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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Swap $\limsup$ and $\mathbb{E}$ of sequence of functions evaluated at a random variable?

I am a postgrad with more of a background in the functional analysis point of view of things but am recently needing to get the expectation involved and it has been a few years since I've done much ...
1 vote
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21 views

Closed-form solution to a recurrence relation of polynomials

A few days ago I asked a question about a probability puzzle. After a lot of working out, I found that the answer to the puzzle involves a pair of recurrence relations of polynomials. To be explicit, ...
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1 answer
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sequence of functions for which the derivative is bounded converges uniformly?

I'd like to prove that for a sequence of differentiables functions $(f_n)$ on $[0;1]$ which converges to $0$, actually converges uniformly. We only know that $\vert f'_n(x)\vert \leq 2015 + \cos(x) $ ...
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2 answers
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The sequence $\{x_{n}\}$ by $x_{n}=\frac{1}{2\pi}\int_{0}^{\pi/2}\tan^{\frac{1}{n}}t \ dt$ is such that $\{x_{n}\}$ converges to $1/4$

For $n\geq 2$, define the sequence $\{x_{n}\}$ by $$x_{n}=\frac{1}{2\pi}\int_{0}^{\pi/2}\tan^{\frac{1}{n}}t \ dt$$ Then prove that the sequence $\{x_{n}\}$ converges to $1/4$. My Attampt: I think it ...
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1 answer
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Study the convergence of the series $\sum_{n\in\mathbb{N}}\left(\frac{e^{it}}{2}\right)^{n}$

I would like to study the series $$ \sum_{n\in\mathbb{N}}\left(\frac{e^{it}}{2}\right)^{n} $$ My idea is to first look at what can be said concerning any type of convergence: $$ \forall{t}\in\mathbb{R}...
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1 answer
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What is difference between Convergence and Fixed point?

i have recently studied fixed points but it left some questions to me. i saw a function $x^{x^{x^{.^{.^{.}}}}}$, leading to $x^y=y$ when it converges to the value $y$. for instance, we can find fixed ...
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1 answer
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Uniform convergence of $f_n(x)=\int_0^x f(t^n)dt$, where $f$ is continuous on $[0,1]$.

How to prove uniform convergence of $f_n(x)=\int_0^x f(t^n)dt$ on $[0,1]$, where $f$ is continuous on $[0,1]$. My attempt. By changing variables $t^n=s$, we see $|f_n(x)|\leq x\max_{[0,x^n]}|f|$. ...
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Convergence of arguments in optimization

This question is motivated by non-parametric maximum likelihood estimation is statistics but I guess it applies more generally to any optimization problem. Let $\{x_1,x_2,\dots,x_n\}$ be a data sample ...
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If $f_n$ converges to $f$ over an interval $I$ then it also converges over $I\setminus \{p\}$

I was given the following question as part of my Calculus 2 assignment: Let $\{f_n\}$ be a sequence of functions that converge to $f$, but doesn't converge uniformly, over an interval $I$. Given $p\in ...
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Uniform convergence of the sequence of derivative functions

I am self-learning Real Analysis from the text Understanding Analysis by Stephen Abbott. I would like for someone to verify if my proof is technically correct and sufficiently rigorous. Theorem 6.3.2 ...
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1 vote
1 answer
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If $(f_n)$ converges pointwise to $f$, then $f$ is uniformly continuous

I am preparing for my Analysis $2$ final and am looking over some problems on past homeworks and exams that I could not solve. This is one of them: Prove that if $(f_n)_{n\in\mathbb{N}}$ is ...
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Prove that the sequence of functions is increasing [duplicate]

Can anyone help me out here? I am trying to find the way to prove that the following sequence of functions is increasing: $$F_n(x)=\left(1+\dfrac{x}{n}\right)^n \text{ for } 0 \leq x \leq n$$ Thanks a ...
1 vote
1 answer
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Sequence of uniformly bounded functions over the positive integers has a certain convergent subsequence

This is Theorem 1.24 from Reed and Simon's Methods of Modern Mathematical Physics, Vol 1: Functional Analysis Let $f_n(m)$ be a sequence of functions on the positive integers which is uniformly ...
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On Bartle's 'Elements of Integration' Exercise 10.J

