Uniform convergence Definition and 164 Threads

Homework Statement Consider f(x)=\sum\frac{1}{n(1+nx^{2})} from n=1 to n=infinity. On what intervals of the form (a,b) does the series converge uniformly? On what intervals of the form (a,b) does the series fail to converge uniformly? Homework Equations Weierstrass M-test...

I don't know, I started from the definition of uniform convergence and it seems pretty obvious to me , can anybody start me at least towards right direction?

I would like to know if there's a counterpart to the single variable theorem, that if f is a differentialble function with a bounded derivative, is uniformly continuous. I think the counterpart should be, if f(x1,...,xn) is continuous function, and differentiable, and each f'_xi are bounded...

Homework Statement Is the series \sum \frac{x^n}{1 + x^n} uniformly convergent on [0,1) ? Homework Equations The Attempt at a Solution I'm guessing that it is NOT uniformly convergent but I don't know how to show it. But I would note that it is quite easy to show that the series is uniformly...

Let (X,µ) be a measure space with µ(X) infinite. I'm trying to find an example of a pair (X,µ) and a sequence {f_n} of L^1(X,µ) functions converging uniformly to a function f such that we do not have f_n --> f in L^1(X,µ).

Homework Statement Let (f_n) be a sequence of continuous functions on [a,b] that converges uniformly to f on [a,b]. Show that if (x_n) is a sequence in [a,b] and if x_n \to x, then \lim_{n \to \infty} f_n (x_n) = f(x) Homework Equations None The Attempt at a Solution I just want...

What can I conclude using the following theorem? Let the functions u_n (x) be continuous on the closed interval a \le x \le b and let them converge uniformly on this interval to the limit function u(x) . Then \int_a^b u (x) \, dx=\lim_{n \to \infty} \int_a^b u_n (x) \, dx Can...

Homework Statement Suppose the infinite series \sum a_v is NOT absolutely convergent. Suppose it also has an infinite amount of positive and an infinite amount of negative terms. Homework Equations The Attempt at a Solution Say we want to prove it converges by proving...

Homework Statement Prove that the sequence f_n(x) = x^{\frac{2^n-1}{2^n}} converges uniformly in the interval [0,1]. Homework Equations The Attempt at a Solution First notice that the f_n do converge to f(x) = x for all x in [0,1]. By definition, if for every positive...

For the sequence of functions fn(x)=n^(2)e^(-nx) on [0,infinity), what is the pointwise limit of this sequence?is the converbence uniform?

For the sequence of functions fn(x)=ne^(-xn^2) on [0,infinity), what is the pointwise limit of this sequence?is the converbence uniform?

could someone please explain if f_n(x)= 1-x^n all over 1-x is uniformly convergent? can someone show why its independent of x if it is and on what intervals? Also, can someone explain if it isn't uniform convergent then can they show a proof and explain. Thanks any help will be greatly...

I shall prove that this integral is uniformly convergent or not: and if it is convergent, i must describe its uniform convergence

Does \sum_{n=0}^{\infty} x^n(1-x) converge uniformly on (0,1)? S_n(x)=\sum_{k=0}^{\infty} x^k(1-x)=\frac{1-x^{n+1}}{1-x}(1-x)=1-x^{n+1}} \rightarrow 1 as n \rightarrow \infty so we get pointwise convergence to 1 now we test for uniform convergence d_{\infty}(S_n,1) = sup |S_n(x)-1|=...

Hi, how would I write out the definition of "uniform convergence" of a function f(x,y) with as few a possible words and using symbols like \forall?

Homework Statement Let \phi_n(x) be positive valued and continuous for all x in [-1,1] with: \lim_{n\rightarrow\infty} \int_{-1}^{1} \phi_n(x) = 1 Suppose further that \{\phi_n(x)\} converges to 0 uniformly on the intervals [-1,-c] and [-c,1] for any c > 0 Let g be any...

Suppose you know that a sequence \{f_n\} of functions converges pointwise to 0 on the whole real line. If there is a subsequence \{f_{n_k}\} of the original sequence that converges uniformly to a limiting function f on the whole real line, does that limiting function have to be 0?

Let X be a set, and let fn : X \rightarrow R be a sequence of functions. Let p be the uniform metric on the space Rx. Show that the sequence (fn) converges uniformly to the function f : X \rightarrow R if and only if the sequence (fn) converges to f as elements of the metric space (RX, p)...

