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Math Help - Proof of Convergence of a Series

  1. #1
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    Proof of Convergence of a Series

    I'm trying to prove that, if |z| > 1 for any complex z, then \displaystyle \sum^{\infty}_{n = 1} \frac{1}{1 + z^{n}} converges. I got this far but no farther:

    If we try by comparison test, |\frac{1}{1 + z^{n}}| \leq \frac{1}{|z|^{n} - 1} = \frac{1}{|z|^{n}} \cdot \frac{1}{1 - \frac{1}{|z|^{n}}}. I know the first part of the RHS is decreasing, limit goes to 0. I'll be done if I show that the second part has bounded partial sums, but I can't figure that part out.

    Thanks.
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  2. #2
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    How about the ratio test?
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  3. #3
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    Nope, not unless you see something I don't. In any case, the final exam is upon me, so I won't need to know this in about three hours anyway.
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  4. #4
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    If |z| > 1 you should be able to show

    \displaystyle \frac{1 + z^n}{1 + z^{n + 1}} \to \frac{1}{z}

    since

    \displaystyle \left|  \frac{1 + z^n}{1 + z^{n + 1}} - \frac{1}{z}\right|  = \left|\frac{z - 1}{z(1 + z^{n + 1})} \right| \le \frac{|z| + 1}{|z| (|z|^n - 1)}

    and you can make |z|^n as big as you like.
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