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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1answer
26 views

Finding functions that fulfil a condition and are strictly increasing.

I need to find $\lambda_{n}(t)$ such that it gives $0$ for $t = 0$ and $p + \dfrac{1}{n}$ for $t = p$ and $1$ for $t = 1$. The first case is fulfilled but I can't seem to find a function that would ...
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1answer
40 views

For an infinite sequence of functions $\Bbb{R}\to\Bbb{R}$, each function is a composition of a certain finite set of functions $\Bbb{R}\to\Bbb{R}$.

Given an infinite sequence of functions $\{g_1, g_2, \ldots, g_n, \ldots\}$ where $ g_n : \Bbb R \to \Bbb R$ prove there's a finite set of functions $ \{ f_1, f_2, \ldots, f_M \} $ such that any $ g_n ...
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2answers
13 views

Convergence of minimum of the sequence of functions and its limit function

Suppose ${f_n(x)}$ converges to $f(x)$ pointwise. Now consider the new sequence $g_n(x)=\min\{f_n(x),f(x)\}$. Can I conclude that $g_n$ also converges to $f$ pointwise?
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5answers
79 views

Help in proving that $\bigl (n^\alpha x^n (1-x) \bigr)_{n=0}^{\infty}$ is convergent.

I need some help in the following question: Let $\ 0<\alpha \in \mathbb R$. If $\bigl (f_n(x) \bigr)_{n=0}^{\infty}$ a sequence of functions as $\forall n\in\mathbb N \ \ \ f_n:[0,1]\to \mathbb R$...
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1answer
30 views

Study point wise and uniform convergence. [closed]

Let $(f_n)$ be a sequence of function defined on $\mathbb{R}^+$ by $f_0(x)=x$ and $f_{n+1}(x)=\frac{x}{2+f_n(x)}$ for all $n\in \mathbb{N}$ Study point wise and uniform convergent of $(f_n)$ on $\...
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2answers
31 views

Show that $g_n$ converges to $g$ uniformly.

Problem Let $f:\Bbb{R}\times[0,1]\rightarrow\Bbb{R}$ be a continuous function and $\{x_n\}$ a sequence of reals converging to $x$. Define $g_n(y)=f(x_n,y),\hspace{0.5cm}0\le y\le1$ $g(y)=f(x,y),\...
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1answer
56 views

About convergent subsequence. [closed]

Let $f_n$ be sequence of functions defined by: $$ f_n = \begin{cases} \cos(nx) & x\geq 1/n \\ x & x<1/n \end{cases} $$ My question is that if this sequence have subsequences ...
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2answers
42 views

A sequence of functions $f_n : [0, 1] → R$ which converges uniformly to a discontinuous function $f(x)$.

Give an example or argue that such a request is impossible. I argued that such a request is impossible because by theorem of the continuity of the uniform limit, if $f_n$ converges uniformly then ...
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0answers
27 views

nonnegative measurable function (monoton convergence theorem)

Let {$a_n$} be a sequence of non-negative sequence of real numbers. Define the function $f$ on $E=[1,\infty)$ by setting $f(x)=a_n$ if $n\leq x< n+1$. Show that $\int_Ef=\sum{a_n}$. The summation ...
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1answer
30 views

$L^{\infty}(\mathbb{R}^{N})$ and smallness of function

Let $(f_{n})_{n\in\mathbb{N}}\subset C(\mathbb{R}^{N})$ be a sequence of real valued function such that $\|f_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\to\infty$ as $n\to\infty$. Then, I know that there ...
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1answer
34 views

Check the pointwise convergence and uniform convergence of a sequence of functions.

For each $n\in \mathbb N$, let $f_n(x)=\begin{array}{cc} \Bigg\{ & \begin{array}{cc} nx^2 & 0\leq x\leq \frac{1}{n} \\ x & \frac{1}{n}<x \leq 1 \end{array} \end{...
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1answer
32 views

Proof of the limit for the sequence of function.

