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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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7
votes
4answers
759 views

Finding the limit of a sequence of integrals

Let us define a sequence of function as $$f_n(x)=\frac{2nx^{n-1}}{x+1}\;\;\text{for each $x\in [0,1]$ and for all $n\in\mathbb{N}$}$$ What is $\displaystyle \lim_{n\to \infty} \int_0^1 f_ n(x) dx$ ...
7
votes
5answers
283 views

Summation problem: $f(x)=1+\sum_{n=1}^{\infty}\frac{x^n}{n}$

I want to evaluate this summation: $$S=1+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+...+...$$ where, $|x|<1$ Here it is my approach $$S=1+\sum_{n=1}^{\infty}\frac{x^n}{n}=f(x)$$ $$f'(x)=...
6
votes
2answers
1k views

Uniform Convergence of $(\sin x)^n$

Is the sequence of functions $(\sin(x))^n$ uniformly convergent in $[0,\pi]$? Can you give me a hint or solution, I have already already prove that it is UC in $[0,1]$ but I don't know how to proceed ...
6
votes
1answer
93 views

If $f_n \rightarrow f$ is a sequence of $L^p$ and $g_n \rightarrow g$ a bounded sequence of $L^{\infty}$ then $f_ng_n \rightarrow fg$ in $L^p$

I want to verify is my proof is correct. Let $p \in [1,\infty)$, $f_n$ a sequence of $L^p$ that converges to $f$ and $g_n$ a bounded sequence of $L^\infty$ that converges almost everywhere to $g$. ...
5
votes
4answers
585 views

Pointwise convergence to a uniform continuous function

What can we say about a sequence of functions that is pointwise convergent (over $R$) to a uniform continuous function? Does it converge uniformly? I have tried it using the definition but can't get ...
5
votes
3answers
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 ...
4
votes
1answer
866 views

Is the uniform limit of uniformly continuous functions, uniformly continuous itself?

That sounds a lot like a tongue-twister. I know that there exist sequences of Lipschitz functions whose uniform limit is not Lipschitz (for instance, just use Weierstrass theorem on $[a,b]$). Clearly ...
4
votes
2answers
82 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 ...
4
votes
2answers
162 views

Proving that this function converges uniformly.

I was wondering if there is an easier way to show that this sequence of functions converges uniformly. Also I am almost $100 \% $ sure that my reasoning is not a proof, so yeah, help me please (there ...
4
votes
1answer
82 views

What is the analytical form of a recursive function

This maybe a question answered in some textbooks, I had trouble figure out how to solve it. Here is a function defined recursively, $ F_1(x) = x $ $ F_2(x) = x^2 -2 $ $ F_n(x) = F_{n-1}^{2}(x) -2 $ ...
4
votes
1answer
138 views

Converging sequence of solution to a differential equation

I was working on a problem about a sequence of functions, each of which is a solution to a sequence of differential equations, that converges to a function which is supposed to be the limit of the ...
4
votes
1answer
64 views

Where is the mistake in my counter-example to the statement? (sequences of functions, uniform convergence)

Statement: Let $f,f_n:[a,b]\to\mathbb{R}$ for $n\in\mathbb{N}$ and for all $x\in[a,b]$ let $\lim_{n\to\infty}f_n(x) = f(x)$. If $f$ and all $f_n$ are continuous, then $f_n$ converges uniformly ...
4
votes
2answers
129 views

Infinite points in a line (Contest Math)

Today, the Central American and Caribbean Mathematical Olympiad was held in El Salvador. Problem 6 was as follows: Let $k$ be an integer greater than $1$. Initially, Tita the frog is sitting at ...
4
votes
1answer
95 views

Uniformly convergent on each compact set of $\mathbb R$ but not on $\mathbb R$

As the title says, I am looking for a sequence of function which is uniformly convergent on all compact sets of $\mathbb R$ but not on $\mathbb R$. I thought $f_n(x) = \frac{x}{n}$ is such a ...
4
votes
0answers
111 views

