Let $\{f_n\}^{+∞}_{n=1}$ be a sequence of functions where $f_n(x)=nx^2e^{-nx}$ where $x$ belongs to $[1;+\infty)$ and $n$ belongs to $\mathbb N$ (all natural numbers).
Determine whether the sequences of functions ${f_n}$ is uniformly convergent or not.
I computed $g_n(x)=|f_n(x)−f(x)|=nx^2e^{-nx}$ then solved the equation $g'_n(x)=0$, I get $x=\frac{2}{n}$ or $x=0$. But $0$ and $\frac{2}{n}$ do not belong to $[1;+\infty)$, for all $n \in \mathbb{N}$.
So what I can do to find the supremum of $|f_n(x)−f(x)|$?
Best Answer
Note that $e^{nx} = 1 + nx + \frac{(nx)^2}{2} + \cdots > \frac 12 n^2x^2$ for all $x > 0$. So we have $f_n(x) = nx^2 e^{-nx} < \frac 2n$. Since $2/n$ converges to $0$ we must have pointwise limit function $f(x) = 0$. So $\vert f_n(x) -f(x) \vert < \frac 2n$. So supremum must be less that $2/n$ for each $n \in \mathbb{N}$.
Can you finish it now?