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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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Sequence of holomorphic function converging pointwise on a subset

Let $(f_n \mid n \in \mathbb{N})$ be a sequence of holomorphic functions on an open set $U \subseteq \mathbb{C}$ and $A \subseteq U$. Can we show that if $\lim_{n \to \infty} f_n(x)$ exists for any $x ...
Vinit Sinha's user avatar
1 vote
1 answer
28 views

If $A$ is a discrete set, compact convergence on $\Omega \setminus A$ implies compact convergence on $\Omega$

Let $\Omega\subset \mathbb{C}$ be a domain, $\{f_n\}$, $f_n:\Omega \to \mathbb{C}$ a sequence of holomorphic functions on $\Omega$ and let $A\subset \Omega$ be a discrete set. If $\{f_n\}$ converges ...
Gerschgorin's user avatar
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0 answers
13 views

If $G\in C([0,1])$ and strictly increasing, can we find a sequence $G_n\n C^{\infty}$ with uniformly equicontinuous density?

Let $G:[0,1]\rightarrow [0,1]$ be a strictly increasing and continuous cdf with $G(1)=1$. I have proven some property for $G\in C^{\infty}([0,1])$ that relies on the continuity of $g(x)=G'(x)$. I hope ...
djsteve's user avatar
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-2 votes
1 answer
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HELP me solve this problem, I had been stuck with it for a while [closed]

Consider the function $f(x)=x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}$, then find $\lim_{n\rightarrow \infty}\sum_{x=1}^n \frac{1}{f(x)}$ BTW, the answer is 3/4, can't figure it out...
X_xBABAIx_X's user avatar
-1 votes
0 answers
61 views

The continuous functions $f, g : \mathbb{R} \rightarrow \mathbb{R}$ satisfy $f(x) = g(x)$ for all $x \in \mathbb{Q}$. [duplicate]

(a) Using the definition of continuity, prove that $f(x) = g(x)$ for all $x \in \mathbb{R}$. (b) Use sequential criteria of continuity to redo the problem. I was able to do the part (b) of this ...
Nicholas Gray's user avatar
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36 views

On continuous functions and convergent sequence

The function $f : (0, 1] \to\mathbb R$ is a bounded and continuous on $(0, 1]$. Let $\{x_n\}$ be a sequence in $[0, 1]$. Prove or disprove the following. (a) If $\{x_n\}$ is convergent, then $\{f(x_n)\...
Nicholas Gray's user avatar
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0 answers
32 views

Proving a sequence of functions isn't uniformly convergent

Let $f_n(x) = x^n(1-x^n)$. I need to prove that the sequence is not uniformly convergent in $[0,1]$. I have already proven that there is a pointwise convergence to $f(x)=0$. However, according to my ...
talopl's user avatar
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2 answers
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Uniform convergence of $f_n(x) = \frac{\ln(1+\frac{x}{n})}{x+1}$ [duplicate]

Basically i have a problem proving the sequence in the title 1. Uniformly converges on a closed interval $[0,a]$ where $a > 0$ and 2. Uniformly converges on $[0,\infty).$ So far i have found the ...
Timon Bubnič's user avatar
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3 answers
60 views

Bringing the limit inside the integral with asymptotic equilvance?

I was working on a limit problem and I think I found the solution to it, but I am not sure that what I did is right. I am working with the following sequence of functions $$ f_{n}(x) = n\, e^{\frac1{n\...
Giovanni Petrone's user avatar
2 votes
1 answer
59 views

Schwartz function dense in weighted Sobolev Space

Given a function $u$ on the Gevrey space with norm defined by \begin{equation} ||u||_{G^{\sigma,s}}=||e^{\sigma|\xi|}(1+|\xi|)^s\hat{u}(\xi)||_{L^2} \end{equation} for $\sigma\geq 0$ and $s\in \mathbb{...
heyy's user avatar
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1 vote
1 answer
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Proof this is the only polynomial that satisfies equation

This is question B-2 from the 1985 Putnam competition. It is relatively straight forward, and with a bit of calculation we can see that the polynomial sequence $$f_{n}(x)=x(x+n)^{n-1}$$ works and we ...
Riccardo Caiulo's user avatar
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0 answers
21 views

Sequence index family

Consider a geometric sequence $a_q=n×2^{q-1}$ where $a_1=n$ is odd. Basically the sequence above implies the following representation: $(n, 2n. 4n, 8n, ...)$ What I'm interesting in, is to study the ...
OSMANI MOURAD's user avatar
1 vote
1 answer
61 views

For which $a$ does $f_n(x) = \frac{x(1-x^2)^n}{n^a}, n=1,2,3,...$ converge pointwise and uniformly on $[0,1]$?

