Introduction: “One of the more beautiful results related to approximating is the integral
Since the integrand is nonnegative on the interval , this shows that is strictly less than , the well known approximation to . The first published statement of this result was in 1971 by D. P. Dalzell [Eureka 34, 10–13 (1971)], although anecdotal evidence [see J. M. Borwein, The life of Pi, history and computation, seminar presentation 2003, available from http://www.cecm.sfu.ca/~jborwein/pi-slides.pdf, March 2005] suggests it was known by Kurt Mahler in the mid-1960s. The result (1) is not hard to prove, if perhaps somewhat tedious. A partial fraction decomposition leads to a polynomial plus a term involving , which integrates immediately to the required result. An alternative is to use the substitution , leading to a polynomial in powers of . We then apply the recurrence relation for taking the integrals of powers of . Of course, the simplest approach today is to simply verify (1) using a symbolic manipulation package such as Maple or Mathematica.
An obvious question at this point might be whether similar elegant integral results can be found for other rational approximations for . A particularly good approximation is 355/113, which is accurate to seven digits. Our aim here is to find a variety of such integral results.”
However, despite several variations of the style of integrand, no simple and elegant result was found.
The article is highly recommended as a basis of an undergraduate project.