0
votes
0answers
41 views

Prove probing to be permutation

So I have been taught that probe sequences (h(k, 0), h(k, 1), ... , h(k,m - 1)) are meant to be a permutation (0, 1, ... ,m - 1), but how does one prove that? I was asked this question in an ...
1
vote
1answer
29 views

True or False, limit, functions questions. Does limit exist?

True or False Let a be a real number, and let f and g be real functions defined at all points x in some open interval containing a except possibly at x = a. a) For each natural n, the function ...
1
vote
1answer
35 views

Problem with itinerary of a coding problem with infinite 1's

If $f(x)=2x \ mod \ 1$ on $[0,1)$. Then if we code $x \in [0,1)$ with its itinerary w.r.t. the partition $P_0=[0,1/2)$ and $P_1=[1/2,1)$. Can you show that there is no point $x$ whose itinerary has ...
0
votes
0answers
39 views

On the limit of a special kind of function

Let $(a_n)$ be a strictly increasing sequence such that $ \sum \dfrac 1{a_n}$ converges ; then off-course $\lim \bigg(\dfrac 1{a_n}\bigg)=0$ , so $\lim (a_n)=\infty$ , in particular $(a_n)$ is ...
0
votes
1answer
35 views

Proving uniformly convergence on a Banach Space

Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$ and $$ {\cal L}_0^2(\mathbb R)=\left\{f:\mathbb R\mapsto\mathbb R\ |\ ...
0
votes
0answers
41 views

Proving inequality equation

Let $a_1, a_2,....,a_n$ be positive numbers such that $\sum_{i=1}^n a_i = 1$ Then for any vector $(x_1,x_2,...x_n) \ge (0,0,...,0)$ I want to show that $$x_1^{a_1}*x_2^{a_2}*...*x_n^{a_n} \le ...
1
vote
1answer
19 views

Series expansions and perturbation

My professor said that $ f \left( y_1(x)+ \epsilon y_2(x)+... \right)= f(y_1(x)) +f'(y_1(x))\> (\epsilon y_2(x)+...) + ...$ but I have no idea how the series continues. Has anyone seen this ...
0
votes
1answer
15 views

On the existence of a non-constant sequence whose differentiable image converges [duplicate]

Let $f: [a,b] \to \mathbb R$ be a function differentiable in $(a,b)$ , then is it true that there is a non-constant sequence $(x_n)$ in $(a,b)$ such that the sequence $\big(f(x_n)\big)$ is ...
1
vote
1answer
23 views

How can I construct a specific sigmoid function?

The simple sigmoid function $$f(x)=1/(1+e^{−x})$$ approaches zero as x tends to negative infinity, and approaches $1$ as x tends to positive infinity. But I want to set $1$ and $20$ instead of $0$ and ...
0
votes
1answer
22 views

How to determine a general arithmetic sequence formula for two intersecting trig function

I have equations out of two trigonometric functions. For example $\cos(4\alpha$) = -$\sin(5\alpha)$ $\tan(0.5\alpha$) = 2 $\sin(\alpha)$ How can I determine a general arithmetic sequence formula ...
1
vote
1answer
39 views

Viewing a sequence as a function on the space of positive integers

I see the following lines in a book : " Consider a bounded sequence of real or complex numbers $\{\eta_n\}$. Such a sequence $\{\eta_n\}$ defines a function $x(n) = \{\eta_n\}$ defined on the ...
2
votes
0answers
34 views

rewriting the inverse image

If $\phi_k:\mathbb{R}^2\rightarrow \mathbb{R}$ are continuous functions, for all $k\geq0$ and $$\phi=\limsup_{n\rightarrow \infty }\phi_n$$ Let $A\subset \mathbb{R}$, is possible to write ...
1
vote
1answer
51 views

Is there a uniformly continuous function such that $a_{n+1} = f(a_n)$?

Let $a_{n+1} = a_n - a_n^2$ and $a_1 = \frac{2}{3}$. I already proved that $a_n \to 0$ Now I was asked, is there a uniformly continuous function such that $a_{n+1} = f(a_n)$? All I can think of is ...
2
votes
1answer
37 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
2
votes
1answer
29 views

Convergent Sequence of Analytic Functions, Do Their Derivatives Converge?

