Wallis product

The wallissche product also Wallis - product is a product representation of the circle constant, ie it is a product with infinitely many factors, whose limit is. It was discovered in 1655 by the English mathematician John Wallis.

Formula

It is customary to display the product in the form:

About a forming, the shorthand notation of the Wallis product is given as follows:

For the inverse follows:

The convergence of this product follows from the convergence of the infinite series

Convergence speed

For the efficient computation of an approximation of pi formula is not suitable. If we calculate approximately the first 5 terms of the Walliser product, and doubles the result, we obtain the approximation for Pi:

With this approximation could not even be determined correctly the first decimal place.

After multiplying out the first 50 terms results in a quotient of two 160 - digit numbers, but only the approximation for pi yields 3.126, indicating that is not even 2 decimal places correctly. As 3.126 / 3.14159 = 0.9950 is, the relative error is about 0.5%. The rate of convergence is slower than linear.

The adjacent table shows, for some selected values ​​of how well the approximation of Pi is obtained by multiplying out the terms within the wallis between product. The table suggests that the error after multiplying of terms in such amounts (eg after 100 terms: 0.25 % =).

This can be proved by the following mathematical reasoning: The quotient between the approximation and the desired value is equal to the infinite product

Using the calculation rules for logarithms, the estimate ( for small ) and by approximating an infinite sum by an integral we see that this product approximately has the following value:

Thus, the first two digits are correct, one needs therefore an accuracy of about 0.3 % ( 3.13 / 3.14 = 0.997 ), ie approximately for 3 decimal places you need for 4 decimal places, etc.

Sketch of proof

One defines, for which the recursion formula applies. In particular, we obtain for the formula.

Calculate and. Well true, and therefore.

In particular, therefore, from which one obtains by squaring the usual formula.