absolute convergence of infinite product and series

Theorem.  The infinite product

$$\prod _{n=1}^{\mathrm{\infty }}(1+c_n)$$

converges absolutely (http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct) if and only if the series

$$\sum _{n=1}^{\mathrm{\infty }}{c}_n$$

with complex $c_n$ converges absolutely (http://planetmath.org/AbsoluteConvergence).

Proof.  The theorem follows directly from the theorems of the entries absolutely convergent infinite product converges and infinite product of sums $1+a_i$ (http://planetmath.org/InfiniteProductOfSums1a_i).

Title	absolute convergence of infinite product and series
Canonical name	AbsoluteConvergenceOfInfiniteProductAndSeries
Date of creation	2013-03-22 18:41:18
Last modified on	2013-03-22 18:41:18
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	6
Author	pahio (2872)
Entry type	Axiom
Classification	msc 30E20
Related topic	AbsoluteConvergentInfiniteProductConverges
Related topic	InfiniteProductOfSums1a_i
Related topic	OrderOfFactorsInInfiniteProduct