3. Theorem 2.12.1 - convergence with logarithm

Suppose $1 + a_{n} \in C ∖ (- \infty, 0]$ . Then $\prod_{n = 1}^{\infty} (1 + a_{n})$ converges iff
:?:

\sum_{n = 1}^{\infty} \ln (1 + a_{n})

converges, with the logarithms have their imaginary part in $(- π, π)$ .

PROOF:
Backwards Direction - the complex logarithm has a left inverse (exponential)

Forwards Direction - Let $1 + a_{n} = p_{n} = | p_{n} | e^{i ϕ_{n}}$ , $\prod_{n = 1}^{\infty} (1 + a_{n}) = p = | p | e^{i ϕ}$

We do need to be careful with our logarithm branches here!