Math is evil

I was reading about Borwein integrals.

Here’s a nice result:

$$\int_0^\infty dx\,\frac{\sin x}{x}=\frac{\pi}{2}.$$

Neat, is it not. Here’s another:

$$\int_0^\infty dx\,\frac{\sin x}{x}\frac{\sin (x/3)}{x/3}=\frac{\pi}{2}.$$

Jumping a bit ahead, how about

$$\int_0^\infty dx\,\frac{\sin x}{x}\frac{\sin (x/3)}{x/3}…\frac{\sin (x/13)}{x/13}=\frac{\pi}{2}.$$

Shall we conclude, based on these examples, that

$$\int_0^\infty dx\,\prod\limits_{k=0}^\infty\frac{\sin (x/[2k+1])}{x/[2k+1]}=\frac{\pi}{2}?$$

Not so fast. First, consider that

$$\int_0^\infty dx\,\frac{\sin x}{x}\frac{\sin (x/3)}{x/3}…\frac{\sin (x/15)}{x/15}=\frac{935615849426881477393075728938}{935615849440640907310521750000}\frac{\pi}{2}\approx\frac{\pi}{2}-2.31\times 10^{-11}.$$

Or how about

\begin{align}
\int_0^\infty&dx\,\cos x\,\frac{\sin x}{x}=\frac{\pi}{4},\\
\int_0^\infty&dx\,\cos x\,\frac{\sin x}{x}\frac{\sin (x/3)}{x/3}=\frac{\pi}{4},\\
…\\
\int_0^\infty&dx\,\cos x\,\frac{\sin x}{x}…\frac{\sin (x/111)}{x/111}=\frac{\pi}{4},
\end{align}

but then,

$$\int_0^\infty dx\,\cos x\,\frac{\sin x}{x}…\frac{\sin (x/113)}{x/113}\approx\frac{\pi}{4}-1.1162\times 10^{-138}.$$

There is a lot more about Borwein integrals on Wikipedia, but I think even these few examples are sufficient to convince us that, never mind the actual, physical universe, even the Platonic universe of mathematical truths is fundamentally evil and unreasonable.