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.

251 questions
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Under given conditions whether $\lim\limits_{n\to \infty} \int_{-\infty}^{\infty}f_n(t)dt=\int_{-\infty}^{\infty}f(t)dt$ or not?

Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of continuous real-valued functions defined on $\mathbb R$ which converges pointwise to a continuous real-valued function $f$. Which of the following ...
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$\lim_{n\to \infty}\int_{0}^{1}\frac{2nx^{n-1}}{1+x}dx=?$ [duplicate]

For $n=1,2,...,$ let $f_n(x)=\frac{2nx^{n-1}}{1+x},x\in[0,1].$ Then $$\lim_{n\to \infty}\int_{0}^{1}f_n(x)dx=?$$ Here $f_n(1)=n$. So the limit function of $f_n(x)$ is not continuous. Also I was ...
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Bounded sequence of functions implies convergent subsequence

Here you can see my attempt at the proof. I am sure I did something wrong because my prof asked me to show it for rationals and I "somehow" showed it for all reals. I would appreciate it if someone ...
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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 ...
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Uniform convergence of $\frac{y/(2N)}{\sin(y/(2N))}$ towards 1

I can't come up with a proof, why $f_N(y) := \frac{\frac{y}{2N}}{\sin\left(\frac{y}{2N}\right)}$ converges uniformly against $1$ for $y\in(0,\pi),\ N\to\infty$. I would be thankful for any advice.
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$g_n = \max \{\min (f_n, g), -g\} \to f$

I am currently self studying Mathematical analysis by M. Apostol. I got stuck in trying to understand $\\$ Theorem 10.30 $\ \$Let ${f_n}$ be a sequence of functions in $L(I)$ which converges ...
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Limiting total variation attached to sequence of uniformly vanishing functions of bounded variation

Let $(f_n)_n$ be a sequence of real functions of a single real variable with compact support in $[0,1]$ and of bounded variation all of them. Let the sequence be uniformly convergent to $0$. Is it ...
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Uniform convergence of $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$, solution verification

Is my reasoning right? I have $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ for $x \in \mathbb{R}$, so I conclude that it's pointwise convergent $f_n \to |x|$, and moreover it's uniformly convergent to $|x|$, ...
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Sequence of continuous function converging pointwise to continuous function is equicontinuous?

I've proven the following "theorem": Let $I \subset \mathbb{R}$ be an interval, $(f_n: I \rightarrow \mathbb{R})_{n \in \mathbb{N}}$ be a family of continuous functions converging pointwise to a ...
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Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$?

Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$? Intuitively, I think that the answer is no. I know that the statement holds for ...
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Series extension of the superposition principle for ODEs

Take the superposition principle for linear ODEs of the form $y'(t)=A(t,y(t)) + g(t)$ ($y\in \mathbb{R}^n$, $A$ a linear function in y). If $g(t)=\sum _{k=1}^N g_k(t)$ then $y(t)=\sum _{k=1}^N y_k(t)$ ...
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Sequence of entire function that converges uniformly over on sets with empty interior

I have to prove that the sequence of entire functions: $$f_n(z)=\frac 1n \sin(nz)$$ converges uniformly over $\mathbb{R}$ (and this I managed to verify) but doesn't on every set with non-empty ...
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Limit of a locally uniformly convergent sequence of continuous functions

I have two questions: 1. I know that the uniform limit of a continuous functions is continuous. But I'm wondering whether this is true if the convergence is locally uniform. That is the uniform ...
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Applying theorem to disprove uniform convergence

I recently read this theorem in real analysis:(Actually a corollary to a theorem) {$f_m$} is a sequence of continuous functions defined on $D$ such that $f_m$$\to$$f$ uniformly on $D$ then for every ...
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Is $(f_n)$ pointwise convergent?

Let $f_n(x)$, for all n>=1, be a sequence of non-negative continuous functions on [0,1] such that $$\lim_{n→\infty}\int^1_0 f_n (x)dx=0$$ Which of the following is always correct ? A. $f_n→0$ ...
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Exercise of sequence of continuous functions

Let $(f_n)_n$ be a sequence of continuous functions on $D\subset \mathbb{R}^{N} \to \mathbb{R}$ which is monotone decreasing. If $\lim_{n\to\infty }f_n(c))=0$ for some $c\in D$ and $\epsilon >0$ , ...
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Finding values of $x$ such that a sequence of functions converges.

$(f_n)$$_n$$_\in $$_\mathbb N is a sequence of functions where f_n : [0,2\pi] \to \mathbb R \ \forall n \in \mathbb N. Find all values of x \in [0,2\pi] such that (f_n)$$_n$$_\in$$_\mathbb N$ ...
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How to prove two sequence have a common limit.

We have \begin{align} U_{0} &= 1 &&\text{and} & V_{0} &= 2 \\ U_{n+1} &= \frac{U_{n}+V_{n}}{2} &&\text{and} & V_{n+1} &= \sqrt{U_{n+1}V_n} \end{align} How to ...
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A sequence of functions decreasing to 0 is equicontinuous in a compact metric space.

How to proof that? Let M be a compact metric space and $\{f_n\} \subset C(M,\mathbb{R})$, so that $\{f_n\}$ is decreasing and $lim f_n(x)=0$, then $\{f_n\}$ is equicontinuous
The function $f_n(x)=3n^3(x-1/n)^2$ for $x\in[0,2]$ is given, and I need to show whether the sequence of functions $(f_n)_{n\in\Bbb{N}}$ converges uniformly to $f(x)=\lim\limits_{n \to \infty}f_n(x)$, ...
Let $I$ be a bounded interval of $\mathbb{R}$, and let $\{f_{n}:I\to\mathbb{R}\}$ be a sequence of differentiable functions. Suppose that a sequence $\{f_{n}(x_{0})\}$ converges for some $x_{0}\in I$ ...