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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37 views

Question on dominated convergence theorem.

I don't understand a statement in a paper I'm reading: We have a function $f:\mathbb{R}^n \rightarrow\mathbb{R}$. $f$ is convex and $\exp(-f)$ is a probability density function but I don't think that ...
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31 views

Prove that $\lim_{n\to\infty}Q_n(x)=g(x)$ for some polynomial $Q_n$.

Suppose that for any complex function it continues on $ [0,1] $ with $ f (0) = f (1) = 0 $. there exists a sequence of polynomials $ P_n $ such that $ \lim_{n \to \infty} P_n (x) = f (x) $ uniformly ...
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62 views

The set that $(f_n)$ converges to a real or a rational number is measurable

Let $(f_n(x)):\Omega\rightarrow \mathbb{R}$ be a sequence of measurable functions, the sets $$A=\{x \in \Omega: (f_n(x)) \text{ converges to a real number}\}$$ $$B=\{x \in \Omega: (f_n(x)) \text{ ...
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1answer
56 views

Pointwise convergent subsequence of a sequence of bounded functions on a countable domain

I was studying analysis on Abbot's understanding analysis, and I came across this exercise: Let $A = \{x_1, x_2, x_3, \cdots\}$ be a countable set. For each $n \in\mathbb N$, let $f_n$ be defined on $...
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1answer
74 views

The reciprocal of Abel's Theorem for Limits

Let $\sum_{n=0}^{\infty} a_n x^n$ be a series of powers with a convergence radius $r \in (0, \infty)$, $a_n \ge 0$ for all $n \in \mathbb Z_ +$ and $$f(x)=\sum_{n=0}^{\infty} a_n x^n, \quad |x| \lt r.$...
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1answer
25 views

Sequence of functions, uniform convergence, and dirac

Define the sequence of functions $$\delta_n(x) = \begin{cases} n \quad & \frac{-1}{2n} < x < \frac{1}{2n} \\0 & \text{else} \end{cases}$$ and define the polynomial $g(x)=c_1 + c_2x + \...
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1answer
117 views

Find a formula for this sequence of integer-valued polynomials

Let $f_n: \Bbb N_0 \rightarrow \Bbb N_0$ be a function for every $n\in \Bbb N_0$. Let $(f_n)_{n=1}^\infty$ be a sequence given by the following recurrence-relation: $$ f_1(k) = 1 $$ $$ f_2(k) = k $$ $$...
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Integral of polynomials & $f_k(x) = \begin{cases} 2k& x \in[\frac{-1}{k},\frac{1}{k}] \\ 0&x \in[-1,1]\setminus[\frac{-1}{k},\frac{1}{k}] \end{cases}$

Define a sequence of functions as follows: $$f_k(x) = \begin{cases} 2k& x \in[\frac{-1}{k},\frac{1}{k}] \\ 0&x \in[-1,1]\setminus[\frac{-1}{k},\frac{1}{k}] \end{cases}$$ Does ${f_k}$ converge ...
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42 views

A sequence $x_n(t)\in C^1[0,1]$ with $x(0)=0$ such that $x_n(t)\to x(t)$ in $C[0,1]$ and $x'_n(t)\to y(t)$ in $L^2[0,1]$ but $x'(t)\neq y(t)$

Terminology/Definition: The set $$ \Gamma(A):=\{(x,Ax):x\in\text{Dom}A\}\subseteq X\times Y $$ is called the graph of $A$. We define the norm in $X\times Y$ by $||(x,y)||:=||x||_X+||y||_Y$. We say ...
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Does the sequence of power-functions coverge to f (point wise, uniformly) and does its derivative exist?

I have the following questions for my upcoming exam and I am having trouble answering them. Let $x_0 ∈ \mathbb{R}$ and $f : (x_0 −R,x_0 +R) → \mathbb{C}$ be the function $f(x)=\sum_{k=0}^\infty a_k (...
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40 views

$\displaystyle\sum_{n=1}^{\infty} | f_n(x)|<\infty$ a.e. $\Rightarrow$ $\displaystyle\lim_{N\to \infty}\sum_{n=1}^N f_n (x)$ exist (at least a.e.)

