find all intervals of uniform convergence

Question

I would like to find all intervals on which sequence $$ f_n = \frac{x^{2n}}{n+x^{2n}} $$

converges uniformly.

I was able to show that it converges uniformly to $0$ when $|x|<1$.

But I am not sure about the cases $|x|=1$ and $|x|>1$. In these cases it converges pointwise to $0$ and $1$ respectively. But I cannot show that it converges uniformly on these intervals

Answer 1

It converges uniformly to $0$ on $|x| \le 1$, because $|f_n| \le 1/n$ there.

It converges uniformly to $1$ on any interval of the form $[1+\epsilon, \infty)$ or $(-\infty, -1-\epsilon]$ for $\epsilon > 0$, because on those intervals $$\left|f_n - 1\right| = \dfrac{n}{n + x^{2n}} < n (1+\epsilon)^{-2n}$$

It does not converge uniformly on $(1, \infty)$, because for any $n>2$ you can take $x > 1$ so close to $1$ that $f_n < 1/2$ there. Similarly for $(-\infty, -1)$.

Of course, if it converges uniformly on an interval, it converges uniformly on any subinterval.