product

Product ( $\prod$ )

$k = m \prod n a_{k} := a_{m} \cdot a_{m + 1} \dots a_{n}$
$k$ … index
$m, n$ … lower/upper bounds
$a_{k}$ … factor
Empty product ( $n < m$ ): $\prod := 1$ , the neutral element of multiplication.

Where does the $\prod$ stop?

Same scope rule as the sum: the $\prod$ binds the factors immediately right of it and stops at the first top-level $+$ or $-$ . Parenthesise to override.

Calculation rules:

Multiplicativity

Distribute over multiplication:
$k \prod (a_{k} b_{k}) = (k \prod a_{k}) (k \prod b_{k})$

Constant factor becomes a power

$k = m \prod n c a_{k} = c^{n - m + 1} k = m \prod n a_{k}$
Each of the $n - m + 1$ factors contributes one $c$ . In particular $\prod_{k = 1}^{n} k = n!$ (see factorial).

Range splitting and index shift

Work exactly as for the sum.

telescoping product

$k = m \prod n \frac{b _{k + 1}}{b _{k}} = \frac{b _{n + 1}}{b _{m}}$
Consecutive factors cancel, only the ends survive.

Log turns products into sums

$lo g k \prod a_{k} = k \sum lo g a_{k} (a_{k} > 0)$

A constant pulls out as a power, not a factor

$\prod_{k = 1}^{6} (- 2 k) = (- 2)^{6} \prod_{k = 1}^{6} k$ , not $- 2 \prod k$ .

Products do not split over addition

$\prod (a_{k} + b_{k}) \neq = \prod a_{k} + \prod b_{k}$

$\prod_{k = m}^{n} c a_{k} = ?$

$c^{n - m + 1} \prod_{k = m}^{n} a_{k}$ . One factor of $c$ per term, so the constant becomes a power.

$lo g \prod_{k} a_{k} = ?$ (for $a_{k} > 0$ )

$\sum_{k} lo g a_{k}$

Graph View

product

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