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Math Help - convergent series problem

  1. #1
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    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..
    Last edited by Plato; November 29th 2009 at 05:32 AM.
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  2. #2
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    Quote Originally Posted by gizmo View Post
    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?
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  3. #3
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    can we use that because we know it converges?
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  4. #4
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    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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