Re: [HM] Les nombres remarquables
John F Harper (harper@kauri.vuw.ac.nz)
Tue, 19 Jan 1999 17:41:15 +1300 (NZD)
On 22/7 - pi = integral from 0 to 1 with respect to x of
x^4[(1-x)^4]/(1 + x^2):
whenever it was discovered, this intriguing fact can be guessed at from
1. the usual partial fraction method shows the integrand = 6th degree
polynomial + 4/(1+x^2),so the integral must = R + pi, where R is rational;
3. integral 0 to 1 of x^6 dx = 1/7, so the denominator of R seems likely
to have a factor 7,
2. x^4[1-x)^4] is very small throughout the domain (0,1), so that R must
be near pi. It is pleasing that when one does the calculation R = 22/7
(not, say, 1023*22/(7*1024) which nothing above would exclude)
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