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# 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 $\displaystyle\lim_{{k\to\infty}}u_{1}u_{2}\ldots u_{k}$.

## Mathematics Subject Classification

30E20*no label found*

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