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.

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 ...
2
votes
0answers
57 views

$ \lim_{n\to \infty}\int_{0}^{1}\frac{2nx^{n-1}}{1+x}dx=?$ [duplicate]

For $n=1,2,...,$ let $f_n(x)=\frac{2nx^{n-1}}{1+x},x\in[0,1].$ Then $$ \lim_{n\to \infty}\int_{0}^{1}f_n(x)dx=?$$ Here $f_n(1)=n$. So the limit function of $f_n(x)$ is not continuous. Also I was ...
0
votes
1answer
37 views

Bounded sequence of functions implies convergent subsequence

Here you can see my attempt at the proof. I am sure I did something wrong because my prof asked me to show it for rationals and I "somehow" showed it for all reals. I would appreciate it if someone ...
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 ...
2
votes
1answer
37 views

Uniform convergence of $\frac{y/(2N)}{\sin(y/(2N))}$ towards 1

I can't come up with a proof, why $f_N(y) := \frac{\frac{y}{2N}}{\sin\left(\frac{y}{2N}\right)}$ converges uniformly against $1$ for $y\in(0,\pi),\ N\to\infty$. I would be thankful for any advice.
1
vote
1answer
26 views

$g_n = \max \{\min (f_n, g), -g\} \to f$

I am currently self studying Mathematical analysis by M. Apostol. I got stuck in trying to understand $\\$ Theorem 10.30 $\ \ $Let ${f_n}$ be a sequence of functions in $L(I)$ which converges ...
1
vote
1answer
39 views

Limiting total variation attached to sequence of uniformly vanishing functions of bounded variation

Let $(f_n)_n$ be a sequence of real functions of a single real variable with compact support in $[0,1]$ and of bounded variation all of them. Let the sequence be uniformly convergent to $0$. Is it ...
0
votes
1answer
78 views

Uniform convergence of $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$, solution verification

Is my reasoning right? I have $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ for $x \in \mathbb{R}$, so I conclude that it's pointwise convergent $f_n \to |x|$, and moreover it's uniformly convergent to $|x|$, ...
0
votes
1answer
39 views

Sequence of continuous function converging pointwise to continuous function is equicontinuous?

I've proven the following "theorem": Let $I \subset \mathbb{R}$ be an interval, $(f_n: I \rightarrow \mathbb{R})_{n \in \mathbb{N}}$ be a family of continuous functions converging pointwise to a ...
1
vote
2answers
32 views

Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$?

Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$? Intuitively, I think that the answer is no. I know that the statement holds for ...
1
vote
1answer
41 views

Series extension of the superposition principle for ODEs

Take the superposition principle for linear ODEs of the form $y'(t)=A(t,y(t)) + g(t)$ ($y\in \mathbb{R}^n$, $A$ a linear function in y). If $g(t)=\sum _{k=1}^N g_k(t)$ then $y(t)=\sum _{k=1}^N y_k(t)$ ...
0
votes
2answers
32 views

Sequence of entire function that converges uniformly over on sets with empty interior

I have to prove that the sequence of entire functions: $$f_n(z)=\frac 1n \sin(nz)$$ converges uniformly over $\mathbb{R}$ (and this I managed to verify) but doesn't on every set with non-empty ...
0
votes
1answer
34 views

Limit of a locally uniformly convergent sequence of continuous functions

I have two questions: 1. I know that the uniform limit of a continuous functions is continuous. But I'm wondering whether this is true if the convergence is locally uniform. That is the uniform ...
2
votes
0answers
43 views

Write a series of piecewise linear functions that converges to $f(x) = x^2$ on the interval $[0,1]$.

Write a series of piecewise linear functions that converges to $f(x) = x^2$ on the interval $[0,1]$. A sequence of piecewise linear functions that converges to $f(x) = x^2$ is as follows: For $k \in ...
1
vote
1answer
95 views

composition of bounded uniformly convergence sequences

I'm hoping to make a generalization of the answer to this question. Let's say that instead that we're composing two uniformly continuous function sequences, does this composition converge uniformly ...
3
votes
1answer
81 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 ? ...
1
vote
1answer
56 views

Convergence of sequence of Riemann-Stieltjes integrals to Riemann-Stieltjes integral

In connection with my post Convergence to Riemann-Stieltjes integral of sequence of Riemann-Stieltjes-like sums with changing integrand and integrator, an alternative approach to my main objective ...
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 ...
1
vote
0answers
23 views

Using ∞-BinFractions to define a topological space homeomorphic to the positive real numbers?

