Uniform convergence Definition and 164 Threads

Homework Statement f_{n} is is a sequence of functions in R, x\in [0,1] is f_{n} uniformly convergent? f = nx/1+n^{2}x^{2} Homework Equations uniform convergence \Leftrightarrow |f_{n}(x) - f(x)| < \epsilon \forall n>= n_{o} \inN The Attempt at a Solution lim f_{n} = lim...

Homework Statement Let f,g be continuous on a closed bounded interval [a,b] with |g(x)| > 0 for all x in [a,b]. Suppose that f_n \to f and g_n \to g uniformly on [a,b]. Prove that \frac{1}{g_n} is defined for large n and \frac{f_n}{g_n} \to \frac{f}{g} uniformly on [a,b]. Show that this is...

How do I determine whether the following sequences of complex functions converge uniformly? i) z/n ii)1/nz iii)nz^2/(z+3in) where n is natural number

Hello!A little problem:With the given series,$$Y(x)= \sum_{n=1}^{\infty}(-1)^n\frac{x^n\ln^nx}{n!} $$ ,why $Y(x)$ is Uniformly converges for all $x\in(0,1]$ ?Ok, I know that $Y(x)$ is u.c by M-test: $$\max{|x\ln{x}|}=\frac{1}{e}$$ And, $$ \sum_{n=0}^{\infty}\frac{(\frac{1}{e})^n}{n!} $$ Is...

Homework Statement f(x) is defined within [a,b]. f_n(x)=\frac{\big\lfloor nf(x) \big\rfloor}{n} Check if f_n(x) is uniform convergent. The Attempt at a Solution This one seems to be easy however since I didn't touch calculus for quite a time I'm not confident with my solution. |\frac...

Homework Statement I have a solution to the following problem. I feel it is somewhat questionable though If fn converges uniformly to f, i.e. fn\rightarrowf as n\rightarrow∞ and gn converges uniformly to g, i.e. gn\rightarrowf as n\rightarrow∞ , Prove that fngn...

Probably is a silly question, but how could I prove that the function (expressed in polar coordinates) \left(\rho^4\cos^2{\theta} + \sin^3{\theta}\right)^{\frac{1}{3}} - \sin{\theta} converges to 0 as rho->0 uniformely in theta (if it is true, of course)?

I am given f_n(x)=\frac{nx}{nx+1} defined on [0,\infty) and I have that the function converges pointwise to 0 \ \mbox{if x=0 and} 1\ \mbox{otherwise} Is the function uniform convergent on [0,1] ? No. If we take x=1/n then Limit_{n\rightarrow\infty}|\frac{1/n*n}{1+1/n*n}-1|=0.5...

Homework Statement Find the minimum number required (value of n) for the average deviation of the Fourier Series to fall below 2% Homework Equations Use the Uniform Convergence of Fourier Series. Where Sm is the partial sum of the Fourier Series. C is constant. Here C is ∏^2 So...

Homework Statement This is a homework question for a introductory course in analysis. given that a) the partial sums of f_n are uniformly bounded, b) g_1 \geq g_2 \geq ... \geq 0, c) g_n \rightarrow 0 uniformly, prove that \sum_{n=1}^{\infty} f_n g_n converges uniformly (the whole...

The problem statement Let f:[a,b]→\mathbb{R} be differentiable and assume that f(a)=0 and \left|f'(x)\right|\leq A\left|f(x)\right|, x\in [a,b]. Show that f(x)=0,x\in [a,b]. The attempt at a solution It was hinted at that the solution was partly as follows. Let a \leq x_0 \leq b. For all x\in...

Is it true that a cauchy sequence of continuous functions defined on the whole real line converges uniformly to a continuous function? I thought this was only true for functions defined on a compact subset of the real line. Am I wrong?

Hi, I was always troubled by the relationships between these modes of convergence (L^1, L^2, and L^{\infty} convergences, to be precise), so I took some books and decided to establish some relations between them. For some, I succeeded, for others I did not. Here's what I did so far: If I...

Homework Statement Prove that the series \sum_{n=0}^\infty e^{-n^2x^2} converges uniformly on the set \mathbb{R}\backslash\ \big] -\epsilon,\epsilon\big[ where \epsilon>0Homework Equations n/aThe Attempt at a Solution I have tried using Weierstrass M-test but I have not been able to find a...

Homework Statement For each of the following sequences (fn), find the function f such that fn --> f. Also state whether the convergence is uniform or not and give a reason for your answer. Homework Equations a.) fn(x) = 1/xn for x greater than or equal to 1 b.) f[SUB]n[SUB](x) =...

Homework Statement show a function f_n is not uniformly convergent using a theorem: Homework Equations if f_n converges uniformly to F on D and if each f_n is cont. on D, then F is cont. on D The Attempt at a Solution not really sure what to do. use the contrapositive? would that...

