Show that a function is well-defined and continuous Prove that $$ h(x) = \sum_{n \ge 1} \frac{\exp\left(-nx^2 [\arctan(x^n)]^2 \right)}{|x| + n^2}$$ is well-defined and continuous on $\mathbb{R}$.
I think using the M-test could work here, but not exactly how to do it. Could someone please show me?

The numerator is $e^{- \text{non-negative quantity}}$ (which is non-negative), so is maximized when the non-negative quantity is $0$ (achievable at $x=0$), giving an upper bound of $1$. A lower bound on the denominator is $n^2$ (and the denominator is non-negative). Thus, each term can be upper bounded by $\frac{1}{n^2}$ (and thus in magnitude, by non-negativity) which is absolutely summable. Thus, by the $M$-test, uniform convergence.
Now apply uniform limit of continuous functions is continuous.
