# uniform convergence

• Feb 18th 2011, 12:58 PM
mathsohard
uniform convergence
(a) Fn(x) = nxe^(-nx), x>= 0. Prove that the convergence is uniform for x>= @, where @ is any positive number. Why is the convergence not uniform on the interval 0<= x<= @??

(b) Fn(x) = (tan^-1)nx, all x. What conditions must a and b satisfy if the convergence is to be uniform on a <= x <= b ?
• Feb 19th 2011, 04:47 AM
FernandoRevilla
Quote:

Originally Posted by mathsohard
(a) Fn(x) = nxe^(-nx), x>= 0. Prove that the convergence is uniform for x>= @, where @ is any positive number. Why is the convergence not uniform on the interval 0<= x<= @??

For $\displaystyle x\in [0,+\infty)$ the function $\displaystyle f_n(x)=nxe^{-nx}$ has an absolute maximum at $\displaystyle x=1/n$ so, if $\displaystyle n\geq 1/a$ then, $\displaystyle f_n$ is decreasing. For $\displaystyle n\geq 1/a$ and $\displaystyle x\geq a$ we have $\displaystyle 0\leq f_n(x)\leq f_n(a)$ .

But $\displaystyle \lim_{n\to +\infty}f_n(a)=0$ so, $\displaystyle f_n\to 0$ in $\displaystyle [a,+\infty)$ uniformly .

Try the other part.

Fernando Revilla