Here I need to prove / disprove that the limit for the sequence of functions is one. Here is what I have found: $$\lim_{n \to \infty} \sqrt[n]{\left(1+\frac{1}{n}\right)^n -e}=1 \\ \text{sequence } \...
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2answers
42 views

Find the limit of the sequence of functions: $f_n (x) = \frac{\ln (2^n +x^n)}{n}, x \ge 0$

So, the given function: $f_n (x) = \frac{\ln(2^n +x^n)}{n}, x \ge 0$. For $|x| < 1$ the limit would be: $\lim_{n \to \infty} \frac{\ln(2^n +x^n)}{n} = \lim_{n \to \infty}\frac{\ln (2^n + 0)}{n} = \...
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1answer
43 views

Weierstrass approximation polynomial with $p^{(i)}(0)=0$

Given a continuous real function $f:[0,1]\to\mathbb{R}$, for $k\in\mathbb{N}$, we need to find a rational polynomial $p$ satisfying $p^{(i)}(0)=0$ $(1\leq i\leq k-1)$ such that for $\epsilon>0$, $\...
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0answers
13 views

Belonging VS converging in a LP space

I'm reflecting on the meaning and the differences between belonging and converging to a $L^p$ space for a sequence of functions. Let's take this sequence as an example (please correct anything that is ...
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1answer
25 views

Backwards direction of Cauchy Criterion for Sequences of Functions

I am reviewing the proof of the Cauchy Criterion for sequences of functions and have a question regarding the backwards direction. Statement: Let $A\subseteq \mathbb{R}$ and $(f_n)$ be a sequence of ...
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1answer
35 views

Uniform convergence with sine functions

Let $f_n = \sin(\frac{3n}{4n+5}x) $ as $f_n:\mathbb{C} \rightarrow \mathbb{C} $. I try to determine if $f_n$ is uniformly convergent on the open intervals $I_1=(0; 1)$ and $I_2(1; \infty)$. Let $\...
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1answer
22 views

Why is a simple application of the Bolzano-Weierstrass Theorem not sufficient to prove Helly's Selection Theorem?

Helly's Selection Theorem Problem (From Rudin): Assume that $\{f_n\}$ is a sequence of monotonically increasing functions on $\mathbb{R}^1$ with $0 \le f_n(x) \le 1$ for all $x$ and all $n$. Prove ...
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1answer
57 views

Weierstrass' M-test in reverse

Weierstrass' M-test says that the series of functions on some set $X$: $$\sum_{n=1}^\infty f_n(x)$$ if $\forall n \in \mathbb{N}, \exists M_n$, \forall x\in X where $M_n \geq |f_n(x)|$, so the ...
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1answer
43 views

How to find limit of a sequence of functions $f_n(x)=\frac{x^n e^x} {n+1}$?

How to find limit of a sequence of functions $f_n(x)=\frac{x^n e^x} {n+1}$? $$\lim_{n\to \infty} \frac{x^n e^x}{n+1}$$ I have no idea how to evaluate this limit. I thought maybe I should rewrite $e^...
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2answers
32 views

$(f_n)$ integrable sequence of functions that converges uniformly to $f$, then $f$ is integrable

I'm reading Abbot's Understanding Analysis, and have stumbled across this problem, but there is a step that I don't fully understand. The problem is: Assume that for each $n$, $f_n$ is an (Riemann) ...
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1answer
51 views

Finding a recurrent relation or formula

The formulation of the original problem: "How many words can be made up of N sticks, if two sticks can make the letter i, and three sticks can make j?" Final task: create a formula (or recurrent ...
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1answer
40 views

What am I doing (simple uniform converge problem) $f_n(x)=n^2x^2e^{-xn}$

consider the function $f_n(x)=n^2x^2e^{-xn}$ I am asked if it uniform converge on $A=(a,\infty) \quad a>0$ So it easy to see that in converge to $0$, but when I wanted to check uniform converge I ...
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1answer
56 views

Point-wise convergence of $f_{n,m}(x) = \cos^{2n}(m!)\pi x$ [closed]

I have recently been trying some questions on convergence of sequence of functions.I got stuck in one of the problems in which I am supposed to find the point-wise limit and discuss the uniform ...
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1answer
34 views

Show that the sequence $g_{n}$ has a uniformly convergent subsequence.