1. Let g_n (x)=nx*exp(-nx). Is the convergence uniform on [0, ∞)? On what subsets of [0,∞) is the convergence uniform? 3. I am looking for a proof of how the convergence is uniform (possibly using Weierstrass' M Test?). I understand that the subset that determines uniform convergence is...

Homework Statement Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f '_n converges to zero uniformly in D = {z : |z| < 1/2}.Homework Equations Cauchy inequalities (estimates from the Cauchy integral formula)The...

F: C(Omega) -> D'(Omega); F(f) = F_f -- O = Omega Introduce the notion of convergence on C(Omega) by f_p -> f as p -> inf in C(O) if f_p(x) -> f(x) for any xEO Show that then F is a continuous map from C(O) to D'(O) Hint: Use that if a sequence of continuous functions converges to a...

Homework Statement show if has a uniform convergence of pointwise also we know that x gets values from 0 to 1The Attempt at a Solution for the pointwise I think its easy to show that limfn(x) as n->infinity is 0 but I am really stuck in uniform convergence I know that fn converges...

Homework Statement Show that there are continuous functions g:[-1,1]\to R such that no sequence of polynomials Q_n satisfies Q_n(x^2)\to g(x) uniformly on [-1,1] as n\to\infty The Attempt at a Solution Suppose there is a sequence Q_n such that Q_n(x^2)\to g(x) uniformly for g(x)=x. Then...

Homework Statement This is a nice one, if it's correct. Show that if (fn) is a sequence of elements of C(X, Y) (where Y is a metric space) which converges uniformly, then the collection {fn} is equicontinuous. The Attempt at a Solution Let ε > 0 be given and let x0 be a point of X...

Homework Statement Hi All, I've been having great difficulty making progress on this problem. Suppose gn converges to g a.e. on [0,1]. And, for all n, gn and h are integrable over [0,1]. And |gn|\leqh for all n. Define Gn(x)=\intgn(x) from 0 to x. Define G(x)=\intg(x) from 0 to x...

Homework Statement Prove the series converges uniformly on R to functions which are continuous on R. \sumn\geq0 (-x)2n+1/(2n+1)! Homework Equations The Attempt at a Solution I'm having trouble actually figuring out what to use for this series.. It looks like a Taylor series...

Homework Statement As in the question - Suppose that f_n:[0,1] -> Reals is a sequence of continuous functions tending pointwise to 0. Must there be an interval on which f_n -> 0 uniformly? I have considered using the Weierstrass approximation theorem here, which states that we can find...

I like thinking of practical examples of things that I learn in my analysis course. I have been thinking about functions fn:[0,1] --->R. What is an example of a sequence of piecewise functions fn, that converge uniformly to a function f, which is not piecewise continuous? I've thought of...

Homework Statement Let fn(x) = (1-x)^(1/n) be defined for x an element of [0,1). Does the sequence {fn} converge pointwise? Does it converge uniformly? Homework Equations The formal definition of pointwise convergence: Let D be a subset of R and let {fn} be a sequence of real valued...

Homework Statement Alright, here is the problem. Given a compact metric space X, and a sequence of functions fn which are continuous and f_{n}:X->R (reals), also f_n->f (where f is an arbitrary function f:X->R). Also, given any convergent sequence in X x_{n}->x, f_{n}(x_{n})->f(x). The problem...

Homework Statement Does the following series converge uniformly? [sum from n=1 to inf] \frac{e^{-nx}}{n^2} on [0, inf) Homework Equations I know I need to use the M test or Cauchys Principle of uniform convergence. My tutor suggests using the former if there is uniform convergence...

A function is pointwise bounded on a set E if for every x\in E there is a finite-valued function \phi such that |f_n(x)|<\phi(x) for n=1,2,.... A function is uniformly bounded on E if there is a number M such that |f_n(x)|<M for all x\in E, n=1,2,.... I understand that in uniform...

Homework Statement We know that f is uniformly continuous. For each n in N, we define fn(x)=f(x+1/n) (for all x in R). Show that fn converges uniformly to f. Homework Equations http://en.wikipedia.org/wiki/Uniform_convergenceThe Attempt at a Solution I know that as n approaches infinity...

Homework Statement \mbox{Check whether } \sum_{n=0}^\infty \frac {1}{e^{|x-n|}} \mbox{ is uniform convergent where its normaly convergent} The Attempt at a Solution \mbox{I choose } \epsilon = 1/2 a_n=\frac {1}{e^{|x-n|}}\ ,\ b_n= \frac {1}{n^2} \lim_{n\rightarrow\infty} \frac {a_n}{b_n}=0\...

Homework Statement Suppose: f(x)\ and\ f'(x)\ are\ continuous\ for\ all\ x \in R For\ all\ x \in R\ and\ for\ all\ n \in N\ f_n(x)=n[f(x+\frac{1}{n})-f(x)] Prove\ that\ when\ a,b\ are\ arbitrary,\ f_n(x)\ is\ uniform\ convergent\ in\ [a,b] The Attempt at a Solution\lim_{n\rightarrow...

