# Uniform convergence

## Main Question or Discussion Point

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 isnt uniform convergent then can they show a proof and explain. Thanks any help will be greatly appreciated!

mathman
Your question is incomplete. What variable (n???) is changing and what is it going to?

n is changing, and its going to infinity

HallsofIvy
Homework Helper
If x lies between -1 and 1, then xn converges to 0 so 1- xn converges to 1. If x> 1 or x< -1 then xn does not converge at all so 1- xn does not converge. If x= 1, 1- xn= 0 for all x while if x= -1, 1- xn= 0 for even n and 2 for odd n and so 1- (-1)n does not converge. If a sequence of functions converges on a compact (closed and bounded) set, then it converges uniformly on that set so 1- xn converges uniformly on $[\epsilon, 1]$ for any $\epsilon> 0$ but does NOT converge uniformly on (0, 1].

thank you

mathman
If x lies between -1 and 1, then xn converges to 0 so 1- xn converges to 1. If x> 1 or x< -1 then xn does not converge at all so 1- xn does not converge. If x= 1, 1- xn= 0 for all x while if x= -1, 1- xn= 0 for even n and 2 for odd n and so 1- (-1)n does not converge. If a sequence of functions converges on a compact (closed and bounded) set, then it converges uniformly on that set so 1- xn converges uniformly on $[\epsilon, 1]$ for any $\epsilon> 0$ but does NOT converge uniformly on (0, 1].
I believe the interval of convergence should be$[\epsilon-1, 1]$