Absolutely Convergent Series is Convergent/Complex Numbers

Theorem
Let $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ be an absolutely convergent series in $\C$.

Then $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ is convergent.

Proof
Let $z_n = u_n + i v_n$.

We have that:

and similarly:


 * $\cmod {z_n} > \size {v_n}$

From the Comparison Test, the series $\displaystyle \sum_{n \mathop = 1}^\infty u_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty v_n$ are absolutely convergent.

From Absolutely Convergent Series is Convergent/Real Numbers|Absolutely Convergent Series is Convergent: Real Numbers]], $\displaystyle \sum_{n \mathop = 1}^\infty u_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty v_n$ are convergent.

By Convergence of Series of Complex Numbers by Real and Imaginary Part, it follows that $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ is convergent.