Homework Statement The sequence fn: [-1,1] -> R, fn(x)= nxe-nx2 converges pointwise to f(x)= 0, x in [-1,1]. Can you verify the following: limn->\infty (\int^{1}_{0}fn(x)dx) = \int^{1}_{0} (limn->\infty fn(x))dx Homework Equations Theorem: If fn is continuous on the interval D for every...

I have a question that I just don't know how to go about. " Let Fn = x/(1+(n^2)(x^2)) where n=1,2,3,... show that Fn converges uniformly on (-infinity,infinity)" To be honest, I don't even know where to start. Is this a series? How would I solve this. Would the Abel's test apply?

Homework Statement Say we have a sequence of functions {fn: [a,b] -> R} such that each fn is nondecreasing. Suppose that {fn} converges point-wise to f. Prove that if f is continuous, then {fn} converges uniformly to f. The attempt at a solution Fix an x in [a,b] and let e > 0. Then we can...

Let fn(x)=nx^2/1+nx ; x lies in [0,1] Is the convergence uniform? Since lim as n-->infinity of fn is x, I can see that fn(x) converges pointwise to f(x)= x But I get stuck when I try to show the convergence is uniform or not.

does {fn} converge uniformly? fn(x)=nx^2/1+nx I can see that fn converges pointwise to f(x)=x. I know, for epsilon>0, I need to find N st for n >or equal to N, |fn(x)-f(x)|<epsilon. |fn(x)-f(x)|=x/1+nx but then I am stuck.

Homework Statement Theorem to be proved: fn is uniformly convergent on E iff for every epsilon >0 there is an N, where n,m >= N then | fn(x) - fm(x) | < epsilon, for all x in E.Homework Equations Definition: fn is uniformly continuous on E if there is an f such that for every epsilon >0, there...

Homework Statement A stereotypical problem in measure theory. fn(x)=1 when 0<=x<=1/n =0 when 1/n<x<=1 f(x)=0 0<=x<=1 Under the lebesgue measure Does fn converge to f? a. for all x b. almost everywhere c. uniformly on [0,1] d. uniformly almost everywhere on [0,1] e. almost uniformly...

Howdy Ho, partner. I have a series of functions {f_{n}} with f_{n}(x) := x^{n} / (1 + x^{n}) and I am investigating the pointwise limit of the sequence f_{n} over [0, 1] to see if it converges uniformly. I found the pointwise limit f(x) to be f(x) = lim_{n\rightarrow\infty} x^{n} / (1 +...

Homework Statement f is integrable on the circle and satisfies the Lipschitz condition (Holder condition with a=1). Show that the series converges absolutely (and thus uniformly). i literally spent about 20 hours on this problem today but i just could not figure it out. i have a feeling...

Homework Statement Suppose g and f_n are defined on [1,+infinity), are Riamann-integrable on [t,T] whenever 1<=t<T<+infinity. |f_n|<=g, f_n->f uniformly on every compact subset of [1,+infinity), and \int^{\infty}_{1} g(x)dx<\infty. Prove that lim_{n->\infty} \int^{\infty}_{1} f_{n}(x)dx...

Homework Statement {f_n} is a sequence of continuous functions on E=[a,b] that converges uniformly on E. for each x in E, set g(x)=sup{f_n(x)}. Prove that g is continuous on E Homework Equations The Attempt at a Solution I've got an idea to prove it: Let e>0 be given, there is a...

Homework Statement http://img384.imageshack.us/img384/1643/60357312ro9.png Homework Equations http://img440.imageshack.us/img440/1935/11582858vp9.png The Attempt at a Solution I didn't know if I had to use the definition of uniform convergence or its Cauchy variant but I choose the Cauchy...

Homework Statement Find sequences {f_n} {g_n} which converge uniformly on some set E, but such that {f_n*g_n} does not converge uniformly on E. Homework Equations The Attempt at a Solution I looked at some sequences of functions known to be convergent but not uniformly convergent...

(Problem 64 from practice math subject GRE exam:) For each positive integer n, let f_n be the function defined on the interval [0,1] by f_n(x)=\frac{x^n}{1+x^n}. Which of the following statements are true? I. The sequence \{f_n\} converges pointwise on [0,1] to a limit function f. II. The...

(Problem 64 from practice math subject GRE exam:) For each positive integer n, let f_n be the function defined on the interval [0,1] by f_n(x)=\frac{x^n}{1+x^n}. Which of the following statements are true? I. The sequence \{f_n\} converges pointwise on [0,1] to a limit function f. II. The...

