Convergence of Series of Complex Numbers by Real and Imaginary Part

Theorem
Let $(z_n)$ be a sequence of complex numbers.

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ converges to $z\in\C$ iff the series $\displaystyle \sum_{n \mathop = 1}^\infty\Re(z_n)$ and $\displaystyle \sum_{n \mathop = 1}^\infty\Im(z_n)$ converge to $\Re(z)$ and $\Im(z)$ respectively.