Tool to perform formal calculations of products with the operator ∏ (uppercase pi), allowing infinite products or defined arithmetic multiplication from 1 to n.

Product ∏ - dCode

Tag(s) : Arithmetics

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In mathematics, the product, denoted $ \prod $, is the result of the multiplication $ \times $ of a series of numbers.

In arithmetic, the product notation $ \prod_1^n $ allows you to calculate a finite multiplication repeated from $ 1 $ up to $ n $ with an increment of 1 (unless otherwise stated).

__Example:__ The product of the first $ 4 $ integers (also called factorial of 4) is written $$ 1 \times 2 \times 3 \times 4 = \prod_{i=1}^{4} i = 4! = 24 $$

__Example:__ The increasing factorial (or Pochhammer function) is written $ (x)_n=x(x+1)(x+2)\cdots(x+n-1) $ and $ (x+1)_n=(x+1)(x+2)\cdots(x+n) = \prod_{k=1}^{n} (x+k) $

Sometimes the product does not converge towards a value, it can diverge and have no result.

The notation $ \prod_1^\infty $ (sometimes shortened to $ \prod $) indicates the calculation of an infinite multiplication repeated from $ 1 $ to infinity $ \infty $ by an increment of 1 (unless otherwise stated).

__Example:__ $$ \prod_{n=2}^{\infty} \left( 1-\frac{1}{n^2} \right) = \frac{1}{2} $$

Some functions can be described as an infinite product

__Example:__ The sine function $$ \sin{x} = x \prod_{n=1}^\infty \cos\left( z/2^n \right) $$

__Example:__ The logarithm function $$ \ln{x} = (x-1) \prod_{n=1}^\infty \frac{2}{1+x^{1/2^n}} $$

The notation $ \prod \prod $ is read $ \prod \left (\prod \right) $ so the inner product (between the parenthesis) is calculated first, then the outer product is calculated.

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Cite as source (bibliography):

*Product ∏* on dCode.fr [online website], retrieved on 2024-09-14,

product,multiplication,pi,index,time,infinite

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