For which $a$ does $f_n(x) = \frac{x(1-x^2)^n}{n^a}, n=1,2,3,...$ converge pointwise and uniformly on $[0,1]$? I start with $\lim_{{n \to \infty}}f_n(x) = \lim_{{n \to \infty}} \frac{x(1-x^2)^n}{n^a} =...
Karl Johan's user avatar
1 vote
0 answers
54 views

Show existence of polynomial that estimates $f$ and whose derivative estimates $f'$.

I'm self-studying from Abbott Analysis and am working on this question: Assume $f$ has a continuous derivative on $[a,b]$. Show that there exists a polynomial $p(x)$ such that $|f(x) - p(x)| < \...
Cole's user avatar
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2 votes
1 answer
148 views

Variant of Baby Rudin problem 7.20

I am attempting to prove the following, which is a variant of Baby Rudin problem 7.20: Let $f:[0,1]\to\mathbb R$ be a Riemann-integrable function with $|f(x)|\le1$ for all $x\in[0,1]$. Suppose $$\...
Jimmy Yang's user avatar
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1 answer
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First n-element Sum of the First Order Polynomial Series

in order to sum a first order polynomial series that goes like: $a_0 = 100,$ $a_1 = a_0 + 55 + 10*(1)$ (where n is the index) $a_2 = a_1 + 55 + 10*(2)$ $a_3 = a_2 + 55 + 10*(3)$ ... $a_n = a_{n-1} + ...
barack's user avatar
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0 answers
21 views

Pointwise convergence, continuity of limit function

I'm self-studying from Abbott's Real Analysis. They begin with an attempt to prove that the pointwise limit of continuous functions is continuous, and show that it can't be done. Assume $(f_n)$ is a ...
Cole's user avatar
  • 66
1 vote
1 answer
25 views

Proving that a sequence of function does not uniformly converge

I am trying to prove that the sequence of functions $f_n (t) = n^2 t e^{-nt}$ does not converge uniformly on $\mathbb{R}_{\geq 0}$. I was able to show that it converges point-wise to the zero function....
JohnT's user avatar
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32 views

The sequence of algebraic numbers $C_n$ in $\int_0^{2 \pi} \ln(\sin(x)^{2n+1} + C_n) dx = 0$

Consider the following integrals $\int_0^{2 \pi} \ln(\sin(x)^{2n+1} + C_n) = 0$ All of the $C_n$ are algebraic numbers. In fact all these $C_n$ can be given as zero's of some integer polynomial with ...
mick's user avatar
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0 votes
0 answers
33 views

convergence of functions.

Let a sequence of functions $v_m$ that converges to $v$ in the space $L^{10/3}(\Omega)$, Where $\Omega$ is a bounded domain. Additionally, $\sup\limits_{m} |v_m|<C$, where $C$ is a constant. Can I ...
Andrés Ortiz 's user avatar
1 vote
0 answers
65 views

How to correctly choose number for Cauchy sequence for infinitesimal functions? Will it be $x=y=[sin(t)]$ or $x=y=[t]$?

I read about Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, page 4) and there was case with Cauchy sequence for infinitesimal functions: ...
Mike_bb's user avatar
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0 votes
0 answers
30 views

Rate of convergence of a threshold defined with sequences of functions

Let $f$ be a real decreasing function (resp. $g$ a real increasing function), and $(f_n)$ be a sequence of real decreasing functions (resp. $(g_n)$ be a sequence of real increasing functions) such ...
Skywear's user avatar
  • 192
1 vote
1 answer
57 views

Are convergent sequences closed under uniform convergence?

Setting: Let $Y$ be a metric space and let $a_{n,k}\in Y$ for all $n\in\mathbb{N}$ and $k\in\mathbb{N}$. Suppose $a_{n,k}\to a_{\bullet, k}$ uniforly as $n\to\infty$. Suppose the sequences $\{a_{n,k}\...
John Frank's user avatar
2 votes
2 answers
67 views

Find the rate of a sequence solving a polynomial inequality

Let $x\geq1$ be an integer and suppose that $n\to \infty$. Show that $x=O(n^c)$ if $$x^{1+a} \leq n + x^an^{-b}$$ for some $a>0$ and $b\geq1$. It looks to me that $x=O(n^{1/(1+a)})$, but I need to ...
John Smith's user avatar
1 vote
0 answers
42 views

Proving $f_n=nxe^{-nx}$ is not Uniformly convergent

To prove that $f_n= nxe^{-nx}$ is not uniformly convergent on $(0,\infty)$ I came up with a proof, But need to check whether that is correct or not... Proof: It is easy to see that $f_n$ converges ...
Praveen Kumaran P's user avatar
3 votes
1 answer
70 views

