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 δ ≈ 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*) = π/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 π/2 − 0.0000352.

So while it would be wrong to conclude that *B*(*n*) = π/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.

If you don’t append sinc(x/15), are there instead other sinc(x/(2k+1)) terms you could append to the integral, for other values of k, that still keep the result at exactly pi/2 ?

3Blue1Brown (Grant Sanderson) did a nice explainer video on this topic: “Researchers thought this was a bug (Borwein integrals)”

https://www.youtube.com/watch?v=851U557j6HE