Let y_n be a sequence of functions in \mathcal{C}([0,1], \mathbb{R}) Suppose that every subsequence of y_n has a subsequence that converges uniformly. Prove that they all converge to the same limit.

1.kn (x) = 0 for x ≤ n x − n, x ≥ n, Is kn(x) uniformly convergent on R? I can show that it is uniformly convergent on any closed bounded interval [a,b], but I don't think it is on R. Could anyone please give me some hints how to prove it? 2.Fix 0 < η < 1. Suppose now...

let f is a continuous, real-valued function on [a,b] then, for any e, there exist a polygonal function p such that sup|f(x)-p(x)|<e using uniform convergence, this might be shown... but i cannot figure it out...

Homework Statement Show that \sum(-1)^(n+1) / (n + x^2) converges uniformly but not absolutely on R. Homework Equations Using Dirichlet's Test for uniform convergence. (fn) and (gn) are sequences of functions on D satisfying: \sum fn has uniformly bounded partial sums gn -> 0...

Hi all Came across this question which i have attempted to answer but am sure i have gone wrong somewhere, any help would be appreciated... Suppose fn(x):R-->R, where fn(x)=x+(1/n) (n belongs to N). Find the pointwise limits of the sequences (fn), (1/n fn) and (fn2). In each case determine...

Homework Statement For k = 1,2,\ldots define f_k : [0,1] \to \mathbb{R} by \begin{align*} f_k(x) = \left\{ \begin{array}{ll} 4k^2x & 0 \leq \displaystyle x \leq \frac{1}{2k} \\ 4k(1 - kx) & \displaystyle \frac{1}{2k} < x \leq \frac{1}{k} \\ 0 & \displaystyle \frac{1}{k} < x \leq 1...

"Let fk be functions defined on Rn converging uniformly to a function f. IF each fk is bounded, say by Ak, THEN f is bounded." fk converges to f uniformly =>||fk - f||∞ ->0 as k->∞ Also, we know|fk(x)|≤ Ak for all k, for all x But why does this imply that f is bounded? I don't see why it...

Homework Statement Homework Equations The Attempt at a Solution This is not graded homework, but optional exercises I found in my textbook. It's days before my exam, but I'm still not sure how to do problems like this. I would really appreciate if someone can teach me how to solve...

Homework Statement Theorem: Let (X,d_X),(Y,d_Y) be metric spaces and let f_k : X \to Y, f : X \to Y be functions such that 1. f_k is continuous at fixed x_0 \in X for all k \in \mathbb{N} 2. f_k \to f uniformly then f is continuous at x_0. Homework Equations If all f_k are...

Homework Statement Define the sequence \displaystyle f_n : [0,\infty) \to \left[0,\frac{\pi}{2}\right) by f_n(x) := \tan^{-1}(nx), x \geq 0. Homework Equations Prove that f_n converges pointwise, but not uniformly on [0,\infty). Prove that f_n converges uniformly on [t, \infty)...

Homework Statement \text{Show that the series }f(x) = \sum_{n=0}^\infty{\frac{nx}{1+n^4 x^2}}\text{ converges uniformly on }[1,\infty]. Homework Equations The Attempt at a Solution I think we should use the Weierstrass M-test, but I'm not sure if I'm applying it correctly: For...

Homework Statement 1.) Prove that if { f_{n} } is a sequence of functions defined on a set D, and if there is a sequence of numbers b_{n}, such that b_{n} \rightarrow 0, and | f_{n}(x) | \leq b_{n} for all x \in D, then { f_{n} } converges uniformly to 0 on D. 2.) Prove that if { f_{n} } is a...

Homework Statement h:[0,1] -> R is continuous Prove that t(x) = \sum^{infinity}_{n=0} xnh(xn) is uniformly convergent on [0,s] where 0<s<1 Homework Equations The Attempt at a Solution I have the definition of h being continuous but after this I am pretty clueless about how to...

Homework Statement fn(x) = 0 if x \leq n = x-n if x\geq n Is fn(x) uniformly convergent on [a,b]? Is it uniformly convergent on R? Homework Equations The Attempt at a Solution I think limn -> infinity fn(x)= 0 However I do not know what supx in [a,b] |fn(x)| would be, I...

Hi. Now you probably know that if a function fk(x) converges uniformly to f(x) then we are allowed to certain actions such as limn-> \inf \int f (of k) dx = \int f dx In other words we are allowed to exchange limit and integral. Now say we have any sequnce valued function fk(x) . And we...

I'm trying to better understand convergence so I made upa problem for myself based on an example from class. I want to know if I'm answering my own questions correctly. Define a sequence of functions fn(x) = 1 if x is in {r1, r2, ... , rn} and 0 otherwise. Where r1, r2, ... , rn are the first...

Homework Statement Let fn(x) = 1/(nx+1) on (0,1) where x is a real number. Show this function does not converge uniformly. Homework Equations The Attempt at a Solution I know why it is not uniformly convergent. Even though fn(x) goes to zero monotonically on the interval (0,1), it's not...