calculus and analysis - Integral of the Sinc product

Let us consider the following integral $$ B_n = \int_0^\infty \prod_{k=1,3,5,\dots}^n\frac{\sin (x/k)}{x/k}dx $$

By definition, Sinc[x] == Sin[x]/x, therefore

B[n_?OddQ] := Integrate[Product[Sinc[x/k], {k, 1, n, 2}], {x, 0, Infinity}]

B /@ Range[1, 13, 2]

{Pi/2, Pi/2, Pi/2, Pi/2, Pi/2, Pi/2, Pi/2}

OK, everything is fine. But...

B[15]

467807924713440738696537864469 Pi/935615849440640907310521750000

What's going on?

Answer

As Eckhard wrote in comments B[n] is the n-th Borwein integral.

(The letter B was not accidental :) )

This funny properties of Borwein integrals is related to the Fourier transform of Sinc function

FourierTransform[Sinc[x], x, k]

1/2 Sqrt[Pi/2] (Sign[1 - k] + Sign[1 + k])

Plot[%, {k, -2, 2}, Filling -> 0]

which is the box function. The result is $\pi/2$ while the sum $$ 1/3+1/5+\dots+1/n < 1. $$ If $n \ge 15$ the sum exceeds $1$ and the result becomes

$$ B_n = \frac{\pi}{2} - \pi \bigg(\sum_{k=3,5,\ldots}^n\frac{1}{k} -1\biggr)^\frac{n-1}{2}\prod_{k=3,5,\dots}^n\frac{k}{k-1}. $$

For $n=15$ it is equal to

$$ \frac{467807924713440738696537864469}{935615849440640907310521750000}\pi. $$

As a prank, Jonathan Borwein reported this to Maple, claiming there was a bug in the software. Maple computer scientist Jacques Carette spent 3 days trying to figure out the problem. Then he realized: There was no bug! That's what these integrals really equal!

The Borwein brothers are the same guys who noticed that the integral

$$ \int_0^\infty \cos(2x) \cos(x) \cos(x/2) \cos(x/3) \cos(x/4) \dots dx $$

matches $\pi/8$ up to $43$ decimal places, but is not equal to $\pi/8$. So you've got to be careful with these guys!