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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.

0
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1answer
20 views

Determine the sequence generated by the following exponential generating functions:

a) $f(x)=3e^{3x}$ I have \begin{align}f(0)&= 3 \\ f(1)&=3e^3 \\ f(2)&=3e^6 \end{align} So would my sequence be $a_n=3e^{2n}$? Or by recurrence $a_n=a_{n-1}(e^3)$? Or should I find a ...
0
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2answers
24 views

Understanding proof of sequential continuity?

I'm trying to understand proof of the following statement: Q. Let $f$ be a function on a closed bounded interval $[a,b]$. Prove that $f$ is continuous at $ c \in [a,b]$ if and only if $f(x_n) \to c$...
0
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1answer
23 views

Probably dumb limit

I have a sequence of continuous functions $f_n : I^k \rightarrow I^k$ converging uniformly to a continuous function $f$. Then for each $n$ I choose a point $x_n$ and since they're chosen in $I^n$ ...
-1
votes
1answer
52 views

Is that statement true or false?

Are following statements true or false ? If function $f$ is differentiable at $x_0$, then the sequence n.$( f(x_0+(1/n)) - f(x_0))_{n\in\mathbb{N}}$ is convergent. I am really not sure, but I think ...
1
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1answer
54 views

Does $\lim\limits_{x \rightarrow c} f(x)$ exist if the sequence $\{ f(x_n)\}_{n=1}^\infty$ is Cauchy?

I'm struggling a little with this question: Let c be a cluster point of $A ⊂ \mathbb{R}$, and $f : A → \mathbb{R}$ be a function. Suppose for every sequence $\{x_n \}$ in A, such that $\lim x_n = c$,...
3
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1answer
21 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 ...
1
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0answers
20 views

Iterating a sequence and verifying its convergence

I am given a sequence $(f_n)_n$ where $n\in N$. $f_n : \Re \rightarrow \Re: x \mapsto 1$ $f_1:\Re \rightarrow \Re$ is defined as follows $$f_1 (x) = 1 + \int_0^x f_0 (t) dt$$ One sees that the ...
1
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0answers
33 views

Combining little-o notation

Suppose we have $$f_{m}(n) = \frac{1}{n}g_{m}(n) + o(\frac{1}{n})$$ where the little-o notation is uniform in the variable $m$ as $n \rightarrow \infty$. Under what conditions is $f(m,n) = o(\frac{1}{...
2
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2answers
73 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^...
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1answer
32 views

Uniform convergence and the supremum theorem

I know that there are at least 3 other questions with the similar problem. I hope, though, you won't flag this as a duplicate since I've got some specific questions here that haven't been answered ...
1
vote
1answer
34 views

Uniform convergence of $f_n(x) = \frac{nx}{(2+nx)(4+x^2)}$

I need to study the uniform convergence of $$f_n(x) = \frac{nx}{(2+nx)(4+x^2)}$$ on the interval $[2,+\infty)$ I've shown that on $[0,+\infty)$: at $x =0$ $f_n(0)=0 \xrightarrow{} 0$ at $x \neq 0$...
0
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1answer
23 views

Uniform convergence of $f_n(x) = n(x-1)e^{-nx}$

I need to study the uniform convergence of $f_n(x) = n(x-1)e^{-nx}$ on the interval $[0,+\infty)$ I've shown that : at $x =0$ $f_n(0)=-n \xrightarrow{} -\infty$ at $x =1$ $f_n(1)=0 \xrightarrow{} ...
0
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1answer
20 views

Uniform and pointwise convergence of sequence of function of $f_n(x) = [\log(1+x)]^n$

I need to study the pointwise convergence of $f_n(x) = [\log(1+x)]^n$ for every $x$ of the domain of the functions. After i have to prove that the sequence of functions $f_n(x)$ is uniformly ...
1
vote
1answer
25 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 \...
3
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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 ...
1
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0answers
55 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
29 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
63 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 $f_n(x) = \frac{x}{n}$ 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
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1answer
22 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
35 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
45 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
29 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
31 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
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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
18 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 ...
1
vote
1answer
79 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
59 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
46 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
54 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
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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
32 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
votes
1answer
52 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
29 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
26 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
51 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
66 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
34 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
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$ ...
1
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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
73 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
15 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
40 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
76 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 $1-\dfrac{1}{e}.$ ...