Logarithm of Infinite Product of Complex Numbers

Theorem
Let $(z_n)$ be a sequence of complex numbers with real part $>-1$.

Then the following are equivalent:


 * The infinite product $\displaystyle\prod_{n=1}^\infty(1+z_n)$ converges to $z\in\C$.


 * There exists an integer $k\in\Z$ such that the series $\displaystyle\sum_{n=1}^\infty\log(1+z_n)$ converges to $\log z+2k\pi i$.

Proof
Conversely, suppose $\displaystyle\sum_{n=1}^\infty\log(1+z_n)=\log z+2k\pi i$.

Because $\exp$ is continuous, $\displaystyle\prod_{n=1}^\infty(1+z_n)=z$.