convergent series problem

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November 29th 2009, 04:26 AM
gizmo

convergent series problem

For what values of z is $\sum_{n=0}^{\infty} \{\frac{z}{1+z}\}^n$ convergent?

I don't know how to start this, any help would be great to get me going..
November 29th 2009, 05:36 AM
Plato

Quote:

Originally Posted by gizmo

For what values of z is $\sum_{n=0}^{\infty} \{\frac{z}{1+z}\}^n$ convergent?

I don't know how to start this

Start by solving $\left| {\frac{z}{{1 + z}}} \right| < 1$ . WHY?
November 29th 2009, 05:42 AM
gizmo

can we use that because we know it converges?
November 29th 2009, 09:19 AM
hjortur

You use the root test:

A series converges absolutely if:

$\lim_{n\to\infty}\left\vert\sqrt[n]{\left(\frac{z}{1+z}\right)^n}\right\vert=\left\ve rt\frac{z}{1+z}\right\vert<1$

So you start like, like Plato to solve

$\left\vert\frac{z}{1+z}\right\vert<1$

Check this out: Radius of convergence - Wikipedia, the free encyclopedia

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