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
votes
3answers
19 views

Pointwise convergence of $f_n(x) = \left(\frac{3^{x/n} + e^{x/n}}{2} \right)^{-n}$

I am given the sequence $f_n : [0,\infty) \to \mathbb R$ such that $$ f_n(x) = \left(\frac{3^{x/n} + e^{x/n}}{2} \right)^{-n} $$ and I am asked to discuss its convergence almost everywhere w.r.t. the ...
0
votes
1answer
23 views

Convergence of Sequences of Functions Evaluated at Sequences in Their Domain

Suppose (1) A sequence of continuous functions$f_n(x)$ converges to a continuous function $f(x)$ pointwise on some set $I$, and (2) a sequence $\{x_n\}$ converges to $x \in I$ (****) Is it true ...
0
votes
0answers
18 views

Differentiable limit of a (uniformly convergent) sequence of differentiable function (again, but not exactly !)

If one has a uniformly convergent sequence of differentiable functions $f_n$ (say from $\mathbb{R}$ to $\mathbb{R}$), we know that the limit $f$ is not always differentiable. Even if it is ...
-1
votes
1answer
19 views

sequence of continous function

There exist a sequence of continuous functions {$f_n$} on $\mathbb{R}$ such that {$f_n$} converges to $f$ uniformly on $\mathbb{R}$,but $$\lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}f_n(x)dx ...
0
votes
1answer
20 views

Proof for uniform convergence of a sequence of functions

I tried to prove the uniform convergence of the sequence of functions $$f_n \left( x \right) = \dfrac{nx}{1 + n^3x^2}$$ defined on $\left[ 0, 1 \right]$ in the following manner:- First, it is easy ...
0
votes
1answer
29 views

Uniform convergence in open intervals

I am doing Uniform convergence of functions and in this section I notice one thing that we most of the time talk about uniform convergence of sequence of functions in Closed Interval. Can we talk ...
0
votes
1answer
41 views

Is there a sequence $(f_n)_n \subset C_b(\mathbb{R})$ with $f_n \to f \in C_b(\mathbb{R})$ pointwise but $f_n(1/n) \not \to f(0)$?

Can we construct a sequence of functions in the space $C_b(\mathbb{R})$ (continuous and bounded functions) such that $$ f_n(x) \to f(x) \ \ \forall x \in \mathbb{R} $$ where $f \in C_b(\mathbb{R})$ ...
2
votes
2answers
76 views

Calculate the limit of $\frac{n^{n+1}}{n!}\int_0^xe^{-nt}t^n\,\mathrm dt$

Set $R_n(x)$ the quantity in the title, for $n\in\mathbb{N}$ and $x > 0$. I'm trying to prove that if $x < 1$, then $R_n(x)\underset{n\rightarrow +\infty}{\longrightarrow} 0$, if $x > 1$, ...
0
votes
1answer
19 views

show that $T_{k}(z)$= $\frac{p_{k}(z)}{(1-z)^{k+1}}$ for all $k \geq 1$ with $T_{0}(z)=\frac{1}{1-z}$ and $T_{k+1}(z)=1+zT'_{k}(z)$ $k \geq 0$

if $(T_{k}(z))_{k \geq 0}$ is sequence defined by $T_{0}(z)=\frac{1}{1-z}$ and $T_{k+1}(z)=1+zT'_{k}(z)$, $k$ $\geq 0$ show that $T_{k}(z)=\frac{p_{k}(z)}{(1-z)^{k+1}}$ for all $k \geq 1$ with $p_{...
0
votes
0answers
25 views

Uniform Continuity implies boundedness in sequence of functions

We want to show that, letting $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions on $[0,1]$ that's uniformly convergent, then there exists $M>0$ such that $\forall n\in\mathbb{N}$, $|f_n(x)|\leq M$...
0
votes
1answer
34 views

Proving that a sequence of functions has a convergent subsequence

Let $X$ be a metric space and $(f_n)$ an equicontinuous sequence of functions from $X$ to $\mathbb{R}$. We suppose that $A=\{x\in X| \{f_n(x),n=1,2,...\} \text{ is bounded}\}$, is not empty, and that $...
2
votes
1answer
42 views

A weird inequality regarding integrals, limits, as well as sequence of functions.