First the proposed theory: Here $0 \in \mathbb N$. Definition: A non-constant function $f: \mathbb N \to \mathbb N$ is called a ∞-BinFraction if it satisfies the following: $\tag 1 \forall n \; f(n+...
0
votes
0answers
34 views

My possibly fraudulent proof of $f_n$ Cauchy in measure => $f_n -> liminf$ in measure

This result seems too convenient and I feel like Folland would have used this to prove proposition 2.30 if this was true. My "proof": Let $f_n$ be Cauchy in measure. By definition of $\liminf f_n(x)...
0
votes
1answer
27 views

Deciding convergence of a sequence of functions

I am given the sequence of functions $f_n(x)=x^n - x^{2n}$ on $[0,1]$. I must define a function $f(x)$ as a pointwise limit function on the indicated interval. If it is uniform, I must then find a ...
0
votes
1answer
77 views

Find a sequence of Lipschitz continuous functions on $[0,1]$ whose uniform limit is $\sqrt{x}$.

Find a sequence of Lipschitz continuous functions on $[0,1]$ whose uniform limit is $\sqrt{x}$, which is a non-Lipschitz function.
0
votes
0answers
30 views

Finding sequence of functions with compact support for Integral of given function

Let $a>0$ and $f(x):\mathbb{R}\rightarrow\mathbb{R}$ with $$f(x):\begin{cases}\frac{1}{\sqrt{a^2-x^2}},& x\in(-a,a)\\ 0,&\text{else} \end{cases}$$ I now have to construct sequences of ...
0
votes
0answers
30 views

Ways of checking pointwise convergence

According to the definition of pointwise convergence: A sequence $f_m(x)$ of real valued functions defined on D a subset of real numbers is said to be pointwise convergent to$f(d)$ at a point $d\...
1
vote
1answer
66 views

Applying theorem to disprove uniform convergence

I recently read this theorem in real analysis:(Actually a corollary to a theorem) {$f_m$} is a sequence of continuous functions defined on $D$ such that $f_m$$\to$$f$ uniformly on $D$ then for every ...
0
votes
3answers
82 views

Is $(f_n)$ pointwise convergent?

Let $f_n(x)$, for all n>=1, be a sequence of non-negative continuous functions on [0,1] such that $$\lim_{n→\infty}\int^1_0 f_n (x)dx=0$$ Which of the following is always correct ? A. $f_n→0$ ...
0
votes
1answer
36 views

Exercise of sequence of continuous functions

Let $(f_n)_n$ be a sequence of continuous functions on $D\subset \mathbb{R}^{N} \to \mathbb{R}$ which is monotone decreasing. If $\lim_{n\to\infty }f_n(c))=0$ for some $c\in D$ and $ \epsilon >0$ , ...
0
votes
1answer
23 views

Finding values of $x$ such that a sequence of functions converges.

$(f_n)$$_n$$_\in $$_\mathbb N$ is a sequence of functions where $f_n : [0,2\pi] \to \mathbb R$ $\ \forall n \in \mathbb N$. Find all values of $x \in [0,2\pi]$ such that $(f_n)$$_n$$_\in $$_\mathbb N$ ...
1
vote
0answers
18 views

Constructing a uniform convergence sequence

Let $f : \overline{\Omega} \subset \mathbb{R}^{N} \to \mathbb{R}$ be a $C^{2}(\overline{\Omega})$ function. Can we always construct a sequence $f_{n}$ such that $f_{n} \to f$ uniformly in $\overline{\...
0
votes
1answer
20 views

a notation for convergeence.

Suppose $\{f_n\}$ is a sequence of complex functions and $|f_n(x)-f(x)|\to 0$ for all $x$. If we put "for all $x$" behind the $|f_n(x)-f(x)|\to 0$, does it show that the convergence is uniformly ...
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 $\{f_n\}_{n\in\mathbb{N^+}}$...
0
votes
0answers
21 views

Prove that the sequence of derivative functions converges uniformly on every interval [-M,M].

The sequence is: $f_n(x) = \dfrac{nx^2+1}{2n+x}$ with derivative $f_n'(x) = \dfrac{4n^2x+nx^2-1}{4n^2+4nx+x^2}$. We know that $f'(x) = x$. We are asked to show that the sequence of derivatives, $f_n'(...
1
vote
0answers
16 views

Prove that the sequence of derivative functions converges uniformly on every interval [-M,M].

The sequence is: $f_n(x) = \dfrac{nx^2+1}{2n+x}$ with derivative $f_n'(x) = \dfrac{4n^2x+nx^2-1}{4n^2+4nx+x^2}$. We know that $f'(x) = x$. We are asked to show that the sequence of derivatives, $f_n'(...
0
votes
1answer
23 views

Convergence of $f_n(x)=nx$

Consider the sequence of functions $$f_n(x)=nx$$ As $n$ gets larger, so does the gradient of the line passing through the origin. Graphically, as $n$ goes to infinity, this will converge to the ...
0
votes
1answer
43 views

Can I use the extreme value theorem to prove uniform convergence of a sequence of functions on a compact interval?

