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Episode 457

Infinite product
Fri, 2018-Aug-03 01:39 UTC
Length - 3:56

Welcome to popular Wiki of the Day where we read the summary of a popular Wikipedia page every day.

With 578,841 views on Thursday, 2 August 2018 our article of the day is Infinite product.

In mathematics, for a sequence of complex numbers a1, a2, a3, ... the infinite product

n

=

1

a

n

=

a

1

a

2

a

3

{\displaystyle \prod _{n=1}^{\infty }a_{n}=a_{1}a_{2}a_{3}\cdots }

is defined to be the limit of the partial products a1a2...an as n increases without bound. The product is said to converge when the limit exists and is not zero. Otherwise the product is said to diverge. A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence an as n increases without bound must be 1, while the converse is in general not true.

The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète (Viète's formula, the first published infinite product in mathematics) and John Wallis (Wallis product):

2

π

=

2

2

2

+

2

2

2

+

2

+

2

2

{\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }

π

2

=

2

1

2

3

4

3

4

5

6

5

6

7

8

7

8

9

=

n

=

1

(

4

n

2

4

n

2

1

)

.

{\displaystyle {\frac {\pi }{2}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdot \cdots =\prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right).}

This recording reflects the Wikipedia text as of 01:39 UTC on Friday, 3 August 2018.

For the full current version of the article, go to http://en.wikipedia.org/wiki/Infinite_product.

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