Uniform convergence of a series

This shows that f_n converges uniformly to f.In summary, we are given that the function f is uniformly continuous. We define a new sequence of functions fn for each natural number n, where fn(x) = f(x+1/n) for all x in the real numbers. Our goal is to show that fn converges uniformly to f. To do this, we can use the fact that if fn is uniformly convergent, then the limit of the supremum of fn(x) as n approaches infinity is equal to 0. By using the definition of uniform continuity, we can show that for any given epsilon, there exists a delta such that for all n greater than 1/delta, |fn(x)-f(x)|
  • #1
WaterPoloGoat
14
0

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_convergence

The Attempt at a Solution



I know that as n approaches infinity, that fn(x)=f(x), which implies that fn converges to f.

I'm currently trying to apply the fact that if fn is uniformly convergent, then

limn->infinity Sup {fn(x): x in R}=0.

But I keep getting stuck on the fact that there's an function in the definition of fn i.e., fn(x)=f(x+1/n). Is there a way to work with it?
 
Last edited:
Physics news on Phys.org
  • #2
It's actually pretty straightforward. Given [tex]\epsilon > 0[/tex], for each x there is a [tex]\delta>0[/tex] such that [tex]|f(x+h)-f(x)|<\epsilon[/tex] whenever [tex]|h|<\delta[/tex]. Hence, for all [tex]n > \delta ^{-1}[/tex] we have [tex]|f_n(x)-f(x)|<\epsilon[/tex].
 

What is uniform convergence of a series?

Uniform convergence of a series is a property of a series of functions that describes how closely the series converges to a specific function. It means that for any given value in the domain of the function, the difference between the value of the function and the sum of the series is small and consistent.

How is uniform convergence different from pointwise convergence?

Pointwise convergence only requires that for each individual value in the domain of the function, the series of functions converges to the corresponding value of the function. Uniform convergence, on the other hand, requires that the convergence is consistent across the entire domain.

What does it mean when a series is not uniformly convergent?

If a series is not uniformly convergent, it means that there exists at least one point in the domain of the function where the series does not converge to the corresponding value of the function. This could result in significant differences between the function and the series at that point, and the convergence is not consistent across the domain.

How is uniform convergence related to continuity?

Uniform convergence of a series is closely linked to the continuity of the function. If a series of continuous functions uniformly converges to a function, then that function is also continuous. On the other hand, if a series does not uniformly converge to a function, it may not be continuous.

What are some applications of uniform convergence of a series?

Uniform convergence of a series is a fundamental concept in analysis and has many applications in mathematics and other fields. It is used to prove the existence of solutions to differential equations, to approximate functions, and in numerical analysis. It also plays a crucial role in the study of power series and Fourier series.

Similar threads

  • Calculus and Beyond Homework Help
Replies
26
Views
804
  • Calculus and Beyond Homework Help
Replies
3
Views
940
  • Calculus and Beyond Homework Help
Replies
7
Views
644
  • Calculus and Beyond Homework Help
Replies
2
Views
640
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
24
Views
2K
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
313
Back
Top