Discuss the uniform convergence of the sequence of functions

Discuss the uniform convergence of the sequence of functions f_n(x) = nxe^-nx^2 on the interval (i) [0, 1] and (ii) on the interval [1, infinity).

Solution

First of all, observe that f_n(0) = 0 for every n in N. So the sequence {f_n(0)} is constant and converges to zero.

(i) Now suppose 0 < x < 1

f_n(0) = nxe^-nx2

But -x²< 0 when 0 < x < 1, it follows that
lim f_n(x) = 0 for 0 < x < 1
n

Finally, f_n(1) = ne^-n for all n.

So,lim   f_n(x) =

n

Therefore, {f_n} is not pointwise convergent on [0, 1].

(ii) for [1, )

For every real number x, we have:
lim f_n(x) = lim nxe^-nx2 = 0
n   n

Thus, {f_n} converges pointwise in interval [1,)