The series $\sum_{n = 1}^{\infty} a_{n}$ with $a_{n} = \frac{1}{\sqrt n - i n}$ does not converge absolutely because, $$|a_{n}| = \frac{1}{|\sqrt n - i n|} = \frac{1}{\sqrt{n + n^2}} \geq \frac{1}{\sqrt 2 \ n } $$ and $\frac{1}{\sqrt 2}\sum_{n=1}^{\infty} \frac{1}{n}$ does not converge.
Therefore the ratio test and other comparison tests dont work. My question is, if this series does converge or if it diverges. I dont managed to show if the series is either a cauchy sequence or not
$$\frac{1}{\sqrt{n}-in} = \frac{\sqrt{n}}{n+n^2}+i\frac{n}{n+n^2} $$ From the definition of a complex series, it doesn't converge, since it's imaginary part doesn't converge. ($\frac{n}{n+n^2} $ is comparable with $\frac{1}{n}$ and $\sum \frac{1}{n} $ doesn't converge)