Arzelá-Ascoli Theorem precompact sets

Is the Arzelá-Ascoli Theorem true in a precompact subset of $\mathbb{R}^n$?. If $S \subset \mathbb{R}^n$ is precompact and we have a sequence $(f_n)$ of functions in $C(S)$ (Space of bounded and ...
4
votes
4answers
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 $n \rightarrow 0$ ...
3
votes
2answers
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 $f(x)=e^{-x}$. ...
3
votes
2answers
156 views

Show that $f_n(x)= \frac{\sin^2(n^{\alpha}x)}{nx}$ is not uniformly convergent for $\alpha \geq 1$ and $x \neq 0$

Can someone help me with proving that \begin{align} f_n: \mathbb{R} \rightarrow \mathbb{R }, \hspace{0.5cm} f_n(x)= \begin{cases} \frac{\sin^2(n^{\alpha}x)}{nx} & \text{ if } x \neq 0, \\ 0 &...
3
votes
3answers
75 views

Calculate $\displaystyle\lim_{n\to\infty}\int_0^1\frac{nx}{1+n^2 x^3}\,dx$.

I need calculate $\displaystyle\lim_{n\to\infty}\int_0^1\frac{nx}{1+n^2 x^3}\,dx$. I showed that $f_n(x)=\frac{nx}{1+n^2 x^3}$ converges pointwise to $f\equiv 0$ but does not converge uniformly in [0,...
3
votes
1answer
81 views

Find $\lim_{n\rightarrow \infty}\int_0^1 f_n(x) dx$

Let $f_n:[0, 1] \rightarrow \mathbb{R}$ be defined by $f_n(x)=\dfrac{n+x^3 \cos x}{n e^x + x^5 \sin x}, n \geq 1$. Find $\lim_{n\rightarrow \infty}\int_0^1 f_n(x) dx$ My answer is $1-\dfrac{1}{e}.$ ...
3
votes
2answers
41 views

Is $f_n \rightrightarrows f $ implies $(f_n(x_n)-f(x_n)) \longrightarrow 0$?

Suppose we have a sequence of functions defined on $S$, $f_n$ that converges uniformly to $f$. Then if $x_n$ is any sequence in the interval can we say that $f_n(x_n)-f(x_n)$ goes to zero? Efforts: ...
3
votes
1answer
962 views

Given sequence of $L-$Lipschitz functions which converges pointwise, prove uniform convergence

Let $f_n:[a.b]\rightarrow \mathbb{R}$ be sequence of $L-$Lipschitz functions, that is: $$\forall x,y\in[a,b]: |f_n(x)-f_n(y)|\leq L|x-y|$$ Suppose $f_n \rightarrow f$ pointwise, prove $f_n \...
3
votes
1answer
40 views

Under given conditions whether $\lim\limits_{n\to \infty} \int_{-\infty}^{\infty}f_n(t)dt=\int_{-\infty}^{\infty}f(t)dt$ or not?

Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of continuous real-valued functions defined on $\mathbb R$ which converges pointwise to a continuous real-valued function $f$. Which of the following ...
3
votes
1answer
85 views

Uniform convergence on two separate intervals

Test the sequence of functions for uniform convergence $$f_n(x)=\frac{nx}{1+n^2x^2}$$ $a) \text{on} \ [0,1]$ $b) \text{on} \ [1,{\infty})$ a) If we differentiate this with respect to ...
3
votes
1answer
112 views

Limit of $\sin(kx)$ as k tends to infinity

I am have been thinking lately of the sequence of functions $$ f_n = \sin nx $$ and its limit as n tends to infinity. I am quite comfortable with the fact that viewing this sequence in $\mathcal{C}([...
3
votes
1answer
69 views

Is this sequence of functions uniformly convergent on [0, 2] ?? [closed]

Define a sequence of functions $f_n : [0,2] \to \Bbb R$ as: $$f_n(x) = \frac {1-x} {1+x^n}$$ Is this sequence of functions uniformly convergent on $[0,2]$?
3
votes
1answer
38 views

