More on Sequences

Besides sums, we also have product notations:$$\prod_{a \in A} a = \prod_{i=1}^n a_i = \prod_{1 \le i \le n} a_i = a_1 \cdot a_2 \cdot a_3 \cdot \dots \cdot a_n.$$

The first notation is written in $\LaTeX$ as

$\prod_{a \in A} a$

("the product of all $a$ in $A$") and the other two as

$\prod_{i=1}^n a_i$

and

$\prod_{1 \le i \le n} a_i$

("the product of all $a_i$ such that $i$ goes from $1$ to $n$".)

Note how the ellipsis $\dots$ (

$\dots$

) turned into middle dots $\cdots$ (

$\cdots$

) when we include it between two $+$s (additions) or two $\cdot$s (multiplications). That is,

$\dots$

adjusts itself to the expression's context.

Fun class activity: How do you write $\prod_{a \in A} a$ as a for loop in Java/C++? What about $\prod_{i=1}^n a_i$?

Example: Let $S = \langle 7, 2, 5, 1\rangle$. Then, $\prod_{k=1}^4 s_k = 7 \cdot 2 \cdot 5 \cdot 1 = 70$.