Equivalence of Definitions of Absolute Convergence of Product of Complex Numbers

Theorem
Let $\left\langle{z_n}\right\rangle$ be a sequence of complex numbers with real part $> -1$.

Then the following are equivalent:


 * The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + z_n}\right)$ converges absolutely.
 * The series $\displaystyle \sum_{n \mathop = 1}^\infty \log \left({1 + \left\lvert{z_n}\right\rvert}\right)$ converges absolutely.
 * The series $\displaystyle \sum_{n \mathop = 1}^\infty \log \left({1 + z_n}\right)$ converges absolutely.
 * The series $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ converges absolutely.

Proof
By Convergence of Infinite Product of Complex Numbers, the first two are equivalent.

By Bounds for Complex Logarithm:
 * $\frac12|z| \leq |\log (1+z)| \leq \frac32|z|$
 * $\frac12|z| \leq |\log (1+|z|)| \leq \frac32|z|$

for $|z|\leq\frac12$.

By Comparison Test the last three are equivalent.