1
vote
1answer
21 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
22 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
21 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
124 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
64 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
22 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
28 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
95 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
61 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
41 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
23 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
21 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
53 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 ...
2
votes
2answers
28 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
43 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
46 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
51 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
40 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
54 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
101 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
30 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
43 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
47 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
46 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
32 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
30 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
51 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
37 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
185 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
44 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 ...
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
66 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: ...
1
vote
1answer
63 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
30 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
85 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
81 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
46 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
65 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
56 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
107 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?
17
votes
1answer
214 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: ...