absolutely convergent infinite product converges

Theorem.  An absolutely convergent (http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct) infinite product

ν=1(1+cν)=(1+c1)(1+c2)(1+c3) (1)

of complex numbersPlanetmathPlanetmath is convergent.

Proof.  We thus assume the convergence of the product (http://planetmath.org/Product)

ν=1(1+|cν|)=(1+|c1|)(1+|c2|)(1+|c3|) (2)

Let ε be an arbitrary positive number.  By the general convergence condition of infinite product, we have


when  n certain nε.  Then we see that

|(1+cn+1)(1+cn+2)(1+cn+p)-1| =|1+ν=n+1n+pcν+μ,νcμcν++cn+1cn+2cn+p-1|
=|(1+|cn+1|)(1+|cn+2|)(1+|cn+p|)-1|<ε  p+

as soon as  nnε.  I.e., the infinite product (1) converges, by the same convergence condition.

Title absolutely convergent infinite product converges
Canonical name AbsolutelyConvergentInfiniteProductConverges
Date of creation 2013-03-22 18:41:15
Last modified on 2013-03-22 18:41:15
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 40A05
Classification msc 30E20
Synonym convergence of absolutely convergent infinite product
Related topic AbsoluteConvergenceImpliesConvergenceForAnInfiniteProduct
Related topic AbsoluteConvergenceOfInfiniteProductAndSeries