convergence condition of infinite product

Let us think the sequencePlanetmathPlanetmathu1,u1u2,u1u2u3,  In the complex analysis, one often uses the definition of the convergence of an infinite productk=1uk  where the case  limku1u2uk=0  is excluded.  Then one has the


The infinite product k=1uk of the non-zero complex numbersPlanetmathPlanetmathu1, u2, … is convergent iff for every positive number ε there exists a positive number nε such that the condition


is true as soon as  nnε.

Corollary.  If the infinite product converges, then we necessarily have  limkuk=1. (Cf. the necessary condition of convergence of series.)

When the infinite product converges, we say that the value of the infinite product is equal to limku1u2uk.

Title convergence condition of infinite product
Canonical name ConvergenceConditionOfInfiniteProduct
Date of creation 2013-03-22 14:37:22
Last modified on 2013-03-22 14:37:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Theorem
Classification msc 30E20
Related topic OrderOfFactorsInInfiniteProduct
Related topic NecessaryConditionOfConvergence
Defines infinite product
Defines value of infinite product