Consider a sequence of continuous function $f_n:[a,b]\to \mathbb{R}$. Suppose there exist constants $\gamma>1$ and $\beta>0$ independent of $n,p$ such that $$\left(\int_a^b|f_n(x)|^{p\gamma}dx\...
1
vote
0answers
65 views

Calculate $\lim_{n \to \infty} \int_0^1nx^nf(x)dx$. [duplicate]

Consider the function $f$ which is continuous. Calculate $\lim_{n \to \infty} \int_0^1nx^nf(x)dx$. Here first I attempted to prove $f_n=nx^n$ is uniformly convergent using sup-norm limit but ...
1
vote
0answers
35 views

Sum of an odd recursive sequence

Let $a_0 = 1$ $a_1 = 1 - \frac{e}{2}$ $a_n = \frac{e}{2^n} - \frac{1 - a_{n-1}}{n - 1}$ for $n > 1$. Find $\sum_{r=0}^{\infty}a_r$.
0
votes
1answer
44 views

Is it a continuous?

For each $n\in\mathbb{N}$, define a function $f_n:[0,1]\to\mathbb{R}$ by $$f_n(x)=\int_{1/n}^{1}\frac{t^{x}}{\sqrt{t+x}}\,dt.$$ Then, is it continuous for each $n\in\mathbb{N}$? I don’t understand ...
0
votes
2answers
26 views

Uniform convergence of a sequence of functions 4

Prove that the sequence $\left((nx)/(1+4n^2x^2)\right)_{n\in\mathbb N}$ is not uniformly convergent on $(-a,a)$, where $a > 0$ My attempt: $\lim_{n\to\infty}(nx)/(1+4n^2x^2) = 0 = f(x)$ Now, ...
0
votes
1answer
16 views

Puntual and uniform convergence of function

I am facing the following problem: Let $H:[0,1] \to R$ a function such thath $H(0) = 0 = H(1)$, H continous in $x = 0$ and $H(x) \neq 0$ for some $x$. Let $H_n:[0,1] \to R$ be such that $H_n (x) = H(...
0
votes
2answers
29 views

Uniform or pointwise convergence of a sequence of functions

My problem is that I find it kind of hard to contrast between uniform and pointwise convergence. For example with this proof I'm not quite sure whether I have proven uniform or poitwise convergence: $...
0
votes
1answer
18 views

Pointwise convergence of a sectionally defined function

I'm asking for a proof-verification of the following: $ f_n(x) := \left\{\begin{array}{ll} 2n^2x, & x\in [0,\frac{1}{2n}) \\ -2n^2x+2n, & x\in [\frac{1}{2n},\frac{1}{n})\\ 0,&...
2
votes
0answers
38 views

Prove that $C_p[-\pi,\pi]$ is complete, knowing that $C[-\pi,\pi]$ is indeed complete

Show that the space $$C_p[-\pi,\pi] = \{f \in C[-\pi,\pi] \; | \; f(-\pi)=f(\pi) \}$$ is complete knowing that the space $C[-\pi,\pi]$, i.e. continuous functions on $[-\pi,\pi]$, is complete. Here is ...
1
vote
1answer
36 views

Uniform convergence of digamma function

Let $F_n$ be a real valued function, $$F_n(r) = \frac{r}{n}(\psi(n) - \psi(r)$$ where $\psi$ is a digamma function. Let the sequence of functions $\{g_n\}$ be defined by $g_n(x) := F_n(nx)$. Show ...
0
votes
1answer
42 views

Prove that $(f_n)$, $f_n =x^n$, $x \in (0,1)$ is not uniformly convergent on $(0,1)$

Question 1. Prove that $f_n:(0,1) \to \mathbb{R}$ is not uniformly convergent on $(0,1)$, where $f_n = x^n , n\in \mathbb{N}$ . Proof: We need to show that, $\forall \ k \in \mathbb{N} $, $\exists \...
0
votes
1answer
26 views

Pointwise convergence of $h_{n}(x)$ on [0,$\infty$)

I know that it converges pointwise to $1$ if $x>0$ and to $0$ if $x=0$ using limits . But I am struggling to show this formally. Any help would be greatly appreciated . Thanks
0
votes
1answer
17 views

How to show a function is locally C^1 implies globally C1?

Actually there is a series problem like f(x)=sum(n=1 to ∞)[sin(nx^2)/1+n^3], the question was whether f(x) is C^1 or not. This question has already answered, but a big issue of mine is I can't find ...
0
votes
1answer
43 views

Let $g_n(x)=\sum_{k=1}^n (-1)^k f_k(x) \forall x\in \mathbb R. $ Then which one of the following are correct answers?

