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Product ∏

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

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Product ∏ -

Tag(s) : Arithmetics

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# Product ∏

## Product ∏ Calculator

### What is a product ∏? (Definition)

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

### How to calculate a finite product?

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.

### How to calculate an infinite product?

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 $$\ln{x} = (x-1) \prod_{n=1}^\infty \frac{2}{1+x^{1/2^n}}$$

### How to calculate a double product?

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.

### How to make the ∏ product symbol?

The product is written with the dedicated mathematical symbol ∏ (Unicode U+220F) which is inspired by the Greek letter pi uppercase Π (Unicode U+03A0).

In Greek, pi corresponds to the letter P (like the first letter of Product).

In LaTeX, the operator is \prod

## Source code

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