Rearrangement of Infinite Series

If every subseries of a series with complex terms converges the series converges absolutely.

Assume that the series $\left|z_{1}\right| + \left|z_{2}\right| + \cdots + \left|z_{n}\right| + \cdots$ diverges. Then there exists a direction of accumulation, that is a real number $\alpha$ such that those terms of the series $z_{1}+z_{2}+\cdots +z_{n}+\cdots$ that are contained in the sector $\alpha - \varepsilon < \arg z < \alpha + \varepsilon$ constitute an absolutely divergent subseries for any $\varepsilon > 0$.

If $\lim_{n\to \infty } z_{n} = 0$ and if the positive real axis is a direction of accumulation of the conditionally convergent series $z_{1}+z_{2}+z_{3}+\cdots$ then there exists a subseries $z_{r_{1}}+z_{r_{2}}+z_{r_{3}}+\cdots$ the real part of which diverges to $+\infty$ and the imaginary part converges to a finite number.

If the series $z_{1}+z_{2}+z_{3}+\cdots$ is convergent, but not absolutely convergent, any value represented by a point of a certain straight line can be obtained as the sum of the series rearranged in a suitable order.