Suppose that $\{f_n\}$ is a sequence of continuous real-valued functions on $[0,1]$ satisfying the following: (A)$\forall x\in \mathbb R,\{f_n(x)\}$ is a decreasing sequence. (B)the sequence $\{f_n\}...
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 ...
0
votes
0answers
26 views

Bounding the maximum of a sequence of continuous functions using integrals

I have absolutely non clue on how to solve this one. First, recall that $$\lim_{p \rightarrow +\infty }\left ({\int\limits_a^b |f(x)|^{p}dx)} \right) ^{\frac{1}{p}} = \max_{x \in [a,b]} |f(x)|$$ ...
0
votes
1answer
27 views

Necessity of compactness in Dini's theorem

In Rudin there's an example given to support the necessity of compactness fn(x)=1/(1+nx) x€(0,1) Which is not uniformly convergent. My question is if the interval would have been compact i.e. if x€[...
0
votes
3answers
35 views

How to evaluate the following limit rearding sequence of functions

Let $g:[0,\frac{1}{2}]\rightarrow\mathbb{R}$ be a continuous function. Define $g_n:[0,\frac{1}{2}]\rightarrow\mathbb{R}$ by $g_1=g$ and $$g_{n+1}(t)=\int\limits_{0}^{t}g_n(s)ds$$ for all $n\geq1$. ...
3
votes
2answers
39 views

Is $f_n \rightrightarrows f $ implies $(f_n(x_n)-f(x_n)) \longrightarrow 0$?

Suppose we have a sequence of functions defined on $S$, $f_n$ that converges uniformly to $f$. Then if $x_n$ is any sequence in the interval can we say that $f_n(x_n)-f(x_n)$ goes to zero? Efforts: ...
1
vote
1answer
35 views

continuity of the function $f(x)=\lim_{n\to \infty}\sum_{k=0}^{n-1} \dfrac{x}{(kx+1)[(k+1)x+1]}$

I need to check the continuity and differentiability of the function $f(x)$ at $x=0$ where, $$f(x)=\lim_{n\to \infty}\sum_{k=0}^{n-1} \dfrac{x}{(kx+1)[(k+1)x+1]}.$$ I tried to check the domain of ...
0
votes
2answers
29 views

Uniform Convergence continuous

True or False: "If $(f_n)$, where each $f_n$ is continuous, converges to $f$ on $S$ and $f$ is not continuous on $S$, then the convergence is not uniform. This statement seem false to me, since I ...
0
votes
0answers
39 views

A function sequence that converges uniformly over $[ a, + \infty[ $ but doesn't for $ ]0,+\infty[$

I have a sequence function that I'm asked to check the point-wise convergence and uniform convergence: $f_n = ne^{-n^2x^2}$ for $x \in R$ The pointwise convergence is pretty easy to do, we find ...
0
votes
2answers
42 views

If $f_n\to f$ pointwise, $f$ is continuous and $f$ is continuous, then $f_n \to f$ uniformly.

Let $(f_n)$ a sequence of continuous function on $[a,b]$ that converges pointwise to $f$. We suppose that $f$ is continuous on $[a,b]$. Prove that the convergence is uniform. I'm stuck at some point :...
1
vote
1answer
36 views

Properties of convergence in $L^{\infty}$

Let $\Omega \subset \mathbb{R}$ be a bounded domain and $\alpha > 0$ be fixed. Assume that $|| u_{n} - v||_{L^{\infty}(\Omega)}\to 0$ as $n\to\infty$. How can I show that $||\, |u_{n}|^{\alpha} - |...
0
votes
1answer
28 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
votes
2answers
47 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
votes
1answer
24 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
56 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
vote
1answer
59 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
votes
1answer
22 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
vote
0answers
23 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 ...
3
votes
2answers
137 views

Show that $f_n(x)= \frac{\sin^2(n^{\alpha}x)}{nx}$ is not uniformly convergent for $\alpha \geq 1$ and $x \neq 0$

Can someone help me with proving that \begin{align} f_n: \mathbb{R} \rightarrow \mathbb{R }, \hspace{0.5cm} f_n(x)= \begin{cases} \frac{\sin^2(n^{\alpha}x)}{nx} & \text{ if } x \neq 0, \\ 0 &...
1
vote
0answers
36 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
votes
2answers
170 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^...
0
votes
1answer
57 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
37 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
votes
1answer
25 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
votes
1answer
21 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
31 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 \...