x^m (1-x)^n /(1+x^2),
for integral values of m and n. Depending on their relative values, one
gets rational approximations to Pi, ln(2), or a linear combination.
Even more remarkably, and indeed embarrassingly, I received a prize for
the Best Short Article of 1995, based on a vote of the readership of the
Gazette!
At the time of writing my article, I did not know of the earlier work of
Dalzell, although this was later pointed out via a letter to the Editor.
I did know of it as a Putnam problem.
Regards to all
Nigel Backhouse
=======================================================================
Dr. Nigel Backhouse Telephone: +44 (0)151-794-4019
Division of Applied Mathematics, FAX: +44 (0)151-794-4061
Department of Mathematical Sciences, email: sx52@liv.ac.uk
Mathematics and Oceanography Building,
The University of Liverpool,
Liverpool L69 3BX
UK
On Tue, 19 Jan 1999, John F Harper wrote:
> On Mon, 18 Jan 1999, Samuel S. Kutler wrote:
>
>> 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