First: Sorry that I wrote "normal" complex coordiantes. No idea how they are properly called =(
So I have this example given, it should show me (i think) about problems occurding when switchting to polar coordinates:
When
$z_n=-2 + i\frac{(-1)^n}{n^2}, \quad n=1,2,...$
we can write
$\lim_{n\to\infty} z_n = \lim_{n\to\infty} (-2) + i \lim_{n\to\infty}\frac{(-1)^n}{n^2}=-2 + i\cdot 0 = -2$
If, using polar coordinates, we write
$r_n = |z_n|$ and $\Phi=Arg z_n, \quad n=1,2,...$
where $Arg z_n$ denotes principal arguments $(-\pi < \Phi \leq \pi)$ of $z_n$, we find that
$\lim_{n\to\infty} r_n = \lim_{n\to\infty} \sqrt{4+\frac{1}{n^4}} = 2$
but that
$\lim_{n\to\infty} \Phi_{2n}=\pi$ and $\lim_{n\to\infty} \Phi_{2n-1}=-\pi$
Evidently, then, the limit of $\Phi$ does not exists as $n$ tends to infinity.
Question: I was wondering what I should take away from this. Does it mean, that the existence of a limit of a complex sequence depends on the chosen coordinate system? If not, how would I properly calculate it in polar coordiantes?