subseteq\mathbb{R}$ containing $0$ that is continuous everywhere differentiable nowhere on $\text{T}$. Define ...
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1answer
35 views

Null sequences and uniform convergence

Def. 1. $\phi:\mathbb{R}^n\to \mathbb{R}$ is called test fuction if $\phi$ is infinitely differentiable ($\phi \in C^{\infty}(\mathbb{R}^n)$) and $\phi$ has compact support (ie the closure of the set $...
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37 views

Does this sequence of functions converge uniformly

Let $h_n : [0,1] \rightarrow \mathbb{R}$ Such that $h_n(x) = 1 − nx$ if $0 \leq x \leq \frac{1}{n}$ and $h_n(x) = 0$ if $\frac{1}{n}$ if $\frac{1}{n} \leq x \leq 1.$ Determine if $h_n$ converges ...
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67 views

Convergence of composition of sequence of functions on a compact subset of $\mathbb{R}$.

Let $\lbrace f_n\rbrace$ and $\lbrace g_n\rbrace $ be two sequences of real-valued continuous functions on a compact subset of $\mathbb{R}$ such that they converge pointwise to $f$ and $g$ ...
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2answers
81 views

Prove that $\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$ is equal to $f(1)$.

$\mathbf{Question}:$ Let $f$ be a continuous function on $[0,1]$. Then prove that the limit $\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$ is equal to $f(1)$. $\mathbf{Attempt}$: First, we try to show ...
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1answer
40 views

On Cartan's remark about interchange of summation and limit

I am reading Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables. On p.18, he wrote: Recall that the limit of a uniformly convergent sequence of continuous ...
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37 views

Divergence of a sequence.

In an example in the book, Thomas Calculus 14e: Q: Show that sequence $\{(-1)^{n+1}\}$ diverges? A: They choose $\varepsilon$ to be $1/2$ and thus, $|L-1|<1/2$ for $+1$ and $|L+1|<1/2$ if ...
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204 views

Limit and Integration are interchangeable

Let $f_n:[0,1]\to\mathbb{R}$ be defined by $f_n(x)=\frac{n+x^3\cos x}{ne^x+x^5\sin x}$, $n\geq 1$. Find $\lim_{n\to\infty}\int_0^1f_n(x)dx$. Approach: Here I found the limit function $f(x)=e^{-x}$. ...
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1answer
18 views

Pointwise Convergence Of Sequence Of Real Functions [duplicate]

Let $f_1:[0,1]\to\mathbb{R}$, $f_1(0)=0$ be continuously differentiable function and $\lambda>1$. Consider the sequence of function defined by $f_k(x):=\lambda f_{k-1}(x/\lambda)$, $k≥2,$ , $x\in [...
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1answer
34 views

Study the uniform convergence of the functional sequence $f_n(x)=\sqrt{n+1}\sin^nx\cos x$

Study the uniform convergence of the functional sequence $f_n(x)=\sqrt{n+1}\sin^nx\cos x$ I found the limit $$\lim_{n\to \infty}f_n(x)=0$$ The solution in the book it says that since $$d_n=\sup_{...
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1answer
36 views

$f_n(x) \rightharpoonup0$ in $L^1(\mathbb{R})$ where $f_n(x) = f(-x+n^3)$

Let $f \in C_c^0(\mathbb{R})$ be a continuous function with compact support. Is it true that $f_n(x)=f(-x+n^3)\rightharpoonup0$ in $L^1(\mathbb{R})$? $f_n(x)=f(x+\frac{1}{e^n})\rightharpoonup0$ in $L^...
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32 views

Uniformly bounded sequence of non-decreasing functions has a convergent subsequence

I came across the following problem in an old notebook. Let $f_n:\mathbb{R}\to [0,1]$ be a sequence of non-decreasing functions, that is, $f_n(x)\le f_n(y$ whenever $x\le y$ for all $n$. Show that ...
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2answers
61 views

Does $(f_n)=(n\sin(\frac{x}{n})-x)$ converge uniformly on $[-a,a]$ for $a\geq0$?

I'm trying to solve the next problem: Let $\left(f_{n}\right)_{n\in\mathbb{N}}$ be a sequence of functions such that $f_{n}\colon\mathbb{R}\to\mathbb{R}$ is given by $f_{n}\left(x\right)=n\sin\left(\...
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2answers
37 views

How to deduce the sup of a function sequence $f_n$ without computing the derivative?

I came across the following function sequence: $$ f_n = e^{-nx}\sin(nx)$$ I'm asked if the following sequence converges uniformly on the set of positive real numbers $[0,\infty[$. I found 2 ...
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1answer
45 views

Validity of interchanging limit and integral under non-uniform convergence!!

