> 22/7 - pi = integral from 0 to 1 with respect to x of
>
> x^4[(1-x)^4]/1 + x^2,
(1 + x^2) I hope.
> Do any of you know it?
It appeared in 1971 in D.P. Dalzell's article "On 22/7 and 355/113"
in Eureka no.34 (1971) 10-13. (Eureka was the approximately annually
published Journal of the Archimedeans, the Cambridge University
Mathematical Society [i.e. of mathematics students]. Is it still being
published?)
Dalzell used it to prove that 22/7 - 1/630 < pi < 22/7 - 1/1260 (both
bounds are better than Archimedes' 223/71 < pi < 22/7), which follows
immediately from 1 < 1 + x^2 < 2 if 0 < x < 1, and then by a suitable
fast-converging series that 355/113 - 33/10^8 < pi < 355/113 - 24/10^8.
John Harper, School of Mathematical and Computing Sciences,
Victoria University, Wellington, New Zealand
e-mail john.harper@vuw.ac.nz phone (+64)(4)471 5341 fax (+64)(4)495 5045