Show the series $\sum_{n=1}^\infty \frac{z^2}{(1+z^2)^n}$ converges uniformly on $|z^2+1|\geq r>1$

Question

Show the series $\sum_{n=1}^\infty \frac{z^2}{(1+z^2)^n}$ uniformly converges on $|z^2+1|\geq r>1$.

I first thought of Weierstrass M-test but I can't control $|z|^2$ part. Could you give any hint?

Here, $z\in\Bbb C$.

Answer 1

If $|z^2+1|\geqslant r$, then you have$$\left|\frac{z^2}{1+z^2}\right|=\left|1-\frac1{1+z^2}\right|\leqslant1+\frac1r,$$and therefore$$(\forall n\in\Bbb N)\left|\frac{z^2}{(1+z^2)^n}\right|\leqslant\left(1+\frac1r\right)\left|\frac1{(1+z^2)^{n-1}}\right|\leqslant\left(1+\frac1r\right)\frac1{r^{n-1}}.$$Can you take it from here?