Borwein integral

A Borwein integral is an integral involving a product of sinc functions, where sinc(x) is defined as sin(x)/x, first presented by the mathematicians David Borwein and Jonathan Borwein in 2001. Borwein integrals are notable for a pattern that appears to hold and then unexpectedly breaks: successive integrals built from more sinc factors keep evaluating to pi/2, as long as the sum of the reciprocals of the factors' denominators stays below 1. The integral built from factors through sin(x/13)/(x/13) still equals pi/2 exactly, but adding the next factor, sin(x/15)/(x/15), pushes the sum of reciprocals past 1 and the value drops to pi/2 minus a tiny amount, about 2.31 times ten to the minus eleven. The integrals are a well known illustration of how a numerical pattern that appears to hold across many cases can fail once a hidden threshold is crossed.