I have a sequence of functions defined by [(n^2)(x^2)]/(exp (nx)).

I know that the limit of this sequence of functions is zero for all x in R, x >= 0.

However, I have to show that for a > 0, this sequence converges uniformly on the interval [a, inf], but it does not converge uniformly on the interval [0, inf].

My textbook says that for n sufficiently large, the uniform norm in the interval [a, inf] is [(n^2)(x^2)]/(exp (ax)), which makes total sense and the limit of this goes to zero. However, it says the uniform norm in the interval [0, inf] is just 4/(e^2) (and obviously this limit does not go to zero). Can someone tell me how you get this second uniform norm and/or show me another way to prove it is not uniformly convergent in the second interval [0, inf]? Thanks for your help.