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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,

  1. $f_n:\Bbb R \to \Bbb R$ defined by $f_n(x)=\frac {1}{1+nx^2}$
  2. $f_n:\Bbb R \to \Bbb R$ defined by $f_n(x)=e^{-nx^2}$
  3. $f_n:\Bbb C \to \Bbb C$ 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.

1010 questions
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Convergence of $f_n(x)=\frac{x}{1+nx^2}$ and its derivative $f'_n(x)=\frac{1-nx^2}{(1+nx^2)^2}$ on $\mathbb{R}$

Consider the sequence $f_n(x)=\frac{x}{1+nx^2}$ on $x\in \mathbb{R}$. Also consider $f'_n(x)=\frac{1-nx^2}{(1+nx^2)^2}$ on $x \in \mathbb{R}$. I want to check their uniform convergence. Convergence of $f_n$: $f_n \to 0$ pointwise on…
Saikat
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If $S_0=1$ and $S_n=1-e^{-S_{n-1}}$ then prove that $0\le S_n\le1$ and $S_n$ converges

If $S_0=1$ and $S_n=1-e^{-S_{n-1}}$ then prove that $0\le S_n\le1$ and $S_n$ converges. My Solution: $S_0=1, \ S_n=1-e^{-S_{n-1}} \implies S_1=1-{1\over e}
Chris
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On uniform convergence of functions with compact support

If sequence of functions $f_{n}$ in $C_C(\mathbb{R})$ converges pointwise to the functions $e^{-x^2}$, then can we conclude $f_n$ converge uniformly.
mathlover
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Problem with pointwise convergence of a sequence of functions.

I'm working with this sequence of functions: $$f_n(x)=\frac{\log x}{\arctan{x^{1/n}}+x^{n}}$$ for $n\geq 2$ and $x \in (0,+\infty)$. I have to find $f$ such that $f_{n} \rightarrow f$ pointwise in $(0,+\infty)$. So, my idea is to compute the…
muserock92
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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 points $y=2$ and $y=4$ when $x=\sqrt{2}$. However…
Goblin
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Calculate the next value?

I'm generating the following sequence , $i = 1 \dots \infty$: $$ y_i = 1/i $$ Given that I have the calculated-value of : $$ y_x $$ how do I calculate the next value $y_{x+1}$ ? I don't know the index '$x$', I know only the value $y_x$ For…
sten
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Let $\{x_n\}_{n\ge1}$ be a sequence of positive real number and if $\{x^2_n\}_{n\ge1}$ is convergent, then is $\{x_n\}_{n\ge1}$ is convergent?

Let $\{x_n\}_{n\ge1}$ be a sequence of positive real number and if $\{x^2_n\}_{n\ge1}$ is convergent, then is $\{x_n\}_{n\ge1}$ is convergent? $\{x_n\}_{n\ge1}$ it's convergent because $\{x^2_n\}_{n\ge1}$ is convergent means $\lim_{n\to…
Chris
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Baire functions question

I have some doubts regarding a question in a Lebesgue theory book by Spiegel. The question is as follows: $\begin{align} &\text{Given}\\ &\quad\text{a) } f_n(x)=e^{-nx^2} \text{ in } \;0\leq{x}\leq{1}\;\text{ and}\\ &\quad\text{b) }…
Atha
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Sequence of a function

Let $A_n$ be a sequence such that $A_0\in\mathbb R$ and $A_{n+1}=f(A_n)$. Is it possible to choose value for $A_0$ such that $A_n$ is a monotonic geometric sequence or a monotonic arithmetic sequence? My attempt: I started from the 3…
Hasan
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Weaker condition than uniform convergence to interchange limit and integral

We know that the relation between uniform convergence and integration in case of interchangeability. I.e. Suppose and are integrable and fn--->f uniformly on [,] . Then lim→∞(a to b)∫fn(x)dx=(a to b)∫f(x)dx. But I want some weaker condition…
SJA
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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$ converges and find pointwise limit if it…
TUC
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Find the limit function for the sequence of function problem

Let us consider the sequence of function $$ f_n(x)= \begin{cases} nx, x\in [0, 1/2^n] \\ 1/nx, x \in (1/2^n, 1]\\ \end{cases} $$ Find the limit function $f(x).$ My work: if $x \in [0, 1/2^n]$ then $f_n(x)$ goes to zero for sufficiently large $n.$…
Manglu
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Prove some facts about a sequence of functions?

I have this simple sequence of functions: $$f_{n}(x)=\frac{\sin(nx)}{nx^{\frac{3}{2}}}$$ I want to see if I can correctly: Prove that $f_{n}\in\ L^{1}(0,+\infty)$ for every $n\in\mathbb N$. Prove that $f_{n}\rightarrow0$ as $n\rightarrow \infty$…
muserock92
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Constructing a sequence of function

Construct a sequence of functions on [0,1] each of which is discontinuous at every point on [0,1] and which converges uniformly to a function that is continuous at every point?
Abdeslem SMAHI
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convergence of a complex sequence of function

$f_n(z)=\frac{z^n}{n}$, where it is uniformly convergent? well, $\frac{z^n}{n}\to 0\forall |z|\le 1$ am I right?
Myshkin
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