# 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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### 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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### 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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### 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 vote
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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 ...
• 4,950
1 vote
51 views

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 $-\ln(\... • 1,147 2 votes 1 answer 46 views ### Interval of convergence of the infinite series$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: $$g( x)... • 4,950 1 vote 0 answers 44 views ### Converse of the Weierstrass M-Test I am self-learning real-analysis from the text, Understanding Analysis, by Stephen Abbott. I would like someone to verify if my below counterexample to the exercise problem 6.4.2 (c) is valid and ... • 4,950 1 vote 1 answer 49 views ### Absolute value of a polynomial fraction I'm trying to understand an exercise on series, and I can't understand the following equality in the exercise:$$\left|\frac{-2(n+1)}{n^2+1}\right| = 2\frac{2n+1}{n^2+1} \leq 2\frac{2}{n} = \frac{4}{... • 113 1 vote 0 answers 47 views ### 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 ... • 4,950 1 vote 0 answers 24 views ### 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$\mathbf{R}$... • 4,950 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$h$... • 1,147 0 votes 1 answer 32 views ### 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 ... • 1,147 2 votes 1 answer 26 views ### 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 ... • 4,950 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}$. Unless some more conditions are placed ... • 1,767 1 vote 0 answers 17 views ### equality of limits related to harmonic numbers Let$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$. I need to prove that \begin{equation} \displaystyle\lim_{n \to \infty} \sum_{k=2}^{\... • 11 0 votes 1 answer 22 views ### Divergence of sequences and associated rate I'm given with two sequences of positive reals$\{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_\...
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 ...