I haven't found a reference solution for this problem on the net, and I'm wondering whether my proof makes sense, and whether it contains mistakes or possesses any other major flaws. Also: For its ...
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Prove that $f_n(x)=\sqrt{(x-1/2)^2+1/n}$ converges uniformly to non-differentiable function $f(x)=|x-1/2|$ on the interval $[0,1].$

I was checking the counter example for sequence of functions $f_n$ in $C^1[0,1]$ such that $\lim_{n\to \infty}f_n$ is not differentiable. I got the counter example from here: Sequence of ...
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Derivatives of the limit of a sequence of functions [duplicate]

I would like to show the following result Let $(f_n)_n$ be a sequence of functions $C^{1}$ of an interval $I$ of $\mathbb{R}$ with real or complex values. We suppose that we have the following two ...
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Continuity of the limit of a uniform convergent sequences of functions

I'm trying to prove the following result. Let $(f_n)_{n}$ be a sequence of functions defined on an interval $I$ with real values. We suppose that all $f_n$ are continuous and that the sequence $(f_n)...
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Two contractions with the same fixed point. Show that this sequence is convergent

We have $f,g:R\rightarrow R$ two contractions with the same fixed point and $(x_{n})_{n\geq1}$ a sequence with real numbers with the propriety that $x_{n+1}\in \left\{f(x_{n}),g(x_{n})\right\}$, for ...
2 votes
1 answer
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Is $f(x)=\sum_{k=1}^{\infty} \frac{\sin (x/k)}{k}$ continuous, differentiable and twice-differentiable?

Exercise problem 6.4.8 from the text Understanding Analysis by Stephen Abbott asks us to investigate the continuity, differentiability and twice-differentiability of the function $f(x)$ defined below. ...
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7 votes
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Exploring the continuous nowhere differentiable function $g(x) = \sum_{n=0}^{\infty} \frac{\cos {2^n x}}{2^n}$

I am self-learning Real Analysis from the text Understanding Analysis by Stephen Abbott. I would like someone to verify if my proof for the below exercise problem on a continuous nowhere ...
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1 vote
1 answer
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Convergence of a sequence of functions with different domains

Consider the sequence $\{f_n\}_{n=1}^\infty$ of functions $f_n: (-\infty, n) \to \mathbb R$ given by $$f_n(t) = \frac{1}{n-t}.$$ On an intuitive level it feels like this sequence should have a limit, ...
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7 votes
1 answer
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Convergence, continuity and differentiability of $f(x)=\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+2}-\frac{1}{x+3}+\ldots$

I am self-learning Real Analysis from the text, Understanding Analysis by Stephen Abbott. I'd like someone to verify, if my below proof and deductions are rigorous and technically correct. [Abbott 6....
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1 vote
1 answer
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Finding the interval of convergence of the infinite series $\sum_{n=1}^{\infty}\frac{2^n + x^n}{1+3^n x^n}$

Exercise problem 3.2.1(c) in Problems in Mathematical Analysis, by Kaczor and Nowak asks to find where the following infinite series converges pointwise: $$\sum_{n=1}^{\infty} \frac{2^n + x^n}{1+3^n x^...
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2 votes
1 answer
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Interval of convergence of the infinite series $\sum_{n=1}^{\infty} \frac{x^n}{1+x^n}$

I am self-learning Real Analysis and solving some exercise problems from Problems in Mathematical Analysis, by Kaczor and Nowak. I would like someone to verify if the arguments below are technically ...
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1 vote
1 answer
51 views

Analyzing convergence of sequence of functions

Consider the sequence of functions $f_n:[a,\infty)\to \mathbb{R}, f_n=x-a-\ln(\frac{nx+1}{na+1})$. For strictly positive $a$, analyze the uniform convergence of the sequence. It's clear that $-\ln(\...
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2 votes
1 answer
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Interval of convergence of the infinite series $g(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{1+x^{2n}}$

I am self-learning real-analysis from the text, Understanding Analysis by Stephen Abbott. I'd like for someone to verify, if my deduction below is rigorous and correct. [Abbott 6.4.4] Define: $$g( x)...
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1 vote
0 answers
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Converse of the Weierstrass M-Test

