-1
votes
0answers
13 views

On sequence and function [on hold]

For a set $S$ we denote its cardinality by $|S|$. Let $e_1,e_2,\ldots,e_k$ be non negative integers. Let $A_k$ (respectively $B_k$) be the sets of all $k$-tuples $(f_1,f_2,\ldots,f_k)$ of inteers such ...
0
votes
1answer
22 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
38 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
35 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
43 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
33 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
22 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
25 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
46 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
24 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
166 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
40 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
50 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 ...
2
votes
1answer
20 views

Constructing sequence of functions

I have to construct a sequence of $\{f_i\}$, where $f_i$ belongs to $C[0,1]$ such that: $$ d(f_i,0) = 1 \\ d(f_i,f_j)=1, \forall i,j \\ \text{Using Sup-Norm metric, i.e.} \mathbb{\|}f\mathbb{\|} = ...
0
votes
0answers
30 views

Fitting a function $f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$

I'm investigating a function $f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ which satisfies the following properties: $f(1,n)=n^2-n-1$ $f(2,n)=n^2-5$ $f(n-1,n)=n^2-2n+1$ $f(n^2,2n)=n^2$ I'd like to ...
1
vote
1answer
61 views

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

$a_{n+1} = a_n - a_n^2$, $a_1 = 2/3$. for $n\ge1$ a) Show the series converges and find its limit. b) find a uniformly continuous $f:\mathbb{R}\rightarrow \mathbb{R}$ such that: ...
0
votes
1answer
46 views

Use monotonic property of subsequences to prove one sided limits at a point of a monotonic function

We were given a two parter homework question: Prove that if every subsequence of a sequence $x_n$ is converging to L then $x_n$ converges to L. Use what you proved above to prove that if $f(x)$ is a ...
1
vote
1answer
29 views

value of the value expression

F (x)=[1+sinx]+[2+sin (x/2)]+[3+sin (x/3)]+.........+[n+sin (x/n)] for x belongs to 0 to pi. Where []denotes greatest integer function. I'd converted f(x) as f (x)=1+2+3+4+5+........+n+sin x+sin ...
0
votes
1answer
34 views

Proof of a continous function?

I would like to know how tho prove or disprove the following: Prove the follwoing statement: Every continous function $f:[a,b] \mapsto \mathbb{R}$ (with a < b) is (from above) bounded. I have to ...
0
votes
1answer
18 views

Behaviour of this function

Consider the function $f(n,a) = n^{-a}-n^{a-1}$ (i)I have to investigate the function for n between 1 to infinity and a between 0 to 1 . I think the function starts from zero at 1 reaches a maxima ...
1
vote
6answers
80 views

Prove that the succession $x_{n}$ where $x_{1}=1$ and $x_{n+1} = \sqrt{3x_{n}}$ is convergent

Well, ive been having a weee bit of problem solving this homework, can anyone give me a hand? Prove that the sequence $x_{n}$ where $x_{1}=1$ and $x_{n+1} = \sqrt{3x_{n}}$ is convergent and calculate ...
3
votes
4answers
78 views

Finding generating functions - how was this jump made?

I'm going through examples of probability-generating functions in a book and am confused by the following example: $$1+2s+4s^2+...=\sum_{n=0}^\infty (2s)^n=(1-2s)^{-1}$$ I understand the summation but ...
0
votes
2answers
41 views

Why is the function sequence $\{\frac{nx}{2 + n + x}\}^\infty_1$ converges uniformly to $x$ in the interval $0 \le x \le1$?

Given function sequence $\{f_n(x)\}^\infty$ defined as $f_n(x) = \frac{nx}{2 + n + x}. (0 \le x \le 1)$ I need to find the limit function and whether it converges uniformly or not uniformly. I found ...
0
votes
0answers
64 views

What was Euler's misconception about functions and infinite series?

I just read this on Strichartz' The way of Analysis: [...] Euler - the leading mathematician of the eighteenth - developed all techniques needed for the study of Fourier series, but he never ...
1
vote
1answer
54 views

Fourier Series Approximations of Functions

From a few examples of smooth functions, discontinuous functions and continuous functions which have a 'kink' (i.e. $|x|$ where left and right limits disagree)... I've seen that the fourier series ...
1
vote
1answer
75 views

supremum and infimum of a bounded and decreasing sequence

Is there supremum and infimum of a bounded sequence? I have a bounded and decreasing sequence. Why does this sequence have infimum?
16
votes
1answer
201 views

Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$

Suppose $f:\mathbb{R}\to\mathbb{R}$ has period 1, and for some $q\in(0,1)$: $$|f(x)-f(y)|\leq q|x-y|\quad \forall x,y$$ Let $g(x)=x+f(x)$, for any $a\in\mathbb{R}$,define the following sequence: ...
2
votes
1answer
35 views

Hypergeometric series of sine function

Can someone please help me express this integral in terms of hypergeometric series? $$\int\frac{\sin⁡((2n-1)x)}{\sin⁡x}\text dx$$
0
votes
0answers
22 views

Prove series of functions is uniformly convergent then $f_n \rightrightarrows 0$

I am doing some exercise on Abbott's textbook and there is one which says: Let $f_n:S \subset \mathbb R \to \mathbb R$. Prove that if $\sum_{k=1}^{\infty} f_n$ is uniformly convergent on $S$, then ...
1
vote
2answers
29 views

Sequence of functions, question regarding notation.