If you have a sequence of real analytic functions that converge on every compact subset in your domain, do their derivatives necessarily converge to the derivatives of the function that they converge ...
0
votes
1answer
21 views

composing one function as a function of another function

I have two functions: $f_1=\sum_i x_iy_i$ and $f_2=\sum_i x_iy_i^2$, where $x_i$s and $y_i$s are positive and smaller than $1$. I want to write one of them as a function of the other (for example ...
0
votes
2answers
37 views

Prove for a monotone function, if $x_0$ is interior on interval $I$ then the one-sided limits exists

The proof goes like this: Lets suppose $f$ is nondecreasing (for nonincreasing we'll observe the function $-f(x)$). Let $x_0$ an interior point in the interval $I$, and $\left\{x_k\right\}$ an ...
1
vote
1answer
22 views

Resolve a recursive series

Consider the following recursive function: $f(n) = 1 + \sum_{i=0}^{n-1} f(i)$ with $f(0)=1$ I need to derive a non recursive form. By simply trying values, I have inferred that it must be ...
3
votes
3answers
151 views

finding the explicit function of a recursive sequence

So I have the recursive sequence $f(0) = 0, f(n+1) = 2f(n)+ (n+1)^2$, and I'm not quite sure how to make it explicit. Substituting $n$ for $n+1$ cleans it up a little, yielding $f(n) = 2f(n-1) + n^2$, ...
2
votes
1answer
35 views

Being g a continuous function show that

$$ (f_n)_{n\in\mathbb N}, \quad x\in \mathbb R $$ $$ f_n(x) = \begin{cases} n+n^2 x & \text{if } x\in\left[-\frac 1 n , 0 \right], \\ n - n^2 x & \text{if } x\in\left[0,\frac 1 n \right], \\ 0 ...
2
votes
3answers
67 views

prove that $f(x)=\sum _{n=0}^{\infty}\frac{\cos(nx)}{2^n}$ is continuous

I refered that each fn is continuous because its the fraction of a continuous function by a number and so $f(x)$ that is the sum of continuous functions is continuous. Is it right?
0
votes
1answer
26 views

problem with limit of functions sequence

Consider $$f_n\left(x\right)\:=\:\sin ^n\left(x\right).$$ How to check what are the points that this sequence is converges pointwise in these domains: $\left[\frac{-\pi }{2},\frac{\pi ...
1
vote
1answer
30 views

conditions for Convergence of sequence of functions

Suppose $\{f_n\}$, $n \in \mathbb{N}$ is a sequence of a positive real-valued functions defined on $[0, T]$ and continuous on $(0, T)$. If {$f_n$} satisfies the following conditions : $f_n( iT/2^n ) ...
1
vote
1answer
35 views

Pointwise convergence of a sequence of functions

Unfortunately my analysis lecturer, as awesome as he is, lacks the structure in his lessons to provide worked out proofs for us to use as guidelines for proving other things. Hence, I am having a ...
8
votes
1answer
112 views

Find $f$ such as $f(x) = \sum_{n=1}^\infty \frac{f(x^n)}{2^n}$

Find $f \in C^0([0,1] , \mathbb{R})$ such as $$f(x) = \sum_{n=1}^\infty \frac{f(x^n)}{2^n}$$ My try : Constant functions work fine. We can notice : $$f(x) = \frac{f(x)}{2}+\sum_{n=2}^\infty ...
6
votes
1answer
73 views

Example of a Lebesgue measurable function which is not a Baire function?

I found the following statement on Wikipedia : "Henri Lebesgue proved that (for functions on the unit interval) each Baire class of a countable ordinal number contains functions not in any smaller ...
1
vote
0answers
43 views

Are there such prime giving functions?

Here let us define a function $f : \mathbb{N} \rightarrow \mathbb{N}$ , such that for every $n$ , The sequence $\{f(n) ,f(n)+1 ,f(n)+2 , f(n)+3, \dots , f(n)+n\}$ contains atleast $1$ prime . Let us ...
1
vote
1answer
33 views

sequence of analytic functions on an open subset of $\mathbb{C}$ that converges uniformly on compact subsets

Let $U$ be an open subset of $\mathbb{C}$ and let $\{f_n\}$ be a sequence of analytic functions on $U$. Suppose that $(f_n)$ converges uniformly on any compact subsets of $U$ to a function $f$. Let ...
0
votes
1answer
22 views

series of functions converge to continuous function

I don't know how to show that the series: sum(1/(ln(n)n^x)) for n=2 to infinity converges to a continuous function at (1,infinity) If it was uniformly converges, I had no problem, but it doesn't. ...
3
votes
1answer
59 views

$f(\frac1{1234}) + f(\frac3{1234}) + f(\frac5{1234}) + … + f(\frac{1231}{1234}) + f(\frac{1233}{1234}) = ?$

If $f(x) = \frac{e^{2x-1}}{1+e^{2x-1}}$, then how to evaluate $$f(\frac1{1234}) + f(\frac3{1234}) + f(\frac5{1234}) + ..... + f(\frac{1231}{1234}) + f(\frac{1233}{1234}) ?$$ I know I'm missing some ...
3
votes
2answers
34 views

uniformly convergence on compact metric space

Let $K$ be a compact metric space. Let $\{f_n\}_{n=1}^\infty$ be a sequence of continuous functions on $K$ such that $f_n$ converges to a function $f$ pointwise on $K$. on Walt. Rudin's book ...
0
votes
1answer
51 views

Uniform convergence for a sequence of function

I have to show that if $f_n$ is a sequence of bounded functions that converges uniformly to $f$ on an interval I, then $f_n$ is uniformly bounded. I understand what uniformly bounded means, there ...
1
vote
1answer
49 views

Do one-sided limits exist for this functi0n?

