Prove that the series $\sum_{n=1}^{\infty}\left\Vert x\right\Vert ^{n} $, $x\in\mathbb{R}^{n} $, does not converge uniformly on the unit ball $\left\{ x\in\mathbb{R}^{n}\mid\left\Vert x\right\Vert <1\right\} $.
I am not sure how to show this. What I got to is that the given series is a geomtric series and hence $f\left(x\right)=\sum_{n=1}^{\infty}\left\Vert x\right\Vert ^{n}=\frac{\left\Vert x\right\Vert }{1-\left\Vert x\right\Vert }$ on the unit ball, which is continuous (on the unit ball). But this doesn't tell us anything.