Given a sequence $f_{n}:[0,1] \rightarrow [0,1]$ of continuous functions, define $g_{n}:[0,1] \rightarrow \mathbb{R}$ by setting $g_{n}(x)=\int_{0}^{1} \frac{f_{n}(t)}{(t-x)^{1/3}}dt$; $x \in [0,1]$. ...
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4answers
33 views

Uniform convergence of sequence of functions $\frac{2+nx^2}{2+nx}$ on [0,1]?

I have recently been trying some questions related to the uniform convergence of a sequence of functions. And meanwhile, I got stuck in one of the problems in which I have been supposed to discuss the ...
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1answer
39 views

The difference between the statements for sequences of function $f_n(x)$

Let I be an interval and c ∈ I. Statement A: For all $\epsilon$ > 0, there is $\delta$ > 0 such that,for all $n ∈ \mathbb{N}$ and for all $x ∈ I$ satisfying $|x−c|≤\delta$, $|f_n(x)−f_n(c)| ≤ \...
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2answers
54 views

How to show that $(f_n)$ uniformly converges to $f$?

Let $f:[0,1]\rightarrow\mathbb{R}$ be continuous. Consider the partition $\big\{0,\frac1n,...,\frac{n-1}{n},1\big\}$ of $[0,1]$. Define $$f_n(t)=\left\{\begin{array}{ll}f\big(\frac{k}{n}\big), & t\...
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2answers
49 views

Show that $\sum_{1}^\infty\frac{\sin(nx)}{n^3}$ is differentiable everywhere

I have recently been trying out some questions on series of functions.In one of the questions, I was given a series $$\sum_{1}^\infty\frac{\sin(nx)}{n^3}$$ and now I am supposed to show that the above ...
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2answers
39 views

Uniform Convergence of a series of functions using the Dirichlet's test

I have recently been trying out some questions on series of functions. I got stuck in one of those problems in which I am supposed to show that the below series of functions is uniformly convergent on ...
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1answer
20 views

Point-wise convergence of function

I have been trying out some questions on sequence of functions.In one of those questions,I am supposed to find the point-wise limit of the following sequence of functions defined on [$0,1$] as $$f_n(x)...
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0answers
36 views

limit of generalized functions $f_{\epsilon}(x) = \frac{sin^2(x/ {\epsilon})}{\pi x^2}$

I am trying to find this limit. Let's conisder $\phi \in C_0(R)$. $$\int^{\infty}_{-\infty} \frac{sin^2(x/ {\epsilon})}{\pi x^2}\phi(x)dx$$ = $$\int^{\infty}_{-\infty} \frac{sin^2(y)}{\pi \epsilon y^2}...
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0answers
36 views

Limit of a function for $\epsilon>1$.

I have been trying some questions on the convergence of a sequence of functions and was wondering about an intermediate step in which we have $\epsilon\gt0$ and an $x$ such that $0<x<1$ and it ...
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1answer
35 views

Uniform convergence of $x^n$ using the definition

I have been trying to prove the uniform convergence of sequence of functions defined by $f_n(x)=x^n$ on $[0,k]$ where $k<1$ by the epsilon definition of uniform convergence. I have found the point-...
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2answers
42 views

Uniform Convergence of $\frac{n}{x+n}$

I was trying an exercise on uniform convergence of sequence of real-valued functions. I got stuck in a problem in which I am supposed to prove that sequence defined by $f_n(x)=\frac{(n)}{(x+n)}$ is ...
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1answer
41 views

Is pointwise convergence permutation-invariant?

I am interested in convergence between countable sequences of real numbers. (Perhaps the definitions to follow are nonstandard. Sorry!) Say that the sequence $\langle \langle x^1_1,x^1_2,x^1_3,...\...
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1answer
32 views

Find $f(x)$ to which the given sequence of functions converges

$$f_{n}(x) = \begin{cases} \sin^{2}\pi x, & n≤ |x|≤n+1, \\ 0,& |x| < n \text{ or }|x|≥ n+1.\end{cases}$$ How can I find $f(x)$ to which $f_{n}(x)$ converges? I do always have problems ...
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1answer
96 views

Show that there exists a real number $R≥0$ such that, for all $x$, $y\in [0$, $1]$ and all $n \in \mathbb{N}$, $|g_n(x)−g_n(y)|\le R|x−y|$.