Given a function $f$ with $f(x) := \begin{cases} \\ 0, & \text{if }x\in\mathbb{R}\setminus\mathbb{Q}\\ 1, & \text{if }x\in\mathbb{Q}\\ \end{cases}$ , what is $\lim\limits_{x\to0^+}f(x)$ and ...
1
vote
3answers
54 views

Prove $\frac{x}{1+n^2x^2}$ is uniformly convergent

I need to prove that \begin{equation}f_n(x)=\frac{x}{1+n^2x^2}\end{equation} Converges uniformly to $0$. I've tried a solution: (Scratchwork) Want ...
3
votes
1answer
44 views

Where does $f_n(x)=\frac{x^n}{1+x^{2n}}$ converge uniformly?

I'm given the function \begin{equation} f_n(x)=\frac{x^n}{1+x^{2n}}, \end{equation} where I can assume $f_n:[0,\infty)\to \mathbb{R}$. I need to determine and show the sets over which the ...
3
votes
1answer
58 views

Weak Convergence of a Sequence of Functions.

To begin my question I wish to first clarify the definition of weak convergence FOR a sequence of functions. We say that given sequence of functions, $\{f_{n}\}_{n=1}^{\infty}$, such that each $f_n$ ...
1
vote
1answer
683 views

Series expansion of square root function

Could I please know how to expand: $$\sqrt{4-3x^2}$$ I simplified it to $\sqrt{1-\frac34x^2}$ but the question is if I let $x = x^2$, what will the coefficient of, say $x^5$ or $x^{10}$ be in the ...
0
votes
1answer
34 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
1
vote
2answers
40 views

Construct a specific function for a given sequence

Given the sequence $A_n=n$, I want to construct a function $f : R \to R$, such that for every $x \in N$: $f(x)=A_x$ $\int\limits_{0}^{x}f(t)dt=\sum\limits_{i=0}^{x}A_i$ How can do that? Thanks
0
votes
1answer
45 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
1
vote
3answers
49 views

Need to know how: $L^{p}$ convergence for function.

I feel terrible at the moment as I have exhausted everything possible to understand this and I am still stuck. I have looked everywhere for some sort of resource to the concept at hand, and yet, they ...
1
vote
1answer
22 views

Series representation of a function. Generating the series formula.

in general say the question is to find the series representation of $ arctan(3x)$ the solution is the $\int \sum (-1)^n * (3x)^{2n}=$ $$ \sum (-1)^n * 3x^{2n+1}/(2n+1) $$ but my confusion is why ...
1
vote
3answers
52 views

Expand a sin(x^3) in Maclaurin's series and find a 30th derivative at (0)

I have a big task and problems with it. I have to expand this function in Maclaurin's series. $$cos(x^{3})$$ I tried expand it as $$\sum_{n=0}^\infty (x^n)/n!$$ but it's for $cos(x)$. So i don't know ...
0
votes
1answer
33 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
1
vote
0answers
31 views

Contraction Mapping Conditions

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ continuous and uniformly bounded, and consider the iteration $$ x^{(k+1)} = \frac{1}{k} \sum_{i=1}^{k} f\left( x^{(i)} \right) + x^{(i)} $$ for ...
0
votes
2answers
56 views

Retrieving $f(x)$ from Taylor Series

I've been trying to extract the $f(x)$ from the following Taylor series: $$ \sum_{k=1}^{\infty} (-1)^{k}\frac{kx^{k+1}}{3^{k}} $$ I moved things around a bit to make it look like this: $$ ...
1
vote
1answer
41 views

Check continuity and find norm of linear functional

I have to check continuity and find norm of the following linear functional $$\ell^{2}\ni \{x_n\}^{\infty}_{n=0} \rightarrow \sum^\infty_{n=0}\frac{x_n}{\sqrt{2^n}}\in \mathbb{K}$$ As for continuity, ...
2
votes
3answers
203 views

Continuity of Fixed Point

For all $a \in \mathbb{R}$, let $f_a: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and contractive, that is, there exists $\epsilon \in (0,1)$ such that $\left\| f_a(x)-f_a(y) \right\| \leq ...
2
votes
2answers
46 views

$S_n=Z_n$? Arithmetic progression.

$S_n$ is the sum of the first "n" numbers of the arithmetic progression "9,16,23..."; $Z_n$ is the sum of the first "n" numbers of the arithmetic progression "4035,4038,4041..." For what values ...
2
votes
1answer
53 views

Any insight about this sequence of numbers?

I don't have a background in math beyond high-school calculus and one course in discrete math. I was hoping you guys might be able to give me some information about the sequence of numbers generated ...