Why $\sum_{n=1}^{\infty}\frac{x^{\alpha}}{1+n^2 x^2}$ doesn't converge uniformly on $[0, \infty)$ for $\alpha > 2$?

Question

I'm trying to understand why $\sum_{n=1}^\infty\frac{x^\alpha}{1+n^2 x^2}$ doesn't converge uniformly on $[0, \infty)$ for $\alpha > 2$.

My book says that $\frac{x^\alpha}{1+n^2 x^2}$ is monotonic and unbounded for $\alpha > 2$ therefore it doesn't converge. I don't get why this means it can't converge exactly, someone care to explain?

Answer 1

Because if a series of functions $\sum_{n=1}^\infty f_n$ converges uniformly, then the sequence $(f_n)_{n\in\Bbb N}$ converge uniformly to the null function. So, the functions $f_n$ cannot be unbounded for every $n\in\Bbb N$.

Answer 2

When $\alpha > 2$, then $\sup\frac{x^{\alpha}}{1+n^2 x^2} = +\infty$ so general member doesn't converged to $0$ uniformly.