Construct a sequence of real analytic functions solving the equation with a condition

Consider the Laplace equation in two dimensions: $$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0$$. Construct a sequence of real analytic functions $(u_k)_{k=1}^\infty$ with $u_k:...
Bagaringa's user avatar
  • 402
0 votes
1 answer
43 views

Uniform convergence of $g_n(x) := \frac{(x+1)^2 e^{-x} \sin x}{n^2x^2+n\sqrt n x +1}$ over $[0, +\infty)$

This is part of a larger exercise: does $$g_n : [0, +\infty) \to \mathbb R ,\ g_n(x) := \frac{(x+1)^2 e^{-x} \sin x}{n^2x^2+n\sqrt n x +1}$$ converge uniformly for $x\ge0$? The sequence of functions ...
user665110's user avatar
3 votes
2 answers
92 views

Does $f_k(x):=\frac{1}{x^{1+\frac{1}{k}}}$ converge uniformly?

Let be $f_k:[1,\infty)\to\mathbb{R}$ with $f_k(x):=\frac{1}{x^{1+\frac{1}{k}}}$. Does this sequence converge uniformly? My approach: It is easy to see that $(f_k)_{k\in\mathbb{N}}$ converges ...
Philipp's user avatar
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0 votes
0 answers
82 views

If $f_n \to f$ pointwise and $f'_n \to g$ pointwise, does it follow that $f'=g$?

I'm trying to solve the following problem (which seems to be the same as in Derivative of a pointwise limit of a sequence of functions but it does not have answers) If $f_n:[a,b] \to \mathbb{R}$ is a ...
MC2's user avatar
  • 751
1 vote
2 answers
134 views

Find all intervals of uniform convergence of $f_n(x)=\frac{nx}{nx^2+1}$

I'm trying to solve the following problem: Let $f_n: \mathbb{R} \to \mathbb{R}$ be the sequence of functions given by: $f_n(x) = \dfrac{nx}{nx^2+1}$. Determine whether the sequence converges pointwise ...
MC2's user avatar
  • 751
1 vote
0 answers
50 views

If $f$ and each $f_k$ are continuous and $f_k \to f$ pointwise then $f_k \to f$ uniformly [duplicate]

I'm trying to show whether the following statement is true or not: If $X$ and $Z$ are matric spaces and $f_k:Z \to X$ is a sequence of continuous functions such that $f_k \to f$ pointwise (for some ...
MC2's user avatar
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2 votes
2 answers
211 views

Exercise on sequence of a function

Subject: Seeking Help with a Mathematics Exercise Hello everyone, I hope this message finds you well. I'm reaching out to seek assistance with a mathematics exercise that has been posing some ...
Henry D's user avatar
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0 votes
1 answer
39 views

The sequence has a stationary accumulation point

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth (continuously differentiable), convex function with a non-empty set of minimizers and $\{x^k\}$ be a sequence such that (a) $\{x^k\}$ has an ...
Dat Ba Tran's user avatar
0 votes
1 answer
118 views

PMA exercise 7.13

I don't understand why all of these steps of the hint to prove a). Is monotone convergence theorem enough to prove a? The sequence is bounded and monotone then $\exists N(\epsilon,x)$ s.t $|f_n(x)- f(...
Mathematics enjoyer's user avatar
0 votes
1 answer
72 views

Is it possible to construct a "divergent" sequence of functions?

I'm trying to understand pointwise versus uniform convergence of functions. So I want to find a sequence $\{f_n\}$ of functions $[0,1] \to \mathbb{R}$ that fails to converge pointwise to a function ...
Mathematical Endeavors's user avatar
3 votes
0 answers
89 views

why integral converges

Let $v$ be a probability measure on $\mathbb{R}$ such that $v$ has no atom. Let $\left(x^{i, N}\right)_{1 \leqslant i \leqslant N}$ be the sequence of real numbers defined by: $$ \begin{aligned} x^{1, ...
Document123's user avatar
0 votes
0 answers
22 views

Pointwise ad unifrom convergence of $g_n : \mathbb R \times \mathbb R^+ \to \mathbb R$.

I have the following exercise: Consider the function $$f : \mathbb R \times \mathbb R^+ \to \mathbb R \,,\ f(x,y) := \frac{\log\left(1+(y-x)^2\right)}{y^2+y}$$ Does there exist the limit of $f(x,y)$ ...
user665110's user avatar
1 vote
0 answers
84 views

Are there other ways to graph a zigzag line?

