The Borwein integrals

The Borwein integrals introduced in [1] are a famous example of how proof-by-example can go wrong.

Define sinc(x) as sin(x)/x. Then the following equations hold.

However

where 2.3 * 10^-11.

This is where many presentations end, concluding with the moral that a pattern can hold for a while and then stop. But I'd like to go just a little further.

Define

Then B(n) = /2 for n = 1, 2, 3, ..., 6 but not for n = 7, though it almost holds for n = 7. What happens for larger values of n?

The Borwein brothers proved that B(n) is a monotone function of n, and the limit as n exists. In fact the limit is approximately /2 - 0.0000352.

So while it would be wrong to conclude that B(n) = /2 based on calculations for n 6, this conjecture would be approximately correct, never off by more than 0.0000352.

[1] David Borwein and Jonathan Borwein. Some Remarkable Properties of Sinc and Related Integrals. The Ramanujan Journal, 3, 73-89, 2001.

The post The Borwein integrals first appeared on John D. Cook.