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**Prove It** It is well known that geometric series are divergent where $\displaystyle \displaystyle |r| \geq 1$.

If you must prove it though, since $\displaystyle \displaystyle |z| = 1$, then we can write $\displaystyle \displaystyle z = e^{i\theta}$.

So $\displaystyle \displaystyle \sum_{k = 0}^{\infty}z^k = \sum_{k = 0}^{\infty}\left(e^{i\theta}\right)^k$

$\displaystyle \displaystyle = \sum_{k = 0}^{\infty}e^{ik\theta}$

$\displaystyle \displaystyle = \lim_{n \to \infty}\left[\frac{e^{i\theta}\left(e^{i(n+1)\theta} - 1\right)}{e^{i\theta} - 1}\right]$.

Can this limit be evaluted? If so, what does it go to?