Plane Geometry - Bruce E. Shapiro

244 SECTION 44. ESTIMATING πn π n10 3.2323158100 3.15149341,000 3.142591710,000 3.1416926100,000 3.14160271,000,000 3.1415937Clearly this method is not very practical: even after a million iterations itis only correct to the 5th decimal place!It turns out, however, that the Taylor series converges much more quicklyfor some values of x, particularly for 1/5 and 1/239. This is useful becauseof the following lemma, which is known as Machin’s formula. 3Lemma 44.1Proof. Use the trigonometric formulaThen using a = b,Å ã Å ãπ1 14 = 4 tan−1 − tan −15239tan −1 a + tan −1 b = tan −12 tan −1 a = tan −1 2a1 − a 2and using 2a/(1 − a 2 ) for both a and b,Setting a = 1/5,a + b1 − ab4 tan −1 a = tan −1 4a(1 − a2 )a 4 − 6a 2 + 14 tan −1 1 5 = 4(1/5)(1 − tan−1 (1/5)2 )(1/5) 4 − 6(1/5) 2 + 1−1 120= tan119−1 1 + 1/239= tan1 − 1/239= tan −1 1 + tan −1 1239The result of the lemma then follows because tan −1 1 = π/4.3 for John Machin(1680-1752) who used it to calculate the first 100 digits of pi in 1706.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012