A Pattern That Breaks Down

In P vs NP I mentioned a pattern that broke down involving Peven numbers and Podd numbers. In doing the research for that I came across some other maths patterns that look solid at first, but eventually break down. Here is my favourite and it involves an unusual trig function called sinc which is defined as sinc(x) = sin(x)/x.

Those numbers at the bottom are just single numbers, I couldn't fit them onto one line because I was ambitious with digit sizing.

So for the first 7 terms the integral always equals exactly Pi/2, but then suddenly the 8th term equals something just a tiny bit smaller. I'm not sure if I find this hideous or beautiful, I think it might be both.

Incidentally the series 1/1+1/3+1/5+...+1/13 = 1.9551…, but the same series with an extra 1/15 clocks above 2 to 2.0218… and it is this threshold that determines the behaviour inside the integral. The specifics get pretty hairy, but if you are interested in this sort of thing then I suspect Fourier Series are going to be the way in and you will meet them during your first year of a maths degree.