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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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2answers
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Let $x_n$ = $2^{2^n}$ +1, n= 1,2,3… [on hold]

Prove that $\frac{2^0}{x_1}$ + $\frac{2^1}{x_2}$ + $\frac{2^2}{x_3}$ +...... $\frac{2^{n-1}}{x_n}$ < 1/3 for all positive integers 'N'
1
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1answer
24 views

Uniform convergence as $\epsilon\to 0^+$

Reading some lectures on Hamilton-Jacobi PDE theory I found some terminology that I really don't understand. Let $\Omega$ be an open subset of $\mathbb{R}^n$. Suppose that $u_\epsilon:\Omega\to \...
2
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1answer
36 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 ...
1
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0answers
50 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
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1answer
19 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 ...
3
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0answers
37 views

Uniformly convergent on each ccmpact 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 $f_n(x) = x/n$ is such a function since ...
2
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1answer
36 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
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1answer
18 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
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1answer
32 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
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1answer
35 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
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1answer
26 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
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2answers
29 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
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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 $y(t)=\sum _{k=1}^N y_k(t)$ ...
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2answers
26 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
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1answer
12 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
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0answers
42 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 ...
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1answer
55 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
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1answer
58 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
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1answer
40 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
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1answer
43 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 ...
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0answers
22 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+...
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0answers
31 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
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1answer
22 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
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1answer
41 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.
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0answers
27 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 ...
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0answers
23 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
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1answer
47 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
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3answers
61 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
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1answer
33 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
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1answer
21 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$ ...
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0answers
17 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
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1answer
19 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
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1answer
68 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 $\{f_n\}_{n\in\mathbb{N^+}}$...
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0answers
19 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'(...
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0answers
12 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
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1answer
22 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 ...
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1answer
37 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
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1answer
74 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 $1-\dfrac{1}{e}.$ ...
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0answers
23 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
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1answer
68 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 ...
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0answers
29 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
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1answer
48 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
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1answer
30 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
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1answer
22 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
46 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
67 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
41 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
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1answer
30 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
94 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
41 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