# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f(n) = 2f(n-1) + n^2$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### $f(\frac1{1234}) + f(\frac3{1234}) + f(\frac5{1234}) + … + f(\frac{1231}{1234}) + f(\frac{1233}{1234}) = ?$

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

I need to prove that $$f_n(x)=\frac{x}{1+n^2x^2}$$ Converges uniformly to $0$. I've tried a solution: (Scratchwork) Want ...
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### Where does $f_n(x)=\frac{x^n}{1+x^{2n}}$ converge uniformly?

I'm given the function $$f_n(x)=\frac{x^n}{1+x^{2n}},$$ where I can assume $f_n:[0,\infty)\to \mathbb{R}$. I need to determine and show the sets over which the ...
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### 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 $f_n$ ...
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### 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 ...
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### 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 $f(\bar{x}) = \bar{x}$). ...
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### 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
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### 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) ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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: ...