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[parent] convergence condition of infinite product (Theorem)

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

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



"convergence condition of infinite product" is owned by pahio.
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See Also: order of factors in infinite product, necessary condition of convergence

Also defines:  value of infinite product
Keywords:  Cauchy sequence

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proof of convergence condition of infinite product (Proof) by fernsanz
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Cross-references: series, necessary condition of convergence, converges, number, positive, iff, convergent, complex numbers, product, infinite, complex analysis

This is version 12 of convergence condition of infinite product, born on 2004-09-21, modified 2006-09-13.
Object id is 6202, canonical name is ConvergenceConditionOfInfiniteProduct.
Accessed 3934 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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