Theorem. The infinite product $$\prod_{n=1}^\infty(1\!+\!c_n)$$ converges absolutely if and only if the series $$\sum_{n=1}^\infty c_n$$ with complex terms $c_n$ converges absolutely.

Proof. The theorem follows directly from the theorems of the entries absolutely convergent infinite product converges and infinite product of sums $1\!+\!a_i$ .