PlanetMath: link between infinite products and sums

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link between infinite products and sums

(Theorem)

Let

$\displaystyle \prod_{k=1}^\infty p_k$

be an infinite product such that

for all

. Then the infinite product converges if and only if the infinite sum

$\displaystyle \sum_{k=1}^\infty \log p_k$

converges. Moreover

$\displaystyle \prod_{k=1}^\infty p_k = \exp \sum_{k=1}^\infty \log p_k.$

Proof.

Simply notice that

$\displaystyle \prod_{k=1}^N p_k = \exp \sum_{k=1}^N \log p_k.$

If the infinite sum converges then (by continuity of $\exp$ function)

$\displaystyle \lim_{N\to \infty} \prod_{k=1}^N p_k = \lim_{N\to\infty} \exp \sum_{k=1}^N \log p_k = \exp \sum_{k=1}^\infty \log p_k$

and also the infinite product converges.

"link between infinite products and sums" is owned by paolini.

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Cross-references: function, sum, converges, product, infinite
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This is version 2 of link between infinite products and sums, born on 2003-06-16, modified 2006-03-17.
Object id is 4368, canonical name is LinkBetweenInfiniteProductsAndSums.
Accessed 3427 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)

Pending Errata and Addenda

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