The property used by Peter Tamaroff is the best way to prove that we do not have uniform convergence. It can also be done directly from the definition.
Suppose that $x\gt 0$. Then the absolute value of the difference between the limit $\dfrac{1}{1+x^2}$ and the sum of the terms up to $\dfrac{x^2}{(1+x^2)^m}$ is $\sum_{k=m+1}^\infty \frac{x^2}{(1+x^2)^k}.\tag{$1$}$ This is an infinite geometric series. The usual formula gives that the sum $(1)$ is equal to $\dfrac{1}{(1+x^2)^m}$.
Let $\epsilon=1/2$. We show that there is no $N$ such that if $m\ge N$, then $(1)$ is $\lt \epsilon$ for every positive $x$. More informally, we show that there is no $N$ that "works" for every positive $x$.
Suppose to the contrary that there is such an $N$, and let $m=N$. Then $\dfrac{1}{(1+x^2)^N}\lt 1/2$. By algebraic manipulation, this is true precisely if $|x|\gt \sqrt{2^{1/N}-1}.\tag{$2$}$
From $(2)$, we see that there are non-zero $x$ for which the inequality does not hold, namely the positive $x$ that are $\le \sqrt{2^{1/N}-1}$. This contradicts the hypothesis of uniform convergence.