PlanetMath: order of factors in infinite product

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order of factors in infinite product

(Theorem)

Theorem 1 If the series $a_1\!+\!a_2\!+\cdots$ with complex terms is absolutely convergent, then the infinite product

$\displaystyle \prod_{k = 1}^\infty (1\!+\!a_k) = (1\!+\!a_1)(1\!+\!a_2)\cdots$

converges and its value, i.e. $\displaystyle\lim_{n\to\infty}\prod_{k = 1}^n (1\!+\!a_k)$ , does not depend on the order of its factors and vanishes only when some factor is zero.

"order of factors in infinite product" is owned by pahio.

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value of an infinite product

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Cross-references: converges, product, infinite, absolutely convergent, terms, complex, series

This is version 6 of order of factors in infinite product, born on 2004-09-22, modified 2006-09-13.
Object id is 6204, canonical name is OrderOfFactorsInInfiniteProduct.
Accessed 1867 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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