absolutely convergent infinite product converges
Theorem. An absolutely convergent (http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct) infinite product
(1) |
of complex numbers is convergent.
Proof. We thus assume the convergence of the product (http://planetmath.org/Product)
(2) |
Let be an arbitrary positive number. By the general convergence condition of infinite product, we have
when certain . Then we see that
as soon as . I.e., the infinite product (1) converges, by the same convergence condition.