For example $f_n(x)=(1+x/n)^n$ converges pointwise on $\Bbb R$ to $f(x)=e^x$, but not uniformly because $f_n(n)\to+\infty$ and $f_n(-2n)$ has no limit. Is it logically sound to say that the ...
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 $1-\dfrac{1}{e}.$ ...
0
votes
0answers
25 views

Can i prove the limit of a function using the definition of sequence limits?

So I'm asked to prove the $\lim_{x\to 1} \frac{4x+3}{x^2+4x-3}$ and of course it can be easily found $\frac{7}{2}$. However, I know I can prove it with epsilon delta definition but frankly that's ...
1
vote
1answer
89 views

Pointwise and uniform convergence of a piecewise sequence of functions on the closed, punctured disk, $\overline{D}\prime(0,1)$.

Consider the sequence of functions $$f_n(z) = \begin{cases} n, & \text{if $0<|z|\leq\frac{1}{n}$} \\ \frac{1}{z^4}, & \text{if $\frac{1}{n}<|z|\leq1$} \end{cases} $$ for $n\geq 1$, on ...
0
votes
0answers
30 views

Understanding the integral of a sequence of functions

I have a integral with the following property: $$ \forall n>0,\quad \int_X f_n(x) dx> 0 $$ I am trying to be able to say that: $$ \int_X \hspace{.5em}\lim_{n\rightarrow 0} \hspace{.5em}f_n(x)dx ...
2
votes
1answer
113 views

Uniform convergence of sequence of function $f_n(x) = \frac{nx^4+1}{nx^4+2x+3}e^{-nx^2}$ on the interval $(1,+ \infty)$

I need to prove that the sequence of functions $f_n(x)$ is uniformly convergent to $f$ on the interval $(1,+ \infty)$. I've already shown that $f_n(x) = \frac{nx^4+1}{nx^4+2x+3}e^{-nx^2}$ is ...
0
votes
1answer
37 views

Convergence of the following sequence of functions.

For $n \ge 1$, let $$g_n(x) = \sin^2 \left (x + \frac 1 n \right ), x \in [0,\infty)$$ and $$f_n(x) = \int_{0}^{x} g_n (t)\ \mathrm {dt}.$$ Then $(1)$ $\{f_n \}$ converges pointwise to a ...
1
vote
1answer
26 views

A question about convergence. Is it possible to find a subsequence increasing in this situation?

Suppose you have a sequence $ f_n \rightarrow f $ (not necessarily increasing), and suppose that for each $n$ there is a sequence of increasing functions $\phi^{(n)}_{m} \rightarrow f_n$. Is it ...
0
votes
2answers
57 views

check uniform convergence of $f_n(x)=\begin{cases} 1-nx &\text{if}\;x \in [0,1/n]\\\\0 &\text{if}\;x \in [1/n,1] \end{cases}$

Let $$f_n(x)=\begin{cases} 1-nx &\text{if}\;x \in [0,1/n]\\\\0 &\text{if}\;x \in [1/n,1] \end{cases}$$ Then $ \lim_{n \to \infty}f_n(x)$ defines a continuous function on $[0,1]$. $\{f_n\}$ ...
0
votes
1answer
107 views

Prove endpoints of Convergence Interval of Power Series are divergent

Part a) of a question required showing that the radius of convergence of the power series $\sum_{} \frac{n!}{n^n}x^n$ and $\sum_{} \frac{n^n}{n!}x^n$ are e and 1/e respectively. This was fairly simple....
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 ...
1
vote
1answer
31 views

Interchanging limit of sequence of functions

Let $\{u_{j}\}_{j\in\mathbb{N}}$ be a bounded sequence in $L^{\infty}(\Omega)$ for a given smooth bounded domain $\Omega \subset \mathbb{R}^{n}$. Assume $u_{j} \to u\in L^{\infty}(\Omega)$ a.e. in $\...
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 ...
1
vote
1answer
57 views

A sequence of functions decreasing to 0 is equicontinuous in a compact metric space.

How to proof that? Let M be a compact metric space and $ \{f_n\} \subset C(M,\mathbb{R})$, so that $\{f_n\}$ is decreasing and $ lim f_n(x)=0 $, then $\{f_n\}$ is equicontinuous
0
votes
1answer
50 views

Determine whether a sequence of functions converges uniformly

The function $f_n(x)=3n^3(x-1/n)^2$ for $x\in[0,2]$ is given, and I need to show whether the sequence of functions $(f_n)_{n\in\Bbb{N}}$ converges uniformly to $f(x)=\lim\limits_{n \to \infty}f_n(x)$, ...
0
votes
1answer
68 views

Differentiability and Uniform convergence on unbounded intervals

Let $I$ be a bounded interval of $\mathbb{R}$, and let $\{f_{n}:I\to\mathbb{R}\}$ be a sequence of differentiable functions. Suppose that a sequence $\{f_{n}(x_{0})\}$ converges for some $x_{0}\in I$ ...