I aim to show that $\{a_n\} = \frac{1+nx^2}{(1+x^2)^n}$ converges to $0$. The following two facts seem obvious:
(1) $\forall n \in \mathbb{N}, a_n \ge 0$ (since each $a_n$ is the product of two positive numbers).
(2) $\forall n \in \mathbb{N}, a_n \ge a_{n+1}$.
Yet this is not quite enough to show that $a_n \rightarrow 0$ since the sequence could very well converge to some other $c > 0$.