Infinite product convergence from logarithmic tails

If a tail of holomorphic factors admits logarithms whose absolute values have a summable bound uniformly on every compact set, their product converges locally uniformly to the exponential of the logarithmic sum. The tail is holomorphic and nowhere zero. Finite initial factors may have zeros; those finite factors determine the locations and orders of all zeros near any given compact set. The scalar sufficient condition ${\sum }_j|u_j|<\infty$ for factors $1+u_j$ follows from $|\log (1+u_j)|\le 2|u_j|$ for $|u_j|\le 1/2$.