On Ramanujan's inequalities for exp(k)

Ramanujan claimed in his first letter to Hardy (16 January 1913) that 1/2e(k) - Sigma(v=o)(k-1) k(v)/v! = k(k/)k! (1/3 + 4/135(k+theta(k))) (k = 1, 2, ...), where theta(k) lies between 2/21 and 8/45. This conjecture was proved in 1995 by Flajolet et al. The paper establishes the following refinement. 1/2e(k) - (Sigma)(k-1)(v=0) k(v)/v! = k(k/)k! (1/3 + 4/135k - 8/2835(k + theta*(k))(2)) (k = 1, 2, ...), where -1/3 < theta*(k) less than or equal to -1 + 4/root21(368 - 135e) = -0.14074... Both bounds for theta*(k) are sharp.