In mathematics, a Borwein integral is an integral studied by involving products of sinc(ax), where the sinc function is given by sinc(x) = sin(x)/x. These integrals are notorious for exhibiting apparent patterns that eventually break down. An example they give is


This pattern continues up to


However at the next step the obvious pattern fails:


In general similar integrals have value π/2 whenever the numbers 3, 5, ... are replaced by positive real numbers such that the sum of their reciprocals is less than 1. In the example above, 1/3 + 1/5 + ... + 1/13 < 1, but 1/3 + 1/5 + ... + 1/15 > 1.