Here's the next one I could use some help on.

The principle argument of a complex number z, denoted by $\displaystyle arg_P(z)$, is defined as the value of $\displaystyle \theta _P = arg(z)$ such that $\displaystyle -\pi < \theta _P \leq \pi$. Give an example of a complex sequence $\displaystyle ( z_n )$ converging to a limit $\displaystyle \alpha$ such that $\displaystyle arg _P (z_n)$ fails to converge.

I know of one such example:

$\displaystyle z_n = \frac{1}{n}~e^{in}$

This has a limit point of 0, but no well defined argument.

However the text says there is at least one other series with a limit point of -1. Not only have I been unable to construct an example (please help), but I can't see the logic behind this one at all. What is so special about -1 that, say, we can't construct one with a limit point of 1?

-Dan