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.

A sequence of functions is denoted as $\{ f_n \}$ or $(f_n)$. For example,

$f_n:\Bbb R \to \Bbb R$ defined by $f_n(x)=\frac {1}{1+nx^2}$
$f_n:\Bbb R \to \Bbb R$ defined by $f_n(x)=e^{-nx^2}$
$f_n:\Bbb C \to \Bbb C$ defined by $f_n(z)=\frac {\sin nz}{\sqrt n}$

are sequences of functions. Usually, our first encounter with them is to study their nature of convergence.

1039 questions

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Is the uniform limit of uniformly continuous functions, uniformly continuous itself?

That sounds a lot like a tongue-twister. I know that there exist sequences of Lipschitz functions whose uniform limit is not Lipschitz (for instance, just use Weierstrass theorem on $[a,b]$). Clearly if the sequence is uniformly Lipschitz, then the…

uniform-convergence uniform-continuity sequence-of-function

asked Jan 12 '18 at 12:32

Paolo Intuito

1,194

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2 answers

Dirac delta sequences

Is it true that any sequence of real functions $(\delta_n)_n$, such that $$\lim_{n\to\infty} \delta_n(x) = 0 \qquad \forall\,x\ne 0$$ and $$\int_{-\infty}^\infty \delta_n(x)\,dx = 1 \ ,$$ tends to a delta function, $$\lim_{n\to\infty} \delta_n(x) =…

real-analysis dirac-delta sequence-of-function

asked Apr 11 '14 at 18:17

Ivica Smolić

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2 answers

If $(f_n')$ converges uniformly on an interval, does $(f_n)$ converge?

Let $(f_n)$ be a sequence of functions that are all differentiable on an interval A, and suppose the sequence of derivatives $(f_n')$ converges uniformly on A to a limit function $g$. Does it follow that $(f_n)$ converges to a limit function f on…

real-analysis uniform-convergence sequence-of-function pointwise-convergence

asked Jan 02 '22 at 23:22

S C

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3 answers

basis of vector space of real sequences over $\mathbb{R}$

It turns out I cannot find a basis for the vector space of all functions from $\mathbb{N}$ to $\mathbb{R}$ (over $\mathbb{R}$). By Zorn's Lemma, there is a basis. So I guess it cannot be written out constructively? What is the dimension then? I am…

linear-algebra functional-analysis set-theory sequence-of-function

asked Nov 17 '13 at 16:56

mez

10,887
5
55
109

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2 answers

Prove that $\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$ is equal to $f(1)$.

$\mathbf{Question}:$ Let $f$ be a continuous function on $[0,1]$. Then prove that the limit $\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$ is equal to $f(1)$. $\mathbf{Attempt}$: First, we try to show that the sequence of functions $\{nx^nf(x)\}_{x\in…

real-analysis limits proof-verification uniform-convergence sequence-of-function

asked Aug 31 '19 at 08:38

Subhasis Biswas

2,660

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1 answer

Converging sequence of solution to a differential equation

I was working on a problem about a sequence of functions, each of which is a solution to a sequence of differential equations, that converges to a function which is supposed to be the limit of the previous sequence I mentioned. The problem is as…

ordinary-differential-equations lipschitz-functions sequence-of-function

asked May 24 '18 at 02:37

Ignacio Rojas

1,212

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1 answer

Given sequence of $L-$Lipschitz functions which converges pointwise, prove uniform convergence

Let $f_n:[a.b]\rightarrow \mathbb{R}$ be sequence of $L-$Lipschitz functions, that is: $$\forall x,y\in[a,b]: |f_n(x)-f_n(y)|\leq L|x-y|$$ Suppose $f_n \rightarrow f$ pointwise, prove $f_n \rightrightarrows f$ I have all the parts of the puzzle…

real-analysis proof-writing uniform-convergence lipschitz-functions sequence-of-function

asked Jul 12 '17 at 17:02

Itay4

2,214

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2 answers

If $f_n$ is a sequence of continuous periodic functions with bounded periods that converges to $f$, is $f$ necessarily periodic?

I came up with the following problem myself. Prove or disprove: Let $f_n$ be a sequence of real-valued continuous periodic functions defined on $\mathbb{R}$. Suppose that there exists a $M$ such that for each $n$, there exists a period $T_n$ of…

real-analysis analysis continuity periodic-functions sequence-of-function

asked Feb 27 '25 at 14:26

19021605

1,049

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1 answer

Is $\left\{\sqrt{n}\sin^{\circ n}\left(\dfrac{z}{\sqrt{n}}\right)\right\}$ locally uniformly convergent over $\mathbb{C}-\{y{\rm i}:|y|\ge\sqrt{3}\}$?

Note $\sin^{\circ n}x$ as the $n$-fold iteration of $\sin x$. It is shown in this great answer that $$ \lim_{n\to\infty}\sqrt{n}\sin^{\circ n}\left(\dfrac{x}{\sqrt{n}}\right)=\dfrac{\sqrt{3}x}{\sqrt{x^2+3}}, \quad\forall x\in\mathbb{R}. $$ I was…

sequences-and-series complex-analysis limits analysis sequence-of-function

asked Oct 25 '23 at 12:22

Jianing Song

2,545
5
25

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0 answers

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 differentiable function is correct. [Abbott 6.4.3] (a) Show…

real-analysis sequences-and-series solution-verification uniform-convergence sequence-of-function

asked Nov 23 '22 at 05:59

Quasar

5,644

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1 answer

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 6.4.6] Let \begin{equation*} f( x) =\frac{1}{x}…

real-analysis sequences-and-series solution-verification uniform-convergence sequence-of-function

asked Nov 20 '22 at 20:34

Quasar

5,644

votes

5 answers

Summation problem: $f(x)=1+\sum_{n=1}^{\infty}\frac{x^n}{n}$

I want to evaluate this summation: $$S=1+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+...+...$$ where, $|x|<1$ Here it is my approach $$S=1+\sum_{n=1}^{\infty}\frac{x^n}{n}=f(x)$$ $$f'(x)=1+x+x^2+x^3+...+...=\frac{1}{1-x}$$ $$f(x)=\int f'(x)dx=\int…

calculus sequences-and-series convergence-divergence power-series sequence-of-function

asked Aug 23 '18 at 19:04

lone student

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6 answers

Show that the sum of reciprocal products equals $n$

I don't even know how to proceed. Please help me with this. (Original at https://i.sstatic.net/DRIX8.jpg) Consider all non-empty subsets of the set $\{1, 2, \ldots, n\}$. For every such subset, we find the product of the reciprocals of each of…

summation sequence-of-function

asked May 10 '18 at 21:42

user390410

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4 answers

Finding the limit of a sequence of integrals

Let us define a sequence of function as $$f_n(x)=\frac{2nx^{n-1}}{x+1}\;\;\text{for each $x\in [0,1]$ and for all $n\in\mathbb{N}$}$$ What is $\displaystyle \lim_{n\to \infty} \int_0^1 f_ n(x) dx$ ? How to find the limit? If I can interchange…

real-analysis limits sequence-of-function

asked Feb 16 '18 at 15:31

mathemagic

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3 answers

Uniformly convergent implies equicontinuous

I'm trying to prove that if I have a sequence of continuously differentiable functions $f_n$ that converge uniformly on $[a,b]$, then $\{f_n\}$ is equicontinuous for all $x_0 \in [a, b]$. My idea is to use uniform convergence to deal with the "tail"…

real-analysis uniform-convergence sequence-of-function equicontinuity

asked Mar 25 '15 at 01:20

Rebekah

1,037

2 3

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