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convergence condition of infinite product
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(Theorem)
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Let us think the sequence $u_1,\,u_1u_2,\,u_1u_2u_3,\,\ldots$ , 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 $\displaystyle\lim_{k\to\infty}u_1u_2 \ldots u_k = 0$ , is excluded. Then one has the
Theorem 1 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 $n \geqq n_\varepsilon$
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 $\displaystyle\lim_{k\to\infty} u_1u_2 \ldots u_k$
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"convergence condition of infinite product" is owned by pahio.
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Cross-references: series, necessary condition of convergence, converges, number, positive, iff, convergent, complex numbers, complex analysis, sequence
There are 24 references to this entry.
This is version 13 of convergence condition of infinite product, born on 2004-09-21, modified 2008-12-28.
Object id is 6202, canonical name is ConvergenceConditionOfInfiniteProduct.
Accessed 5039 times total.
Classification:
| AMS MSC: | 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions) |
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Pending Errata and Addenda
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