Uniform Convergence of Sequences

[LauraLovies]

Homework Statement

For each of the following sequences (f_n), find the function f such that f_n --> f. Also state whether the convergence is uniform or not and give a reason for your answer.

Homework Equations

a.) f_n(x) = 1/xⁿ for x greater than or equal to 1
b.) fn(x) = x/(1+xⁿ) for x element of [0,1]

The Attempt at a Solution

a. I think the limit function is 1 if x=1 and 0 if x?1. I also think it does not converge uniformly but do not know why.

b. I think the limit function is 0 if x=0 and 1/2 if x=1. I think it does not converge uniformly.

 

[LCKurtz]

What theorem do you know which relates continuous functions and uniform convergence?

 

[LauraLovies]

What theorem do you know which relates continuous functions and uniform convergence?

Are you talking about something Cauchy?

 

[LCKurtz]

What theorem do you know which relates continuous functions and uniform convergence?

Are you talking about something Cauchy?

I'm trying to get you to look in your text and find the theorem that is relevant to your problem and is undoubtedly in your text. What theorem(s) do you find?

 

[LauraLovies]

I'm trying to get you to look in your text and find the theorem that is relevant to your problem and is undoubtedly in your text. What theorem(s) do you find?

There is a theorem that states that if the functions are continuous and the sequences converge then there is uniform convergence. I just have no idea if the sequences converge uniformly or not.

 

[LCKurtz]

There is a theorem that states that if the functions are continuous and the sequences converge then there is uniform convergence. I just have no idea if the sequences converge uniformly or not.

Perhaps you should look that theorem up and see what it states exactly.