We know that if a sum is convergent, but it's not convergent absolutely, then we can arrange it's terms so that it converges to any real number.
For complex numbers similar theorem does not hold i. e. if a sum is convergent and not convergent absolutely, then it can happen so that we can't obtain every complex number (which is pretty obvious because for example we can't obtain non-real complex numbers from a sum of real numbers). However there exist sums that have this property, for example $\sum \frac{i^n}{n}$.
My question is, what are the criteria for which a complex sum that converges but doesn't converge absolutely can be rearranged so that it converges to any number on the complex plane.
Question source is my own curiosity. By criteria I mean any sort of necessary or sufficient condition, within reason of course.
