This is an exercise from Remmert.
The power series $\sum\limits_{n=1}^{\infty} \frac {z^{n} }{ n^{2}} \ $ has radius of convergence $1 \ $. Show that the function it represents is injective in $\{ z \in \mathbb{C} | \ \ \lVert z \rVert < \frac{2}{3} \} \ $.
The text gives the hint: $z^n -w^n = (z-w)\ ( z^{n-1}+z^{n-2}w + \ldots + w^{n-1} ) \ $.