# convergence condition of infinite product

Let us think the sequence  $u_{1},\,u_{1}u_{2},\,u_{1}u_{2}u_{3},\,\ldots$  In the complex analysis, one often uses the definition of the convergence of an infinite product$\displaystyle\prod_{k=1}^{\infty}u_{k}$  where the case  $\displaystyle\lim_{k\to\infty}u_{1}u_{2}\ldots u_{k}=0$  is excluded.  Then one has the

###### Theorem.

The infinite product $\displaystyle\prod_{k=1}^{\infty}u_{k}$ of the non-zero complex numbers  $u_{1}$, $u_{2}$, … is convergent iff for every positive number $\varepsilon$ there exists a positive number $n_{\varepsilon}$ such that the condition

 $|u_{n+1}u_{n+2}\ldots u_{n+p}-1|<\varepsilon\quad\forall\,p\in\mathbb{Z}_{+}$

is true as soon as  $n\geqq n_{\varepsilon}$.

Corollary.  If the infinite product converges, then we necessarily have  $\displaystyle\lim_{k\to\infty}u_{k}=1$. (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 $\displaystyle\lim_{k\to\infty}u_{1}u_{2}\ldots u_{k}$.

Title convergence condition of infinite product ConvergenceConditionOfInfiniteProduct 2013-03-22 14:37:22 2013-03-22 14:37:22 pahio (2872) pahio (2872) 16 pahio (2872) Theorem msc 30E20 OrderOfFactorsInInfiniteProduct NecessaryConditionOfConvergence infinite product value of infinite product