<Dd> Srinivasa Ramanujan . </Dd> <P> This converges extraordinarily rapidly . Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π . </P> <Dl> <Dd> 1 π = 12 ∑ k = 0 ∞ (− 1) k (6 k)! (13591409 + 545140134 k) (3 k)! (k!) 3 640320 3 k + 3 / 2 (\ displaystyle (\ frac (1) (\ pi)) = 12 \ sum _ (k = 0) ^ (\ infty) (\ frac ((- 1) ^ (k) (6k)! (13591409 + 545140134k)) ((3k)! (k!) ^ (3) 640320 ^ (3k + 3 / 2)))) </Dd> <Dd> David Chudnovsky and Gregory Chudnovsky . </Dd> </Dl> <Dd> 1 π = 12 ∑ k = 0 ∞ (− 1) k (6 k)! (13591409 + 545140134 k) (3 k)! (k!) 3 640320 3 k + 3 / 2 (\ displaystyle (\ frac (1) (\ pi)) = 12 \ sum _ (k = 0) ^ (\ infty) (\ frac ((- 1) ^ (k) (6k)! (13591409 + 545140134k)) ((3k)! (k!) ^ (3) 640320 ^ (3k + 3 / 2)))) </Dd>

What are three civilizations that used working approximations of pi