An infinite product $\prod _{n=1}^{\mathrm{\infty }}(1+a_n)$is absolutely convergent iff 1;; $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ is absolutely convergent.

Proof:

• Taylor expand $\frac{\ln (1+a_n)}{a_n}$ around the point $1$, and find bounds. Use reverse triangle inequality and finally the comparison test.