Can I interchange the limit and integral of a sequence of functions which is not uniformly convergent in $[0,1]$ i.e $f_n \not\to f$ uniformly is it true that $\int_0^{x_n}f_n \to \int_0^1f$ for $x_n\...
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1answer
29 views

If a sequence of functions $\{f_n\}$ be uniformly convergent on $[a,b]$, would it be uniformly convergent of $(a,b)$?

Let us consider a sequence of functions $\{f_n\}$ on a compact interval $[a,b]$, which is uniformly convergent (to a function, say $f$) on $[a,b]$. Does it ensure the uniform convergence of $\{f_n\}$ ...
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1answer
26 views

Uniform convergence of exponential function

Question: is the sequence of function $(f_n)$ defined by, $f_n(x)=\sum_{k=0}^n \frac{x^k}{k!}$ for all $x\in A$ where $A$ is bounded subset of $\mathbb{R}$, is uniformly convergent on $A$? My ...
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1answer
40 views

Give an example of $h$ and $h_n$ where h is not continuous at $0$.

All of the following have to hold for the function and the sequence that are given as an example: 1) $h(x)=lim_{n\rightarrow\infty}h_n(x)$ 2) $\{h_n(x)\}$ is a sequence of decreasing non-negative ...
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41 views

$\{g_n\}$ be uniformly bounded sequence of functions on $[0,1]$ and converges pointwise to $g$.

My question is -suppose $\{g_n\}$ be uniformly bounded sequence of functions on $[0,1]$ and converges pointwise to a function. Then can I say $\int_{0}^{1} |g_n(x)-g(x)| dx\to 0$ as $n\to\infty$? If I ...
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2answers
46 views

Uniform convergence of the series $\sum_{n=1}^\infty\frac{x}{(1+x)^n}$

I have to determine whether or not the series $$\sum_{n=1}^\infty\frac{x}{(1+x)^n}$$ converges uniformly on $[0,1]$. I attempted to use Weierstrass' M-Test, by finding the maxima of each $a_n$ which ...
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0answers
23 views

Does convergence of a high order derivative implies convergence of lower orders

Assume that $ f_n $ is a sequence of smooth functions over the real line which converges uniformly to some function $ f $. Assume moreover that there is $ p > 1 $ such that the sequence of $ p $-th ...
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2answers
61 views

If $f_n\to f$ pointwise, $f$ is continuous and $f$ is continuous, then $f_n \to f$ uniformly.

Let $(f_n)$ a sequence of continuous function on $[a,b]$ that converges pointwise to $f$. We suppose that $f$ is continuous on $[a,b]$. Prove that the convergence is uniform. I'm stuck at some point :...
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1answer
20 views

Interpretation of a piece wise function with regards to point wise functional convergence.

I was watching a youtube tutorial about pointwise convergence and the author was interpreting the following piecewise function: $$h_{n}(x) = \left\{ \begin{array}{ll} 1,& x\geq \frac{1}{n} \...
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1answer
37 views

Show $f_{n}(x) = n \sin(\frac{x}{n})$ converges point wise on the compact set $[-R,R]$

Show $f_{n}(x) = n \sin(\frac{x}{n})$ converges point wise on the compact set $[-R,R]$ Playing with the function I was able to deduce that $f_{n}(x) \rightarrow f(x) = x$. My question is how to show ...
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1answer
37 views

Limit of a sequence of function using Definition

I have always tried to find these evaluations of limits of functions very very difficult. First of all, I need to know the basic steps of calculating these, is intuition a key factor in calculating ...
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16 views

$f$ nonnegative measurable function on $E$ show there is increasing simple nonnegative sequence converge to $f$ each with finite support

It is problem 24 section 4.4 Royden, real analysis 4th edition What I tried is the following consider $E_n=\{x| \;f(x)\geq n\}$ and let $f_n=f I_{E_n}$ now $f_n$ converges point wise to $f$ and $f_n$...
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3answers
53 views

continuity of sequence of functions

For each $n\in\mathbb{N}$, define $f_{n}:\mathbb{R}\to\mathbb{R}$ by $$f_{n}(x)=\int_{0}^{n}\frac{\sin(xt)}{e^{t}+x^{2}}\,dt.$$ How to ensure that each $f_{n}$ is continuous on $\mathbb{R}$ by ...
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2answers
57 views

Uniform or non uniform.

Consider the sequence of functions $f_n:(-1,1) \rightarrow \mathbb{R}$, $f_n(x) = \sum_{k=1}^n \frac{n}{n^2-k^2x}$. Find the function $f(x)$ to which $f_n$ converges pointwise. Determine if this ...