Let $\{ f_n \}_{n=1}^{\infty}$ be a sequence of functions. If $\displaystyle\sum_{n=1}^{\infty} \big| f_n(x) \big|<\infty$ almost everywhere $x$, then does $\displaystyle\lim_{N\to \infty}\sum_{n=1}...
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64 views

Show that function series is bounded

Let $h$ be a continuous function on $\mathbb{R}$ such that $h(x) = o(|x|^{-\alpha})$ at $\infty$ for some $\alpha > 1$. Let also $(b_k)_{k\in\mathbb{Z}}$ an increasing sequence such that, $$\lim_{k\...
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1answer
34 views

Integral of $L^{p}(\mathbb{R}^{n})$ sequence outside a large ball

I've seen the question Integral of L^1 function outside a large ball Could it be generalized for a sequence? Specifficaly, let $(u_{j})$ a bounded sequence in $L^{p}(\mathbb{R}^{n})$, I would like to ...
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Given $x_{n+1}=x_n(1-x_n)$, find $\lim_{n \to \infty}n.x_{n}$ [duplicate]

Let the sequence $\{x_n\}$ be such that $0<x_{1}<1$ and $x_{n+1}=x_{n}\left(1-x_{n}\right)$. Find $\lim_{n \rightarrow \infty}n.x_{n}$ So far I have only been able to prove that the sequence is ...
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1answer
50 views

Recursive function converges to $e^t-t-1$

The recursive function $\phi_k:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ is defined as: $$\phi_0(t):=0,$$ $$\phi_{k+1}(t):=\int_0^t s+\phi_k(s) ds$$ Prove by Induction that $\phi_k(t)=\frac{t^...
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If $f(x)$ is a bounded function in $[a,b]$, then $-f(x)$ bounded in $[a,b]$

If $f(x)$ is a bounded function in $[a,b]$, then $-f(x)$ bounded in whose supremum is $-a$ and infimum is $-b$, but how can we say it's bounded in $[a,b]$ while its supremum and infimum is out of the ...
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86 views

For which values of $x$ the limit $\displaystyle\lim_{n\to \infty} (-1)^n \frac x{2^n}$ exists?

Let $f_1:[0,4]\to [0,4] $be defined by $f_1(x)=3-\frac x2$. Define $f_n(x) =f_1\left( f_{n-1}(x)\right)$ for $n\ge 2$. Find the set of all $x$ such that $\displaystyle\lim_{n\to \infty} f_n(x)$ exists ...
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If $f(x,t)$ is of class $C^1$ then does $f_n(x):=\frac{f(x,t_n)-f(x,t_0)}{t_n-t_0}$ uniformly converge to $\frac{\partial f}{\partial t}(x,t_0)$?

Uniform convergence Let be $X$ a topological space and $\{f_n\}_{n\in\mathbb{N}}$ a sequence of functions from $X$ to $I\subseteq\mathbb{R}$. So we say that the sequence $\{f_n\}_{n\in\mathbb{N}}$ is ...
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Not sure where my error is for integrating a sequence of functions

EDIT: The first comment on this post brought to my attention the error in my logic. I have since figured out where I went wrong! We have a sequence of functions on the domain $x\in [0,1]$ which looks ...
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32 views

Dini's Theorem Proof on the Reals

Dini's Theorem states that: Let $K$ be a compact metric space. Let $f:K→\mathbb R$ be a continuous function and $f_n:K→ \mathbb R,n∈\mathbb N$, be a sequence of continuous functions. If ${f_n}$ ...
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29 views

Limit of a function sequence

Suppose that: (1) for any $n\geq 1$, $F_{n}(\Delta )$ is a smooth function for $\Delta \in (0,1]$. (2) for any integer $k\geq 0$, \begin{equation*} \text{the limit }\lim_{\Delta \rightarrow 0}\frac{d^{...
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1answer
49 views

Uniform convergence of $f_n(x)=\frac{n\cdot x}{1+n^2}$

I want to check the uniform convergence of $\displaystyle{f_n(x)=\frac{n\cdot x}{1+n^2}}$. We have that \begin{equation*}f^{\star}=\lim_{n\rightarrow \infty}f_n(x)=\lim_{n\rightarrow \infty}\frac{n\...
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45 views

Pointwise convergence of this sequence $f_n: \mathbb{R}^+ \rightarrow \mathbb{R}$

Problem Let the following sequence $f_n: \mathbb{R}^+ \rightarrow \mathbb{R}$ given by $$ f_n(x) = \sum\limits^n_{k=1} \frac{1}{\sqrt{n^2 +k^x}}$$ Show that $f_n$ converges to a function $f$ such that:...
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1answer
32 views

Is the minimum of a sequence of uniformly equicontinuous functions continuous?