Given $\{f_n\}$ uniformly bounded and $f_n \to f$ uniformly on $S.$ Prove $\frac{f_1 + f_2 + \cdots + f_n}{n} \to f$ uniformly on $S.$

I know this is equivalent to showing $\dfrac{f_1 + f_2 + \cdots + f_n} n - f \to 0$. And in turn $\dfrac{(f_1 - f) + (f_2 - f) + \cdots + (f_n - f)}{n} \to 0$. I can split the fraction as follows: ...
3
votes
2answers
65 views

Sequence of uniform convergent sequence $\{f_n\}$ such that $\{f_n'\}$ does not converge [closed]

I would like to find uniformly convergent sequence of differentiable functions $f_n:[0,1]\to\mathbb{R}$ such that the sequence $f_1'$, $f_2'$,$f_3'\ldots$ does not converge.
3
votes
2answers
101 views

On limit of point wise convergent sequence of continuous functions on real line

Let $\{f_n\}$ be a sequence of continuous functions on real line which is point wise convergent . Then is it true that for every $c\in \mathbb R$ , the set $\{x \in \mathbb R : \lim_{n\to \infty} f_n(...
3
votes
1answer
82 views

Given $P_n(x)=\frac{n}{1+n^2x^2},$ Prove $f_n(x)=\frac{1}{\pi}\int_{-\infty}^\infty f(x-t)P_n(t)$ converges uniformly

Let $P_n:\mathbb{R} \rightarrow \mathbb{R}$ be a sequence of functions defined by $P_n(x)=\frac{n}{1+n^2x^2}$. $(a)$ Prove $\int_{-\infty}^\infty P_n(x) \mathop{dx} = \pi ,$ and that $$\forall \...
3
votes
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 ...
3
votes
1answer
22 views

Proof Verification for Uniform Convergence on Sequence of Functions

just looking for a verification on a proof. Thanks in Advance Let $f_n$ be a sequence of functions such that $f_n=\frac{x^{2n}}{1+x^{2n}}$ defined on $[-2,2]$. Prove or Disprove Uniform Convergence ...
3
votes
1answer
94 views

Discontinuity properties of $f_n$ carries over to the limit function $f$

Suppose that $f_n:[a,b] \rightarrow \Bbb R$ and $f_n$ converges uniformly to $f$. Which of the following discontinuity properties of the functions $f_n$ carries over to the limit function ? ...
3
votes
1answer
124 views

Is it true that $\lim_{n\to\infty}\int_{0}^{1}f_n(x)\,dx = \int_{0}^{1}f(x)\,dx$ in general and if $|f_n(x)|\le 2017$?

Let $f_n(x)$ and $f(x)$ be continuous functions on $[0, 1]$ such that $\lim_{n\to\infty} f_n(x) = f(x)$ for all $x \in [0, 1]$. Answer each of the following questions. If your answer is “yes”, then ...
3
votes
1answer
312 views

$\epsilon$/3 - Proof for converging sequence of functions

I tried to prove the following statement and would like to ask if it's okay like that. $\underline{Statement}$: Let [B,$||.||_B$] be a complete normed linear space, $T_n :B \rightarrow B$ a ...
3
votes
0answers
87 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 ...
3
votes
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^...
2
votes
2answers
80 views

Calculate the limit of $\frac{n^{n+1}}{n!}\int_0^xe^{-nt}t^n\,\mathrm dt$

Set $R_n(x)$ the quantity in the title, for $n\in\mathbb{N}$ and $x > 0$. I'm trying to prove that if $x < 1$, then $R_n(x)\underset{n\rightarrow +\infty}{\longrightarrow} 0$, if $x > 1$, ...
2
votes
2answers
279 views

Show that the class $C_c(\mathbb{R^n})$ of continuous functions with compact support is not a complete metric space

I'm asked to show the following: Show that the class $C_c(\mathbb{R^n})$ of continuous functions with compact support on $\mathbb{R^n}$ with the sup-norm metric $d(f,g):= \text{sup}_{x\in \mathbb{R^...
2
votes
4answers
165 views

Elementary proof regarding $\sum_{n=1}^\infty \frac{\sin(nx)}{n}$

I was looking for an elementary proof (without use of Fourier series) that $$\sum_{n=1}^\infty \frac{\sin(nx)}{n}$$ converges to $(\pi-x)/2$ for $x\in (0,2\pi)$. I have managed to demonstrate that ...
2
votes
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(\...
2
votes
4answers
35 views

Is the limit of this sequence of function $1$ or $0$?