I am self-learning real-analysis from the text, Understanding Analysis, by Stephen Abbott. I would like someone to verify if my below counterexample to the exercise problem 6.4.2 (c) is valid and ...
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1 vote
1 answer
49 views

Absolute value of a polynomial fraction

I'm trying to understand an exercise on series, and I can't understand the following equality in the exercise: $$\left|\frac{-2(n+1)}{n^2+1}\right| = 2\frac{2n+1}{n^2+1} \leq 2\frac{2}{n} = \frac{4}{...
1 vote
0 answers
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On Bartle's 'Elements of Integration' Exercise 7.U

I haven't found a reference solution for this problem on the net, and I'm wondering whether my proof makes sense / is correct. Also I believe that it should be possible to find a simpler proof for the ...
2 votes
2 answers
54 views

Showing that $\lim f_n'=f'$ where $f_n(x)=\frac{nx^2+1}{2n+x}$ and $f = \lim f_n = \frac{x^2}{2}$

I am self-learning Real Analysis from the text, Understanding Analysis by Stephen Abbott. I am having some troubles proving part c) of the below exercise problem. Any hints/suggestions without giving ...
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1 vote
0 answers
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Show that the sequence of derivatives $(h_n')$ diverges for every $x\in\mathbf{R}$, where $h_n(x)=\frac{\sin nx}{\sqrt{n}}$

I am self-learning Real Analysis from the text, Understanding Analysis by Stephen Abbott. [Abbott 6.3.4] Let $$h_n(x) = \frac{\sin nx}{\sqrt{n}}$$ Show that $h_n(x) \to 0$ uniformly on $\mathbf{R}$ ...
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0 votes
2 answers
35 views

Uniform convergence of sequence of differentiable functions

Show that the sequence of differentiable functions $x^n/n$ in $[0,1]$ converges uniformly to a differentiable function $f$ in $[0,1]$. Also, show that the sequence $f'_n$ converges to a function $h$ ...
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0 votes
1 answer
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Point wise and uniform convergence of sequences of functions [closed]

Analyze the uniform convergence of the following sequences of functions: $x+(1/n)$. What can we conclude about $(x+(1/n))^2$? $1/(1+x)^n$ in $[0,1]$. Also, study the point wise convergence of this ...
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2 votes
1 answer
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Uniform convergence of the derivative function $h_n'(x)$

I am self-learning Real Analysis from the text Understanding Analysis, by Stephen Abbott. I would like for someone to (1) Verify my proof for part (a) of this exercise problem. (2) Do you have any ...
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3 votes
0 answers
55 views

When does $f_n = f(x + 1/n)$, uniformly converge to $f$?

This source (p.15) claims that a sequence of functions $ f_n : \mathbb{R} \to \mathbb{R},$ with $f_n = f(x + 1/n)$, converges to $f$ uniformly on $\mathbb{R}$. Unless some more conditions are placed ...
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1 vote
0 answers
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equality of limits related to harmonic numbers

Let $H_{n,k}=\sum_{j=1}^{n}\frac{1}{j^k}$ and $\displaystyle\lim_{n \to \infty}H_{n,k}=H_{\infty,k}$ for $k >1$. I need to prove that \begin{equation} \displaystyle\lim_{n \to \infty} \sum_{k=2}^{\...
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Divergence of sequences and associated rate

I'm given with two sequences of positive reals $\{a_n\}_{n \in \mathbb{N}}$ and $\{b_n\}_{n \in \mathbb{N}}$ living in a compact set $\Omega$ so that $\lim_\limits{n \to \infty} a_n = 0$ while $\lim_\...
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1 answer
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No generalization of subadditivity for limit superior like Fatou's lemma.

