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Integral proofs that 355/113>π. (English) Zbl 1181.11077

Introduction: “One of the more beautiful results related to approximating π is the integral

0 1 x 4 (1-x) 4 1+x 2 dx=22 7-π·(1)

Since the integrand is nonnegative on the interval [0,1], this shows that π is strictly less than 22/7, the well known approximation to π. 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 1/(1+x 2 ), which integrates immediately to the required result. An alternative is to use the substitution x=tanθ, leading to a polynomial in powers of tanθ. We then apply the recurrence relation for taking the integrals of powers of tanθ. 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 π. 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.

MSC:
11Y60Evaluation of constants
Software:
Maple