# Uniform Convergence

• Nov 16th 2011, 04:53 PM
veronicak5678
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)
• Nov 16th 2011, 05:33 PM
TKHunny
Re: Uniform Convergence
Can you state 'f' explicitly? Is it constant, linear, what?

Can you find f'_n, generally?
• Nov 16th 2011, 05:45 PM
veronicak5678
Re: Uniform Convergence
f_'n (x)=(1-nx^2)/(1+nx^2 )^2 and f=0
• Nov 17th 2011, 04:15 AM
TKHunny
Re: Uniform Convergence
Very good. You're almost there. Where is the derivative zero?
• Nov 17th 2011, 07:49 AM
veronicak5678
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?
• Nov 17th 2011, 09:33 AM
TKHunny
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?
• Nov 17th 2011, 01:06 PM
veronicak5678
Re: Uniform Convergence
At x=0, f'_n =1
• Nov 17th 2011, 01:44 PM
TKHunny
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)"
• Nov 17th 2011, 01:49 PM
veronicak5678
Re: Uniform Convergence
I guess I thought there was more to it. Thanks for your help.