Pi (Product) notation

Product notation (also called pi notation) indicates repeated multiplication.

For example, the following product notation represents the product of the first three squared numbers (from $i$ = 1 to 3):

$\prod_{n = 1}^{3} i^{2} = 1^{2} \times 2^{2} \times 3^{2}$

This notation is very similar to summation notation. The only difference is that while summation notation expresses a pattern of sums (i.e. a sequence of numbers), product notation shows a pattern in products. Instead of the Greek letter sigma (Σ), we use the upper case Greek letter pi (Π).

Properties

Separability of Constants

$\prod_{i = 1}^{n} c a_{i} = c^{n} \prod_{i = 1}^{n} a_{i}$

Separability of Multiple Products

$\prod_{k = m}^{n} a_{k} \cdot \prod_{k = m}^{n} b_{k} = \prod_{k = m}^{n} a_{k} \cdot b_{k}$

References

Statistics how to
Notes – Introduction to Mathematical Thinking