Convergence of Generalized Sum of Complex Numbers

Theorem
Let $\left({z_i}\right)_{i \in I}$ be an $I$-indexed family of complex numbers.

That is, let $z_i \in \C$ for all $i \in I$.

Denote by $\operatorname{Re} z_i$, resp. $\operatorname{Im} z_i$ the families of real, resp. imaginary parts of the family $z_i$.

Then the following are equivalent:


 * $(1): \qquad \displaystyle \sum \left\{{z_i : i \in I}\right\}$ converges to $z \in \C$
 * $(2): \qquad \displaystyle \sum \left\{{\operatorname{Re} z_i : i \in I}\right\}, \sum \left\{{\operatorname{Im} z_i : i \in I}\right\}$ converge to $\operatorname{Re} z, \operatorname{Im} z \in \R$, respectively

Corollary
Suppose that $\displaystyle \sum \left\{{ z_i: i \in I }\right\}$ converges to $z \in \C$.

Then $\displaystyle \sum \left\{{ \overline{z_i}: i \in I }\right\}$ converges to $\overline z$, where $\overline z$ denotes the complex conjugate of $z$.