Product

The term ``product'' refers to the result of one or more Multiplications. For example, the mathematical statement $a\times b=c$ would be read ``$a$ Times $b$ Equals $c$,'' where $c$ is the product.

The product symbol is defined by

$$\prod_{i=1}^n f_i \equiv f_1\cdot f_2\cdots f_n.$$

Useful product identities include

$$\ln\left({\,\prod_{i=1}^\infty f_i}\right)= \sum_{i=1}^\infty \ln f_i$$

$$\prod_{i=1}^\infty f_i = \mathop{\rm exp}\nolimits \left({\,\sum_{i=1}^\infty \ln f_i}\right).$$

For $0\leq a_i<1$, then the products $\prod_{i=1}^\infty (1+a_i)$ and $\prod_{i=1}^\infty (1-a_i)$ converge and diverge as $\prod_{i=1}^\infty a_i$.

See also Cross Product, Dot Product, Inner Product, Matrix Product, Multiplication, Nonassociative Product, Outer Product, Sum, Tensor Product, Times, Vector Triple Product

References

Guy, R. K. ``Products Taken over Primes.'' §B87 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 102-103, 1994.