Product Notation

Product notation ( $\prod$ ) is to multiplication what Sigma notation is to addition. It is widely used in number theory (factorials, modular arithmetic) and polynomial roots.

Definition

Definition (Product Notation)

i = m \prod n a_{i} = a_{m} \cdot a_{m + 1} \dots a_{n}

Properties

1. Multiplicative Property

i = 1 \prod n (a_{i} b_{i}) = (i = 1 \prod n a_{i}) (i = 1 \prod n b_{i})

2. Power Property

i = 1 \prod n (c a_{i}) = c^{n} i = 1 \prod n a_{i}

(Note: The constant

c

is multiplied

n

times, so it becomes

c^{n}

, not

c

3. Telescoping Products

Similar to sums, if

a_{i} = \frac{b _{i + 1}}{b _{i}}

, terms cancel nicely.

Example (Telescoping Product)

Evaluate

\prod_{k = 2}^{n} (1 - \frac{1}{k ^{2}})

Proof. Factor the term: $1 - \frac{1}{k ^{2}} = \frac{k ^{2} - 1}{k ^{2}} = \frac{( k - 1 ) ( k + 1 )}{k \cdot k}$ . Rewrite the product:

k = 2 \prod n \frac{k - 1}{k} \cdot \frac{k + 1}{k} = (k = 2 \prod n \frac{k - 1}{k}) (k = 2 \prod n \frac{k + 1}{k})

First part:

\frac{1}{2} \cdot \frac{2}{3} \dots \frac{n - 1}{n} = \frac{1}{n}

. Second part:

\frac{3}{2} \cdot \frac{4}{3} \dots \frac{n + 1}{n} = \frac{n + 1}{2}

. Result:

\frac{1}{n} \cdot \frac{n + 1}{2} = \frac{n + 1}{2 n}

. ∎

Useful Identities

Factorial: $\prod_{i = 1}^{n} i = n!$
Geometric Progression: $\prod_{k = 0}^{n} r = r^{n + 1}$
Wallis Product: An infinite product involving $π$ (advanced context).

Practice Problems

Exercise (Problem 1)

Evaluate

\prod_{k = 1}^{10} 2^{k}

Exercise (Problem 2)

Simplify

\prod_{k = 2}^{n} \frac{k ^{3} - 1}{k ^{3} + 1}

Exercise (Problem 3)

Calculate the value of

\prod_{n = 2}^{\infty} (1 - \frac{2}{n ( n + 1 )})