I have this sequence, $$ f_n(x) = \frac{nx}{1+n^2x^2} $$ I have to say if this sequence is uniform convergent,
based on the definition of uniform convergence, a sequence is uniform convergent if this limit exists: $$\lim_{n\to \infty}(\sup(|f_n(x) - f(x)|)) = 0$$
I've found that f(x) = 0, and therefore limit becomes easier: $$\lim_{n\to \infty}(\sup(|f_n(x)|)) = 0$$
now I have to find the absolute maximum of this sequence, and in theory I have to study the function by taking the derivative (using quotient rule in this case), but I think there is a faster approach:
- finding the domain of the function,
- if the upper bound is equal to infinity, then it doesn't converge.
in this case, domain is $${-\infty, +\infty}$$ so it doesn't make sense to find convergence. Is it legit to use this method? or should I always do the longer approach involving the first derivative?