I am trying to solve a problem which says: Let $S \subset \mathbb R^{\mathbb N}$ and let $\{f_n\}_{n \in \mathbb N}$ a sequence of functions $f_n:S \to \mathbb R$ that converges uniformly to a ...
1
vote
2answers
107 views

How do you calculate this sum $\sum_{n=1}^\infty nx^n$? [duplicate]

I can not find the function from which I have to start to calculate this power series. $$\sum_{n=1}^\infty nx^n$$ Any tips?. Thanks.
1
vote
2answers
65 views

Prove that the series converges to the integral

Prove: $\int _0^{1}x^{-x}dx$ = $\sum_{n=1}^\infty\frac{1}{n^n} $ I thought of using: $x^{-x}$ = $e^{-x lnx}$ and then using : $e^{-xlnx}$ = $\sum_{n=1}^\infty\frac{(-xlnx)^n}{n!} $ but I'm stuck from ...
2
votes
2answers
56 views

Dual of a sequence

Let $S$ be the set of all sequences $(a_1,a_2,\ldots)$ of non-negative integers such that (i) $a_1 \ge a_2 \ge \ldots;$ and (ii) there exists a positive integer $N$ such that $a_n=0$ for all $n \ge ...
1
vote
2answers
48 views

Recursive sequence of functions

Recursive sequence of functions: $f_{n+1}= \sqrt{x+f_n}$, $f_1(x)= \sqrt{x}$. this sequence is monotonic, but what is bounding it? thanks
1
vote
1answer
68 views

Minimum of a sum

I have the function $$f(x)= \sum_{i=1}^n (x-a_i)^2 \ x\in R$$ I am asked to find the minimum of it. I am lost so any help would be nice. Thanks in advance!
1
vote
1answer
33 views

Determine all subinterval of $[0,\infty)$ on which a series of function converges unif or pointwise

I have no idea to find all subintervals $[0,\infty)$ on which $\sum_{n=0}^{\infty}\left(\frac{x}{x-2}\right)^n$ converges uniformly or pointwise Using ratio test, I can show the series of $f_n(x)$ is ...
2
votes
1answer
96 views

Evaluate limit of integral of sequence of function

Evaluate $$\lim\limits_{n\to \infty}\int_{0}^{1}\frac{\sqrt{n}(e^{-x/n}-1)}{x}dx.$$
4
votes
2answers
76 views

Prove that $\lim_{n\to\infty}|f(x_n+\xi)-f(x_n)|=0$

Let $f:(0,+\infty)\to\mathbb{R}$ be continuous and bounded. Let $\xi>0$. Show that there is a sequence $(x_n)$ in $(0,+\infty)$ with $x_n\to\infty$ s.t. $$\lim_{n\to\infty}|f(x_n+\xi)-f(x_n)|=0.$$ ...
1
vote
1answer
38 views

How to construct longitudinal from transversal waves and vice versa?

The above construction of a longitudinal wave out of a transversal wave has been encountered somewhere in an old physics textbook. There are several drawbacks with this construction. The maximum ...
1
vote
0answers
39 views

Text Generating Functions: Do they exist?

This is a little far out question, but just curious: is it even possible to have a non-high-degree-polynomial function (as in polynomial regression function) that could generate a sentence of say, 10 ...
0
votes
1answer
25 views

function on a fixed length sequence of positive real number that induces lexicographic order

Let $S$ be the finite set of sequences of length $n$, whose entries are all real positive numbers. Can we define a function $f$ on $S$ such that the order $f$ induces on $S$ i.e $\le$ is the same as ...
2
votes
2answers
32 views

if $f(k/N)\rightarrow0$ as $N\rightarrow\infty$ for any $k$, must $f(h)\rightarrow0$ as $h\rightarrow0$?

If $f$ is a function defined on [$\mathbb{R}$ or $\mathbb{C}$] such that for any [real or complex] $k$, $f(\frac{k}{N})\rightarrow0$ as $N\rightarrow\infty$ in $\mathbb{N}$, must it be true that ...
2
votes
1answer
129 views

Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.

I tried doing this by defining a function $f$ which takes an element (a subset of $\mathbb{N}$) and maps it to an infinite sequence of $0$s & $1$s. A subset, i.e the element of $P(\mathbb{N})$ ...
0
votes
5answers
65 views

What is the simplification of sum of $i(i+1)$?

I am trying to simplify $$\frac{\sum_{i=1}^n i(i+1)}{n(n-1)}$$. I am not sure how to simplify ${\sum_{i=1}^n i(i+1)}$ part. How can I simplify it?
0
votes
1answer
67 views

Problem of sequence

I have the function $f_n(x) = \frac{x^n e^{-x}}{n!}$ and a sequence that is defined by $u_{n}=f_{n}(n)$. Additionally, I find that $\frac{u_{n+1}}{u_{n}}\le e^{-\frac{1}{4n}}$ with the precedent ...
3
votes
1answer
139 views

How does a taylor series of a binomial function equals a trigonometric function? [closed]

Any proof or derivation for the sinx and cosx function would be help. Image taken from http://en.wikipedia.org/wiki/Taylor_series
30
votes
0answers
618 views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
0
votes
0answers
79 views

Function from series summation?

There are two variables in time series $Z$ and $S$ depending on $t$. They are formulated as summation of series: $$\sum_{t=1}^TZ_t=\sum_{t=1}^TS_t-\beta(S_1-S_2-S_{T-1}+S_T)$$ Is there a way to ...
4
votes
2answers
177 views

How to evaluate a zero of the Riemann zeta function?

Here is a super naive question from a physicist: Given the zeros of the Riemann zeta function, $\zeta(s) = \sum_{n=1}^\infty n^{-s}$, how do I actually evaluate them? On this web page I ...