Pi ($\Pi$) Notation

The capital Greek letter Pi ($\Pi$) is utilized similarily to $\Sigma$ notation, but instead of summation, the product of the terms is computed. For example, below the start, stop, and term are defined as three constants: 1, 4, and 5 respectively:

$\displaystyle\prod\limits_{1}^{4} 5 = 5\cdot5\cdot5\cdot5$

Likewise, we can define variables to the start, stop and term:

$\displaystyle\prod\limits_{n=0}^{3}(n + x) = (0+x)(1+x)(2+x)(3+x)$

Here we see a variable start and stop, as well as a term that contains a yet undefined variable $x$. Since we do not know what value(s) $x$ should take, we evaluate the $\prod$ as far as possible. That is, we apply algebraic simplification if possible.

Finally, $\prod$ notation is able to nest, just like $\sum$ notation. For example:

$\displaystyle\prod\limits_{i=1}^{2}\displaystyle\prod\limits_{j=2}^{4}2ij = (2i_{1}*2 * 2i_{2}*3 * 2i_{3}*4) * (2i_{2} * 2 * 2i_{2} * 3 * 2i_{2} * 4)$

$= (2*2)(2*3)(2*4)(4*2)(4*3)(2*4)$

Factorials

While looking at $\prod$ notation, it is also timely to take a brief detour to look at factorials of non-negative integers:

$5!$

$3!$

$20!$

or more generally:

$n!$

This notation simply translates to: take the product of all positive integers less than or equal to $n$. For example:

$5! = (1)(2)(3)(4)(5)$

The aside is timely, because we can write the factorial using $\prod$ notation. I am intentionally not going to demonstrate how to do this as it will be a question on the weekly assignment. As a hint, remember that we can utilize a variable in the start and then reuse that variable in the term.