Assume that $(f_n)_n$ is a sequence of functions continuous on $[0$, $1]$, differentiable on $(0,1)$, that converge pointwise on that interval to a function $f$, and such that each $f_n$′ is bounded ...
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2answers
21 views

Proof for uniform convergence of sequence of functions

I was given this problem: These are my calculations and I'm asking for verification: Pointwise limit: $\lim_{n \to \infty} f_{n}(x) = \lim_{n \to \infty} \frac{x^{2n}}{1+x^{2n}} = \lim_{n \to \...
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1answer
26 views

uniform convergence of functional series $\sum_{k=1}^{\infty} {(-1)^{k} \frac{k+\sin(x)}{k^{2}}}$

I tried to solve the exercise that ask to define the convergence and the uniform convergence of the functional series $$\sum_{k=1}^{\infty} {(-1)^{k} \frac{k+\sin(x)}{k^{2}}}$$ it is easy to prove ...
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1answer
22 views

Find a limit for a sequence of functions with the domain in $(0, \infty)$

How to find limits for function $f_n = \sqrt{n}\left(\sqrt{x - \frac{1}{n}}- \sqrt{x}\right)$ if $Df \in (0,\infty)$ I think as $n \rightarrow \infty$ it would be $\infty$ and then no matter the $x$ ...
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0answers
22 views

Continuity of the sequence of averages

Suppose you have a sequence of bounded continuous functions $f_n:(a,b]\to\mathbb{R}$. Then, define the function $$S(x)=\liminf_{n\to\infty}\frac{1}{n}\sum_{j=1}^nf_n(x).$$ Is $S(x)$ continuous?
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1answer
48 views

Convergence $\sum_{k=1}^N \frac{a_k}{ \sum_{i=0}^k a_{i}^2} $

I am trying to see whether or not this series converges for $a_k>0,k\geq 0$ and $\sum_k a_k^2 < \infty$. When $a_k \geq 1$, I have this bound: $$ \begin{align*} \sum_{k=1}^N \frac{a_k}{\sum_{i=...
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1answer
39 views

Proving that a function sequence converges uniformly to limit function

I have the function sequence $f_n = n\sinh(x/n)$, $\forall n \in \mathbb{N}, \forall x \in [-1, 1]$ which I believe has the limit function $f = 1$ for $x = 1, f = 0$ for $x \in (-1,1)$ and $f = -1$ ...
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1answer
35 views

How to show that for $p>2$ this sequence of functions is uniformly convergent?

Let $f_n:[0,1] \to \mathbb R$ be defined by $\displaystyle f_n(x)= \frac{nx}{1+n^2 x^p}$ for $p>0$. Find the values of $p$ for which the sequence $f_n$ converges uniformly to the limit. In ...
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1answer
31 views

Dini's theorem (specific case)

Note: I asked this question before but it wasn't well written, So I deleted my previous question and re-wrote it. According to Dini's theorem: If $X$ is a compact topological space, and $\{ f_n \}$...
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0answers
45 views

Let $f_n(x) = {nx\over 1+nx^2}$ on the domain $[-1,1]$

(a) Find the pointwise limit function $f$ on $[−1, 1]$. (b) Show that $\lim\limits_{n \to \infty} \int_{-1}^1 f_n(x)dx$ exists. Is it equal to $\int_{-1}^1 f(x)dx$? This is my solution: For part a, ...
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1answer
29 views

Geometric sequence not correct?

So I was checking some Khan Academy excercise about a sequence and it went something like this... $$4, 25, 100...$$ It said that $f(1)=4$, $f(2)=25$ and $f(3)=f(1)f(2).$ So I was thinking about how ...
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3answers
83 views

If $\forall x \in (0,1] \ \ f_n(x) \to 0, n \to \infty$ then $\exists x_n \downarrow 0$ such that $f_n(x_n) \to 0$.

In the second edition of the book "Probability" (A. N. Shiryaev, R.P. Boas, 1996) there is a problem 2, p. 553 ($\S$ 8, chapter VII) which is equivalent to the next one. Suppose that for every $n \ge ...
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2answers
61 views

How to prove that $X \cap Y$ is closed?

I have to prove that $X \cap Y$ is closed. My idea is to apply the definition of a closed set: Consider a sequence $(x_n)$ of members of $X \cap Y$. If the sequence $x_n$ tends to some limit point ...

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