# Pointwise limit

• Nov 22nd 2009, 03:01 PM
thaopanda
Uniformly Convergent
Determine whether or not the sequence { $f_{n}$} $_n \in N$, where $f_{n}(x) = n^2x^2e^{-nx}$ is uniformly convergent on:

a.) $[0,\infty)$

b.) $(0,\infty)$

c.) $[1,\infty)$

d.) And decide whether or not for $f_{n}: [0,\infty) \rightarrow$R if $f_{n}(x) = \frac{nx}{1+n^2x^2}$ converges uniformly.
• Nov 23rd 2009, 01:50 AM
Opalg
Quote:

Originally Posted by thaopanda
Determine whether or not the sequence { $f_{n}$} $_n \in N$, where $f_{n}(x) = n^2x^2e^{-nx}$ is uniformly convergent on:

a.) $[0,\infty)$

b.) $(0,\infty)$

c.) $[1,\infty)$

d.) And decide whether or not for $f_{n}: [0,\infty) \rightarrow$R if $f_{n}(x) = \frac{nx}{1+n^2x^2}$ converges uniformly.

First, find the pointwise limit, which in this case is the zero function. To test whether or not $f_n\to0$, you need to investigate whether $\sup\{|f(x)|\}\to0.$ The function is positive, so you don't need to worry about taking the absolute value, and you can find the maximum value of the function by calculus (find where the derivative is 0).