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.
1,046 questions
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Find an interval for which $f_n(x) = \frac{x}{1+nx}$ does not converge uniformly
I am studying for my Real Analysis course and one of my practice problems asks us to "prove the sequence of functions $f_n(x) = \frac{x}{1+nx} \to f$ uniformly on certain intervals."
I've ...
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Does Convergence of Arc Length Imply Uniform Convergence?
Define $L_f(a,b)$ denote the arc length of the graph of the function $f$ on $(a,b)$.
For a sequence of functions $f_n(x):D\to \mathbb{R}$ that converges to $f(x)$, even if $f_n$ converge uniformly to $...
- 9,329
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When is it justified to take a limit inside a series? [closed]
Let
$$
f(x)=\sum_{k=1}^{\infty}(-1)^k(k+1)\,\chi_{\left(\frac1{k+1},\,\frac1k\right]}(x),
\qquad x\in(0,1].
$$
Thus $f$ is constant on each interval $\left(\frac{1}{k+1},\frac{1}{k}\right]$, taking ...
- 537
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Does the iteration of the function $N=2^{k+1}\cdot m-1$ , (where $m=\frac{2^{t}\cdot n+1}{3^{k+1}}$) eventually produce all odd multiples of 3?
Starting from from $n=1$ , does the iteration of the function $N=2^{k+1}\cdot m-1$ , (where $t,k$ are whole numbers) eventually produces all odd multiples of $3$ where $n$ is the initial odd number in ...
- 21
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Does this function disprove the statement that uniform convergences preserves no jump discontinuities?
This is question 5 from chapter 4 from Pugh's Real Mathematical Analysis. Part (f) asks if uniform convergence preserves "no jump discontinuities." I believe that I have created a function ...
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Exercise 1 from the Scuola Normale admission exam 2024-25
So I’m trying some problems from previous years of the "bando ordinario of the Scuola Normale" here is the first question from the academic year 2024–25.
Let $f:\mathbb{R}\to\mathbb{R}$ be ...
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Jensen Inequality Exercise - Show that $\lVert x \rVert _{p, \text{avg}} \to e^{\lVert \ln x \rVert _1}$ as $p \to 0$.
Note: I’m in a very beginner step of $L^p$ spaces and norms. Detailed answer would be very appreciated.
Jensen Inequality Exercise - Show that $\lVert x \rVert _{p, \text{avg}} \to e^{\lVert \ln x \...
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If $f_n$ is locally constant around $x$ then $(f_k(x))_{k\geq n}$ is constant where $(f_n)$ converges to Cantor's staircase
Let $\mathcal{C}([0,1])$ the set of continuous functions on $[0,1]$. Let $E = \{f \in \mathcal{C}([0,1])~|~f(0)=0, f(1)=1\}$.
I've then considered the operator $S: E \longrightarrow E$ defined for $f\...
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Equivalent Condition to Eventual Level-Boundedness
Are these two forms of eventual level-boundedness actually equivalent?
Let $\ell_n : \mathbb{R} \to \mathbb{R} \cup \{-\infty\}$ be a sequence of upper semi-continuous functions.
Rockafellar & ...
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Show Eventually Almost Surely Concavity of a Log Likelihood
I have the following Log Likelihood
$$\mathcal{L}_n(\theta)\colon= -\frac{1}{2}\sum_{k=1}^{n}\int_0^T\lambda_{k}^{2\beta-2\gamma}(m_{k,\theta}(t)-m_{k,\theta_{0}}(t))^2 dt-\sum_{k=1}^{n}\int_0^T\...
- 83
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Relatively prime polynomials of $\mathbb{Q}[x]$
Let $(Q_n)_{n\geq1}$ be a sequence of monic polynomials of $\mathbb{Q}[x]$ such that the product $Q_1 \cdots Q_n$ divides $Q_{n+1}$. I need to prove that
$\gcd(Q_2\cdots Q_n + Q_3\cdots Q_n+ \cdots+ ...
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Calculate $\lim_{n \to \infty}[ \int_{3}^{4} \sqrt{-x^2+6x-8}^n \,dx ]$
This is what I've done so far:
I wrote $-x^2+6x-8 = 1-(x-3)^2$, then I did a substitution $t=x-3$ so that $dt = dx$. For $x=3, t=0$ and for $x=4, t=1$.
So that my integral became: $$I_n=\int_{3}^{4} \...
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Sequence of functions is uniformly convergente if is cauchy
The proof that a Cauchy sequence of functions $f_n:X \to \mathbb{R}$ is uniformly convergent goes like this:
For each fixed $x\in X$, the sequence $(f_n(x))_{n\in\mathbb{N}}$ is Cauchy. Therefore,...
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If $f_n(x+\frac{1}{n}) = f_n(x)$ and $f_n\rightarrow f$ uniformly, then $f$ is a constant function.
Let $(f_n)$ be a sequence of continuous functions such that
$$f_n:\mathbb{R}\to\mathbb{R}, ~~ f_n(x+1/n)=f_n(x)$$
for all $x\in \mathbb{R}$ and $n\ge 1$.
Suppose $f_n\to f$ uniformly on $\mathbb{R}$. ...
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$ \lim_{x \to x_0} f_n(x) = +\infty $ then $ \lim_{x \to x_0} f(x) = +\infty$
Let $$\ f_n : X \to \mathbb{R} \ $$ converge uniformly to f.
(A) $$\\ $$
we have that for all ( n ):
$$ \lim_{x \to x_0} f_n(x) = +\infty. (B)
$$
I want to show that:
$$ \lim_{x \to x_0} f(x) = +\...
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