We know that if $(a_n)$ and $(b_n)$ are two sequences in $\overline{\mathbb R}$, then we have $\limsup(a_n+b_n)\leq \limsup a_n +\limsup b_n$,So if $\{f_n:\mathbb N\to \overline{\mathbb R}\}$ is a ...
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0 answers
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$\frac{1}{t_n-s_n}(-t_ne^{-t_n}-e^{-t_n}+s_ne^{-s_n}+e^{-s_n}) \to 0$, as $t_n-s_n \to +\infty$

I am trying to show that $$\frac{1}{t_n-s_n}(-t_ne^{-t_n}-e^{-t_n}+s_ne^{-s_n}+e^{-s_n}) \to 0$$ $$\frac{1}{t_n-s_n}(t_n-e^{-t_n}-s_n+e^{-s_n}) \to 1$$ as $t_n-s_n \to +\infty$, with $t_n > s_n$, $\...
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2 votes
1 answer
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limit $\frac{\sqrt{t_n^2+1}-\sqrt{s_n^2+1}}{t_n-s_n}$ as $t_n-s_n \to + \infty$, where $t_n > s_n, \forall n.$

I am trying to show that $$\frac{\sqrt{t_n^2+1}-\sqrt{s_n^2+1}}{t_n-s_n} \to 1$$ $$\frac{t_n-s_n+\arctan(t_n)-\arctan(s_n)}{t_n-s_n} \to 1$$ as $t_n-s_n \to + \infty$, where $t_n > s_n, \forall n.$ ...
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0 answers
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When pointwise convergence implies local uniform convergence [duplicate]

Let $f_{n}:\mathbb{R } \rightarrow \mathbb{R }$ a sequence of continous functions and $f:\mathbb{R } \rightarrow \mathbb{R }$ a continous function. If $f_{n}$ convergens pointwise to $f$, than the ...
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2 votes
1 answer
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convergence of function series using Weierstrass M-test

I am trying to examine uniform convergence of series $$\sum_{n=1}^{\infty}{\frac{x^2}{n^4+x^4}}$$ For $x \ \in \mathbb{R}$. What i've tried: $$\text{assume } x \in [-R, R] \text{ for some } R>0$$ $$...
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1 answer
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Question regarding a step in Wikipedia's proof of the Radon-Nikodym Theorem

I have a question regarding Wikipedia's proof of the Radon-Nikodym theorem for finite measures: Why does there exist a sequence of functions $\{f_n\}$ in $F$ such that $$\lim_{n\to\infty}\int f_n\ d\...
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0 votes
1 answer
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A more succinct way of showing that $\{f_n(x)\}_n$ does not converge for any $x \in [0,1].$

Here is the proof I have: Is there a more succinct (elegant and organized) way of showing that $\{f_n(x)\}_n$ does not converge for any $x \in [0,1]$? Could someone help me with this, please?
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0 votes
0 answers
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Constructing a sequence $\{ f_n \} \subset C_{c}[0,1]$ such that ..

Constructing a sequence $\{ f_n \} \subset C_{c}[0,1]$(the space of continuous functions with compact support in $[0,1]$) such that .. $(a)$ $||{f_n}||_{\infty} \to \infty$ and $||{f_n}||_{1} \to 0.$ ...
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0 answers
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Picard Iteration scheme for an implicit ODE problem

I have been studying the Picard iteration method to solve an ODE and most of the literature that I have read seems to solve an ODE of the following form: $$\frac{dy}{dx} = f(x,y)$$ which is solved ...
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1 answer
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Can’t figure out whether a sequence of functions converges uniformly.

The sequence in question is $ f_n(x)= 1 - \exp (- \frac{nx^2}{nx + 1})$, and the interval is [ 1/2 , 3 ] I tried finding the sup of the difference between this and the limit function, using the ...
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Closed unit ball of a closure of convex hull.

We know that a convex hull is defined as the intersection of all convex sets containing a given subset of Euclidean space, equivalently as the set of all convex combinations of points in the subset. ...
2 votes
1 answer
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Picard Iteration: Proof of point wise convergence for system $f(y)=y^2$

This is essentially an extension of this question: Differential Equation and Picard Iteration. I have the following IVP: $$ \begin{align} y' &= y^2\\ y(0) &=1, \end{align} $$ which is to be ...
0 votes
1 answer
61 views

Find common element of two arithmetic sequences that does not have differential? [closed]

I have the following two sequences: $a^n$ $b+cm$ I want to find the common element where the $n$ and $m$ meet in result.

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