I have a uniformly equicontinuous (you can even assume that they are uniformly Lipschitz if that helps) sequence of functions $(f_n)_{n=1}^{\infty}$ on $[0,1]$ such that for any $x$, we have that $\...
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1answer
47 views

If $f:B\to\mathbb{R},\ B\subset\mathbb{R}$ is increasing then there is a sequence of strictly increasing functions whose pointwise limit is $f$

I have proved the following statement and I would like to know if my proof is correct and/or how it could be improved. Suppose $B\subset\mathbb{R}$ and $f:B\to\mathbb{R}$ is an increasing function. ...
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1answer
112 views

Show that $\frac{1}{f_{n}}$ converges uniformly to $\frac{1}{f}$ given the following

Please point out any flaws in my arguments Let $f_{n}$ be a sequence of functions on $[a,b]$ s.t. $f_{n}(x) \neq 0$ $\forall x \in [a,b], n \in \mathbb{N}$ and let $f : [a,b] \to \mathbb{R}$ and $\...
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1answer
39 views

Theorem about series of functions

If both series $\sum_{n=1}^{\infty} f_{\mathrm{n}}(x)$ and $\sum_{n=1}^{\infty} f_{\mathrm{n}}^{\prime}(x)$ converge uniformly on I and $f_{n}^{\prime}$ is continuous for every $n \in \mathbb{N}$, ...
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1answer
52 views

Finding $\lim_{N\to\infty}\int_{0}^{\infty}1-\left(1-\exp\left(-x^2\right)\right)^{N}\text{d}x$

Is my answer correct? Since $\forall N\in\mathbb{N}^{*}$, $x \geq 0$, $\left| 1-\left(1-\exp\left(-x^2\right)\right)^{N} \right|<1$, so by the Dominated Convergence Theorem, we have $\lim_{N\to\...
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1answer
56 views

Does $f_n(x)=\sin(x)^{1/n}$ uniformly converge in $I=[\frac {\pi}{4},\frac {3\pi}{4}]$

So I need to prove/disprove that $f_n(x)$ converge. I have proved it but I don't really like my solution and I hope to see $\epsilon$ proof ($\exists \epsilon>0$ such that $|f_n(x)-f(x)|<\...
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40 views

Monotonicity of sequence of functions - Dini's theorem

To apply Dini's theorem (https://en.wikipedia.org/wiki/Dini%27s_theorem) we must have monotonicity, i.e. either $f_k(x)\leq f_{k+1}(x)$ or $f_k(x)\geq f_{k+1}(x)$ for all $x$ and from a $k$ onwards. ...
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3answers
65 views

Uniform convergence of the sum $\sum_{n=0}^\infty {(n+1)x^n}$, $|x| < 1$

Originally I needed to prove the continuity of the sum, but $(n+1)x^n$ is continuous, so I only need to show the uniform convergence of the sum. I know that the sum is equal to $\dfrac {1}{(x-1)^2}$ ...
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34 views

Monotone convergence theorem (proof with Darboux sum)

Let $f_n$ be a sequence of increasing and Riemann integrable function (i.e $f_1(x)\le...\le f_n(x) \ \forall x \in [a,b]$. If $f_n\to f$ pointwise and $f$ is Riemann integrable, show that $\lim_{n\to\...
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Show uniform convergence of sequence of functions $f_n(x)=x^n$ on $[0,0.9]$

Show uniform convergence of sequence of functions $f_n(x)=x^n$ on [0, 0.9]. There is a similar question for the interval [0, 1) (https://math.stackexchange.com/questions/2191621/is-the-sequence-of-...
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2answers
61 views

Proving $f_n(x)$ not uniformly convergent

Let $f_n(x)$ be a sequence of functions that converge for $f(x)$ on $x\in[a,b]$ but it's not uniformly convergent on the same range of $x$. Prove that $f(x)$ is not uniformly convergent on $x\in(a,b)$....
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behaviour of sequence of bounded real functions $f_n$ converging uniformly to $f$

I've been going at a statement trying to prove it or find a counterexample: $f_n : R\rightarrow R$ is a sequence of bounded functions converging uniformly to $f: R \rightarrow R$. If each function $...
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23 views

Uniform convergence of the sequence of functions: $g_n(x) = \sum_{k=2}^n \frac{(-1)^k}{k\ln(k)}f(kx)$.

