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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.

7
votes
4answers
707 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
242 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
945 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
84 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
556 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 ...
4
votes
2answers
160 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
67 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
119 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
63 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
127 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
86 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 ...
3
votes
2answers
137 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
1answer
653 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 ...
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
39 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
38 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
81 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
37 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
64 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
99 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
763 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
81 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
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
80 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
119 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
61 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
229 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
82 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 ...
2
votes
2answers
76 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
4answers
159 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
167 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
3answers
105 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
93 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
45 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
2answers
210 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
2answers
54 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
64 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{&}...
2
votes
1answer
85 views

Show that the sequence of functions $\{f_n\}$ convereges uniformly to $f$ on $[0,1]$ by the given condition.

Let $f$ be a continuous function on $[0,1]$. Suppose $\{f_n\}$ is a sequence of continuous functions on $[0,1]$ such that for any sequence $\{x_n\}$ in $[0,1]$, if $x_n \rightarrow x$ then $f_n(x_n) \...
2
votes
1answer
77 views

Convergence to Riemann-Stieltjes integral of sequence of Riemann-Stieltjes-like sums with changing integrand and integrator

I am considering the limiting behavior of a sequence of Riemann-Stieltjes (RS) (or at least RS-like) sums in the sense of their convergence to a Riemann-Stieltjes integral. The general term has the ...
2
votes
1answer
82 views

Study the convergence of the sequence $f_n(x)=\frac{x-n}{x^2}\cdot\chi_{(n,+\infty)}(x)$

For every $n\in\mathbb{N^+}$, let $f_n:(0,+\infty)\to\mathbb{R}$ be as defined: $$f_n(x)=\frac{x-n}{x^2}\cdot\chi_{(n,+\infty)}(x).$$ Study the convergence of the sequence $\{f_n\}_{n\in\mathbb{N^+}}$...
2
votes
1answer
183 views

Equidifferentiable iff derivative is equicontinuous?

Let $\{f_n\}$ a sequence of function differentiable at $x_0$ We have equidifferentiability at $x_0$, if $\lim_{h \to 0} \max_n \left| \frac{f_n(x_0+h) - f(x_0)}{h}- f'_n(x_0)\right| = 0$ Are the ...
2
votes
1answer
45 views

Proving convergence of sequence of functions of sequence

Consider the sequence of functions $f_k :A \rightarrow \mathbb{R}$. Suppose that $\{f_k\} \rightarrow f$ uniformly in A and $f_k$ is continuous in A $\forall k$. Prove that if $x_0 \in A$ and $\{x_k \...
2
votes
2answers
147 views

A fixed point theorem for continuous and increasing function on $[0,1]$

Let $f:\ [0,1] \to [0,1]$ be continuous and increasing. Also let $p \in [0,1]$. Define $$x_n = f^n(p) = f(f(...f(p))...)$$ Prove that either $p$ is a fixed point of $f$ or $x_n$ converges to a ...
2
votes
1answer
81 views

Limit of integrals of trigonometric functions

If we have a sequence of functions defined by $f_n(x) = \frac{\sin((x + \frac 1n)^2) - \sin((x-\frac 1n)^2}{\sin(\frac 1n)},$ how can we find $\lim_{n \rightarrow \infty} \int_0^1 f_n(x) \, d\lambda$ ...
2
votes
2answers
185 views

Sequence of differentiable,equicontinuous functions

I got stuck the other day trying to tackle the following problem : Let $ \left \{ f_n\left. \right \} \right. $ be a sequence of differentiable functions : $ f_n \quad : [0,1] \to \mathbb{R} $ ...
2
votes
1answer
44 views

Pointwise and uniform convergence application

Let $(f_{n})_{n}$ be a sequence of functions $f_{n}: \mathbb{R} \to \mathbb{R} $. They converge pointwise to a function $f: \mathbb{R} \to \mathbb{R} $. We know that the convergence is uniform on ...
2
votes
4answers
83 views

Show that $\underset {x \in [0,1]} {\sup} f_n(x) \rightarrow \underset {x \in [0,1]} {\sup} f(x)$ as $n \rightarrow \infty$.

Let $\{f_n\}$ be a sequence of continuous functions converging uniformly to a function $f$ on $[0,1]$. Then show that $\sup\limits{x \in [0,1]} f_n(x) \rightarrow \sup\limits_{x\in[0,1]} f(x)$ as $n \...
2
votes
2answers
43 views

find the limit function $\lim _{n \rightarrow \infty} P_n(x)$

Let $P_n$ be a sequence of polynomials such that for $n=0, 1, 2,...$ $P_0=0$ and $P_{n+1}(x)=P_n(x)+\frac{x^2-P_n^2(x)}{2}$. Assuming the fact that $\{P_n\}$ is convergent pointwise, find the limit ...