convergence condition of infinite product

Let us think the sequence  $u_1,u_1{u}_2,u_1{u}_2{u}_3,\mathrm{\dots }$  In the complex analysis, one often uses the definition of the convergence of an infinite product  ${\displaystyle \prod _{k=1}^{\mathrm{\infty }}}{u}_k$  where the case  $\underset{k\to \mathrm{\infty }}{lim}{u}_1{u}_2\mathrm{\dots }{u}_k=0$  is excluded.  Then one has the

Theorem.

The infinite product ${\displaystyle \prod _{k=1}^{\mathrm{\infty }}}{u}_k$ of the non-zero complex numbers  $u_1$, $u_2$, … is convergent iff for every positive number $\epsilon$ there exists a positive number $n_{\epsilon }$ such that the condition

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

Corollary.  If the infinite product converges, then we necessarily have  $\underset{k\to \mathrm{\infty }}{lim}{u}_k=1$. (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 $\underset{k\to \mathrm{\infty }}{lim}{u}_1{u}_2\mathrm{\dots }{u}_k$.