Convergence of an infinite product

Summary

An infinite product converges when, after finitely many zero factors are set aside, its partial products tend to a nonzero limit. The nonzero condition makes the theory parallel to series: a product of positive terms converges exactly when the series of logarithms converges. A convergent product has factors tending to $1$, and its value is $0$ only when some factor is $0$. The exact definition is in convergence of an infinite product.