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'convergence condition of infinite product'
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| Title of object: |
convergence condition of infinite product |
| Canonical Name: |
ConvergenceConditionOfInfiniteProduct |
| Type: |
Theorem |
| Created on: |
2004-09-21 17:55:27 |
| Modified on: |
2006-09-13 10:50:29 |
| Classification: |
msc:30E20 |
| Keywords: |
Cauchy sequence |
| Defines: |
value of infinite product |
Revision comment (for changes between this and next version):
Changes for correction #9916 ('use \ldots here too'); comparison to series convergence.
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Content:
In the complex analysis, one often uses the definition of the convergence of infinite product $\displaystyle\prod_{k = 1}^{\infty}u_k$ where the case\, $\displaystyle\lim_{k\to\infty}u_1u_2 \ldots u_k = 0$\, is excluded.\, Then one has the
\begin{thmplain}\, 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
$$\vert u_{n+1}u_{n+2} \ldots u_{n+p}-1 \vert < \varepsilon \quad \forall \,p\in\mathbb{Z}_+$$
is true as soon as\, $n \geqq n_\varepsilon$.
\end{thmplain}
\textbf{Corollary.}\, If the infinite product converges, then we necessarily have\, $\displaystyle\lim_{k\to\infty}u_k = 1$.
When the infinite product converges, we say that the {\em value of the infinite product} is equal to $\displaystyle\lim_{k\to\infty} u_1u_2...u_k$. |
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