# Thread: Uniform Convergence

1. ## Uniform Convergence

Consider the sequence of functions f_n (x)= x / (1+nx^2)

let f = lim f_n. Compute f'_n and find all values of x such that f'(x)=lim f'_n (x)

2. ## Re: Uniform Convergence

Can you state 'f' explicitly? Is it constant, linear, what?

Can you find f'_n, generally?

3. ## Re: Uniform Convergence

f_'n (x)=(1-nx^2)/(1+nx^2 )^2 and f=0

4. ## Re: Uniform Convergence

Very good. You're almost there. Where is the derivative zero?

5. ## Re: Uniform Convergence

The derivative is zero when nx^2 =1, or when x=(1/n)^1/2

But isn't lim f' =0 for all x?

6. ## Re: Uniform Convergence

Let's try to remember which variable is moving. Are we talking about x approaching zero or n increasing without bound?

AT x = 0, what is the limit of f'_n as n increases without bound?

7. ## Re: Uniform Convergence

At x=0, f'_n =1

8. ## Re: Uniform Convergence

Right, and that's not zero (0), is it?

I believe you were done with this problem when you said this: "x=(1/n)^(1/2)"

9. ## Re: Uniform Convergence

I guess I thought there was more to it. Thanks for your help.