1. ## Uniformly Convergent

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

a.) $\displaystyle [0,\infty)$

b.) $\displaystyle (0,\infty)$

c.) $\displaystyle [1,\infty)$

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

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

a.) $\displaystyle [0,\infty)$

b.) $\displaystyle (0,\infty)$

c.) $\displaystyle [1,\infty)$

d.) And decide whether or not for $\displaystyle f_{n}: [0,\infty) \rightarrow$R if $\displaystyle f_{n}(x) = \frac{nx}{1+n^2x^2}$ converges uniformly.
First, find the pointwise limit, which in this case is the zero function. To test whether or not $\displaystyle f_n\to0$, you need to investigate whether $\displaystyle \sup\{|f(x)|\}\to0.$ 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).