Cauchy product

The Cauchy product formula, also Cauchy product or Cauchy- fold, named after the French mathematician Augustin Louis Cauchy allows the multiplication of infinite series.

Are and two absolutely convergent series, then the series

Also an absolutely convergent series, and it is

The series is called Cauchy product of series and.

If we write this formula, so we get:

If you break this series at a certain value of ab, we obtain an approximation of the desired product.

Especially for the multiplication of power series is valid

Examples

Application to the exponential function

As an application example is intended to show how the functional equation of the exponential function of the Cauchy product formula can be derived. The exponential function converges absolutely known. Therefore, one can calculate the product by means of the Cauchy product and receives

By the definition of the binomial coefficient can transform the more than

Where the penultimate equality sign by the binomial theorem is justified.

A divergent series

It is the Cauchy product

Are formed of a conditionally convergent series only with itself.

Here applies

Applied to the inequality of the arithmetic and geometric means following the root in the denominator

Since the thus do not form a null sequence, the series diverges

Generalizations

By the theorem of Mertens it is already sufficient to require that at least one of the two convergent series converges absolutely, that you Cauchy product converges (not necessarily absolute) and its value is the product of the given number of values ​​.

Both series converge only conditionally, so it may be that their Cauchy product does not converge, as the above example shows. In this case, however, converges if the Cauchy product, then its value is true for a set of Abel consistent with the product of the two series of values ​​.