Let $f:\mathbb{R} \to \mathbb{R}$ be bounded. Let $(g_n)_n$ be the sequence of functions defined by \begin{align} g_n(x) = \sum_{k=2}^n \frac{(-1)^k}{k\ln(k)}f(kx). \end{align} I need to prove that $(...
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1answer
72 views

$f_n\rightarrow f$ uniformly in an interval

Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ a sequence of functions and $f:\mathbb{R}\rightarrow \mathbb{R}$. Which of the following statements are correct? (a) If $f_n\rightarrow f$ uniformly in each ...
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1answer
51 views

Show that sequence of antiderivatives has a subsequence that converges pointwise

I have this question from a real analysis assignment, For $n\geq1$, let $f_n:[0,1]\to\mathbb{R}$ be a continuous function with $$|f_n(x)|\leq 1+\frac{n}{1+n^2x^2}$$ Define $F_n:[0,1]\to\mathbb{R}$ via ...
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3answers
76 views

Prove $\sum_{n=1}^\infty(\frac{1}{a_{2n-1}}-\frac{1}{a_{2n}})$ convergent

Let $(a_n)_{n=1}^\infty$ Let be a positive, increasing, and unbounded sequence. Prove that the series: $$\sum_{n=1}^\infty\left(\frac{1}{a_{2n-1}}-\frac{1}{a_{2n}}\right)$$ convergent. We know that ...
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55 views

Limit of a sequence of functions of a sequence of reals

Let $C \subset \mathbb{R}$ be a compact set. Let $f$ be a function $\mathbb{R} \rightarrow C$ which is continuous in $x_0$, and an $\varepsilon$-interval around it, $x_n$ a sequence of numbers s.t. $\...
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1answer
51 views

Uniform convergence in $[0,1]$

I have problems with exercise: Study the uniform convergence on $[0, 1]$ of the sequence $f_n$ defined by: $f_n = \displaystyle\frac{t^2}{t^2+(nt-1)^2}$ My attempt: $\displaystyle\lim_{n \to{}\infty}{ ...
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2answers
48 views

uniform convergence of $f_n(x)=1/n^x$

Consider the sequence of functions: \begin{align} f_n(x) = \frac{1}{n^x}. \end{align} Does $(f_n)_n$ converge uniformaly on $[0,\infty)$. I proved via the Cauchy criterion that $(f_n)_n$ can not ...
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3answers
43 views

differentiable and sequences of functions [closed]

Let $f_n:(a,b)\rightarrow \mathbb{R}$ be differentiable and $f_n\rightarrow f$. Can $f$ be differentiated? Would it be $f'(x_0)=\lim_{n \to \infty}f_n'(x_0)$ for $x_0\in (a,b)$ ?
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1answer
217 views

$f_n(x)= f(x+n)$ show that the limit function is uniformly continuous

Let $f$ be a real-valued continuous function on $I=\{x\in \mathbb{R} | x \geq 0\}$. For a positive integer $n$ the function on $I$ is defined by \begin{align*}f_n(x)= f(x+n)\end{align*} Answer the ...
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1answer
50 views

pointwise convergence does not imply uniform convergence for series

Regarding sequences of functions $(f_n(x))$, I can wrap my head around the idea that uniform convergence $\Rightarrow$ pointwise convergence, but pointwise convergence does not imply uniform ...
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1answer
33 views

Uniform convergence of telescopic function series

I need to find if the series $$\sum_{k=2}^\infty\left(\cos\frac{x}{k}-\cos\frac{x}{k-1}\right)$$ converges uniformly on $(-\infty,\infty)$. My attempt: The partial sum of the series: $$\sum_{k=2}^n\...
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0answers
26 views

Find a sequence of continuous functions that converges to a piecewise function

Let $f(x)= \left\{ \begin{array}{lc} 0, & x \leq 0 \\ \\ 1, & 0 < x \\ \end{array} \right.$ Find a sequence of $f_{n}$ such each one of $f_{n}: \...
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1answer
113 views

Find the pointwise limit of the sequence$f_n(x) = {1\over 1+x} + {2\over 1+x^2} + {4\over 1+x^4}+…+{2^n\over 1+x^{2^n}}$

$f_n:(1,\infty)\to\mathbb R$ is a sequence of functions defined by $f_n(x) = {1\over 1+x} + {2\over 1+x^2} + {4\over 1+x^4}+...+{2^n\over 1+x^{2^n}}$ What is the pointwise limit of the sequence? Since ...
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1answer
37 views

$f_n(x) = (n+1)x^n(1-x)$, on which interval does $f_n$ converge uniformly?

$f_n(x) = (n+1)x^n(1-x)$, on which interval does $f_n$ converge uniformly? Consider $x\in(-1,1]$, it is easy to see that $f_n$ converges to $f(x) = 0$ as $n\to\infty$, but stuck in constructing a ...
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1answer
151 views

Is the pointwise limit of monotonically increasing functions a uniform limit?

For a real-valued sequence of functions $\{f_n\}_{n=1}^\infty$ on the closed interval $[0,1]$, each $f_n$ is monotonocally increasing and suppose that the sequence of functions $\{f_n\}_{n=1}^\infty$ ...

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