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..

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- Nov 29th 2009, 04:26 AMgizmoconvergent series problem
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.. - Nov 29th 2009, 05:36 AMPlato
- Nov 29th 2009, 05:42 AMgizmo
can we use that because we know it converges?

- Nov 29th 2009, 09:19 AMhjortur
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