1. Show that the sequence x^2 (e^-nx) converges uniformly on [0, infinity).
2. Show that if a is greater than zero then the sequence (n^2 x^2 (e^-nx)) converges uniformly on the interval [a, infinity) but does not converge uniformly on the interval [0, infinity).
For problem 2 text gives a hint that if n is sufficiently large, ||f_n||_[a,infinity]=n^2 a^2/e^na , however ||f_n||_[0,infinity)=4/e^2 (A _ is used to denote a subscript).© BrainMass Inc. brainmass.com September 22, 2018, 3:01 pm ad1c9bdddf - https://brainmass.com/math/calculus-and-analysis/uniform-convergence-of-a-sequence-of-functions-340478
Uniform cnfvergence of sequences of functions is examined and illustrated with two examples in the attached PDF file.