If we have that: f_T \left( x \right) \to f\left( x \right) for each x (pointwise convergence) and also that: f_T and f are bounded or/and uniformly continuous functions then can we show that there is also uniform convergence? If no why not? Can you show it with an example? In general...

First of all, hello all this is my first post i think. Congratulations on this great community Please move my post if I'm not posting on the right forum and I'm sorry for any inconvenience. I have this problem that I need to solve and I don't have a clue. I hope you could give me some ideas...

Homework Statement Prove: (1) the series \sum_{n=0}^\infty (-1)^n x^n (1-x) converges absolutely and uniformly on the interval [0,1] (2) the series \sum_{n=0}^\infty x^n (1-x) converges absolutely and uniformly on the interval [0,1] The Attempt at a Solution I have shown, by...

Homework Statement Let (X,d) be a compact metric space, (f_n) be an equicontinuous sequences of functions in C(X, \mathbb{R} )such that, for every fixed x in X, (f_n(x)) \to 0 . Show that (f_n) converges uniformly to the zero function Homework Equations The Attempt at a Solution...

Homework Statement Prove that if fn -> f uniformly on a set S, and if gn -> g uniformly on S, then fn + gn -> f + g uniformly on S. Homework Equations The Attempt at a Solution fn -> f uniformly means that |fn(x) - f(x)| < \epsilon/2 for n > N_1. gn -> g uniformly means that |gn(x) -...

I hate, HATE uniform convergence for series... hate this, really do. I'm trying hard, but our prof gives us very difficult borderline problems. Now, I am almost 100% sure this is not uniformly convergence at x=1, but since it's a stinking series, it's hard to figure out it's summation...

Homework Statement 1) Test the following series for Uniform Convergence on [0,1] \sum\limits_{n = 1}^{\inf } {\frac{{( - 1)^n }}{{n^{x}\ln (x)}}} Homework Equations The Attempt at a Solution Obviously, it's not uniformly convergent since f(n,1) = \sum\limits_{n =...

Homework Statement I need to understand as to why the following series fn(x) = x/(1+n*x^2) is point wise convergent (as mentioned in the book of Ross) and not [obviously] uniformly convergent. Homework Equations The relevant equation used is that lim (n-> infinity) sup|(fn(x) -...

Homework Statement Suppose: fn goes to f pointwise on [a,b] For each c in (a,b) fn goes to f uniformly on (c,b), and for some M, fn(x) is less than M for all n, all x in [a,b] Prove fn goes to f uniformly on [a,b] Homework Equations The Attempt at a Solution I think I proved that...

I have several questions 1. Find an example of two sequences of functions, f_n and g_n such that they both converge uniformly to f, g on some set E but such that f_n * g_n does not converge uniformly on E. Let f_n = x^2 for all n. and g_n=sinx/(xn). Since sinx/x is bounded by 1 it is easy...

5: Let p > 0. Let f_n (x) = x n^p e^{-nx} (i) For what values of p is f_n (x) -> 0 uniformly on [0,1]? (ii) For what values of p is it true that lim_{n->\infty} \int_0^1 f_n (x) dx = 0 For (i), we know that uniform convergence implies that the convergence to 0 depends ONLY on the value...

Homework Statement Hello! I've been tasked with figuring out if the following is uniformly convergent on [0,1] and I could use push in the right direction: sin(nx) Homework Equations The Attempt at a Solution Would picking M=1 in the weierstrass-m test show it not uniformly...

Homework Statement Just trying to get feel for uniform convergence and it's relationship to boundedness. If a sequence of functions (fn) converges uniformly to f and (fn) is a sequence of bounded functions, is f also bounded? Homework Equations The Attempt at a Solution

Hi! Hope this is the right forum, I'm not quite sure myself. Anyway, this problem has been bugging me for what seems way too long for such an apparently simple problem. It's about the difference between Pointwise and Uniform Convergence (Topology). By reading different articles online...

Hi all, I'm learning some calculus theory and I found one point I don't fully understand: \mbox{Let M} \subset \mathbb{R} \mbox{ be non-empty set and let } f, f_{n}, n \in \mathbb{N} \mbox{ be functions defined on M. Then the following is true:} f_n \rightrightarrows f \mbox{ on...

Here is the question from the book --------- For each integer n\geq 1, let f^{(n)}:(-1,1) \to \mathbb{R} be the function f^{(n)}(x):= x^n. Prove that f^{(n)} converges pointwise to the zero function, but does not converge uniformly to any function f:\mathbb{R} \to \mathbb{R}. ---------- I...

If f is uniformly continuous then does that mean f^-1 its inverse is also?