Definition:Convergent Product

Also defined as
Some authors define an infinite product to be convergent the sequence of partial products converges. The phrase:
 * $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$

then becomes:
 * $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converges to $0$.

The definition used here has many desirable consequences, such as:
 * Factors in Convergent Product Converge to One
 * Convergence of Infinite Product Does not Depend on Finite Number of Factors
 * Product of Convergent and Divergent Product is Divergent

which creates an analogy with series.

Also see

 * Logarithm of Infinite Product of Real Numbers
 * Logarithm of Infinite Product of Complex Numbers
 * Definition:Absolute Convergence of Product