# Thread: Proof of Convergence of a Series

1. ## 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.

2. How about the ratio test?

3. 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.

4. 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.