For what values of z is $\displaystyle \sum_{n=0}^{\infty} \{\frac{z}{1+z}\}^n$ convergent?
I don't know how to start this, any help would be great to get me going..
You use the root test:
A series converges absolutely if:
$\displaystyle \lim_{n\to\infty}\left\vert\sqrt[n]{\left(\frac{z}{1+z}\right)^n}\right\vert=\left\ve rt\frac{z}{1+z}\right\vert<1$
So you start like, like Plato to solve
$\displaystyle \left\vert\frac{z}{1+z}\right\vert<1$
Check this out: Radius of convergence - Wikipedia, the free encyclopedia