convergence of function series using Weierstrass M-test

Question

I am trying to examine uniform convergence of series $$\sum_{n=1}^{\infty}{\frac{x^2}{n^4+x^4}}$$ For $x \ \in \mathbb{R}$. What i've tried: $$\text{assume } x \in [-R, R] \text{ for some } R>0$$ $$|\frac{x^2}{n^4+x^4}| = \frac{x^2}{n^4+x^4} \leq\frac{x^2}{n^4}\leq \frac{R^2}{n^4}$$ And since $\sum_{n=1}^{\infty}{\frac{R^2}{n^4}}$ converges, the original series converges uniformly on $[-R, R]$. I am pretty sure that only proves continuity on $\mathbb{R}$. I have not been able to prove it for $\mathbb{R}$, how should I go about it?

Answer 1

If you want uniform convergence over $\mathbb{R}$, observe that: $$n^4+x^4 \geq 2n^2 x^2$$ for any $n \in \mathbb{N}$ and $x \in \mathbb{R}$. This is just the AM-GM inequality. Hence: $$\frac{x^2}{n^4+x^4} \leq \frac{1}{2n^2}$$

In particular, for any $x \in \mathbb{R}$: $$\sum_{n \in \mathbb{N}} \frac{x^2}{n^4+x^4} \leq \sum_{n \in \mathbb{N}} \frac{1}{2n^2}$$

And this proves uniform convergence over $\mathbb{R}$.