infinite product

But in a world of nearly infinite product choices, the margins matter.

It can never define an entire function, because the infinite product does not converge.

One may also consider products of infinitely many terms; these are called infinite products.

If we represent a square-free number as the infinite product:

A necessary condition for convergence of the infinite product in question is that each factor must approach 1 as .

(see Pi) in the form of an infinite product.

(Work is needed since some infinite products of positive numbers equal zero.)

(The product is necessarily finite, since infinite products are not defined in ring theory.

For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.

The mass formula gives the mass as an infinite product over all primes.

The same holds for an infinite product provided that it converges to a function continuous at the origin.

This can be written more concisely as an infinite product over all primes p:

The finite product can be expressed in terms of the infinite product:

Bunyakovsky's property implies for all primes p, so each factor in the infinite product C is positive.

Such infinite products are today called Euler products.

(See for example the infinite product formula for Z below.)

Now we can use Euler's infinite product representation for the sine function:

In particular, finite direct product of amenable groups are amenable, although infinite products need not be.

Then the infinite product converges if and only if the series converges.

Other infinite products relating to e include:

For example, in Leibniz' formula, an infinite sum (all of whose terms are infinite products) would have to be calculated.

The Artin L-function is then the infinite product over all prime ideals of these factors.

(Note that the binary digits are reversed from the ordering in the infinite product.)

Note that if the number of zeros is infinite one may have to define how to take the infinite product.)

The category of metric spaces, Met, is finitely complete but has neither binary coproducts nor infinite products.