I teach precalculus and I was addressing a common misconception that students have when I came across something puzzling to me. When we graph $\sin^{-1}(\sin(x))$ and $\cos^{-1}(\cos(x))$ we get a ...
GhostyOcean's user avatar
4 votes
1 answer
360 views

Particular Integral, with sequence

Subject: Seeking Help for a Computer Science Contest - Integral Estimation Hello everyone, I hope this message finds you well. I am currently preparing for an ongoing computer science contest, and I ...
Henry D's user avatar
  • 153
1 vote
1 answer
41 views

limit of the integral $\int_0^A \frac{1}{n!} [\log(\frac Bx)]^n~dx$ where $A,B$ are positive constants

Let $A,B>0$ be fixed constants. I am trying to compute the limit $$\lim_{n\to \infty}\frac{1}{n!} \int_0^A\left[\log\frac{B}{x} \right]^n~dx, $$ where $[-]$ is the floor function. Can we ...
blancket's user avatar
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0 votes
0 answers
71 views

Sequence of discontinuous functions converge uniformly [duplicate]

Let $X \subseteq \mathbb{R}$ be a subset. Let $\{f_n\}$ be a sequence of real valued functions such that $f_n \to f$ uniformly on $X$. Let $D_n$ denote the set of discontinuities of each $f_n$ and $D$ ...
user_1729's user avatar
0 votes
0 answers
139 views

Is the product of two uniformly convergent sequence of functions uniformly convergent?

I am doing the exercise V.1.7 of Amann and Escher's Analysis. In the exercise I am required to prove that the product of two uniformly convergent sequence of functions is uniformly convergent if just ...
John's user avatar
  • 1
0 votes
1 answer
67 views

Counterexample for a convergent sequence of functions on a $\sigma$-finite measure space

I have already proven following statement, but I am struggling to construct a counterexample, even with the hint. Every kind of help is appreciated: Let $(X, A, \mu)$ be a measure space with $\mu(X) &...
alex.'s user avatar
  • 47
1 vote
0 answers
43 views

A question for uniform convergent of sequence of functions.

Consider the sequence of functions $<f_n(t)>$, defined as $ f_n(t) = \begin{cases} e^{-t^2} & \text{if } -n \leq t \leq n \\ \frac{e^{-n^2}}{[1-n(t-n)]} & \text{if } n \leq t &...
neelkanth's user avatar
  • 6,100
2 votes
0 answers
120 views

Questions and observations regarding Putnam 2020 - A.6.

CONTEXT My starting point is Question A.6 of Putnam 2020 competition, that goes like that For a positive integer $N$, let $f_N$ be the function defined by $$ f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)...
dfnu's user avatar
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0 votes
0 answers
53 views

Show that $B_1(0)$ is a closed set in the space $C([0,1])$

Let be $(C([0,1]),\Vert \cdot\Vert_{\infty})$ the normed space of continuous functions, equipped with the supremum norm $\Vert\cdot\Vert=\sup\limits_{x\in[0,1]}|f(x)|$. Show that $B_1(0):=\{f\in C([0,...
Philipp's user avatar
  • 4,564
0 votes
0 answers
76 views

Does the sum of a sequence of functions $\{f_n(x)\}$ converge if the upper bound $h_n(x)$ with $f_n(x)\le h_n(x)$ converges pointwise to $0$?

I am dealing with a problem on the existence of the limit for a sequence of function $\{g_n(x),n\in\mathbb N\}$ with function $g_n(x):\mathbb R^n\to\mathbb R^m$, i.e. to judge the existence of limit $...
OwnCandy's user avatar
0 votes
0 answers
104 views

Does the existence of limit of a sequence formed by continuous functions at some points imply the existence of the limit at other points?

$\{A_i,i\in\mathbb N\}$ is a fixed matrix sequence with element $A_i\in \mathbb R^{n\times m}$. $\Phi\in\mathbb R^{m\times m}$ is a constant matrix and $d\in\mathbb R^m$ is a vector. The sequence $\{...
OwnCandy's user avatar
1 vote
1 answer
94 views

What is the pattern so we can make the next stars?

I found the following pattern question in a group! It took me a lot of time but unfortunately I don't have any ideas to find any logical thing here. Here's the picture of the question: The question ...
Amirreza Hashemi's user avatar
1 vote
1 answer
41 views

Find a sequence of functions

There are familiar functions whose derivatives are periodic. $e^x$ has a period of $1$, since $$\frac{d}{dx} e^x = e^x$$ And $\sin(x)$ has a period of $4$. I am interested in finding a sequence of ...
Carlyle's user avatar
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