I was given to compute the limit $\lim_{n\to\infty}f^n(x)$ with $x\in(0,1)$, where $$f^n(x)=\begin{cases} 0\,\,& x\leq0\\ x\,\,& 0<x<1/n\\ 1\,\,& x\geq1/n \end{cases}.$$ Would the ...
2
votes
2answers
51 views

Uniform convergence on $\mathbb R$ of the series $\sum_{n=2}^{\infty} \frac{(-1)^{n+1}} {\sqrt n + \cos x}$

Is the following series of functions uniformly convergent on $\mathbb{R}$? $$\sum_{n=2}^{\infty} \frac{(-1)^{n+1}} {\sqrt n + \cos x}$$ My attempt: My answer is No I know that by Leibnitz ...
2
votes
3answers
109 views

How to prove two sequence have a common limit.

We have \begin{align} U_{0} &= 1 &&\text{and} & V_{0} &= 2 \\ U_{n+1} &= \frac{U_{n}+V_{n}}{2} &&\text{and} & V_{n+1} &= \sqrt{U_{n+1}V_n} \end{align} How to ...
2
votes
3answers
94 views

Example of a sequence of integrable functions on $[0,1]$ s.t. $\lim_{n\to\infty}\int_0^1|f_n(x)|\,dx = 0$ but $f_n$ not converges to $0$ a.e? [closed]

I need an example of a sequence of integrable functions on $[0,1]$ s.t. $$\lim_{n\to\infty}\int_0^1 |f_n(x)|\,dx = 0$$ but $f_n$ does not converge to $0$ a.e. Anyone can provide an example with a ...
2
votes
2answers
529 views

Show that the sum of reciprocal products equals $n$

I don't even know how to proceed. Please help me with this. (Original at https://i.stack.imgur.com/DRIX8.jpg) Consider all non-empty subsets of the set $\{1, 2, \ldots, n\}$. For every such ...
2
votes
2answers
278 views

Convergence of $F_n=x^n\sin(nx)$ on $S=(-1,1)$

let $\{F_n(x)\}$ be a sequence of functions where $F_n=x^n\sin(nx)$ on $S=(-1,1)$. Find $\lim_{n\to \infty}F_n(x)$. Show $F_n$ converges uniformly on closed subset of $S$ but not on $S$. My attempt: ...
2
votes
1answer
25 views

Suppose $f_{k}$ defined on $\mathbb{R}^n$ converge uniformly to a function $f$. Each $f_{n}$ is bounded, say by $A_{k}$. Prove that $f$ is bounded.

Suppose that functions $f_{k}$ defined on $\mathbb{R}^n$ converge uniformly to a function $f$. Suppose that each $f_{n}$ is bounded, say by $A_{k}$. Prove that $f$ is bounded. Attempt Want to Show: ...
2
votes
2answers
57 views

Prove that $f_n$ has $n$ distinct roots which are symmetric about $0$

Given $f_0(x)\equiv 1$ and define $f_n$ recursively via $$f_{n+1}(x)=xf_n(x)-f'_n(x).$$ Prove that $f_n$ is a polynomial of degree $n$. Further prove that $f_n$ has $n$ distinct real roots which are ...
2
votes
1answer
67 views

why is $f_{n}\rightarrow f \; \text{point-wise} \iff \limsup_{n\rightarrow\infty}f_{n}=\liminf_{n\rightarrow\infty}f_{n}=f$

in a proof of a lemma that states the following : $f_{n}\rightarrow f \; \text{point-wise} \implies f \; \text{is measurable} $ the author proceeded in 3 steps : proving that $\sup_nf_n\;\text{&}...