On Ramanujan's Inequalities for exp( k )

Ramanujan claimed in his first letter to Hardy (16 January 1913) that1 2 e k − ∑ v = 0 k − 1k vv !=k kk !( 1 3 + 4 135 ( k + θ ( k ) ))   ( k = 1 , 2 , ⋖ ) ,where θ( 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 2 e k − ∑ v = 0 k − 1k vv !=k kk !( 1 3 + 4 135 k − 8 2835( k + θ * ( k ) ) 2)   ( k = 1 , 2 , ⋖ ) ,where− 1 3 < θ * ( k ) ⩽ − 1 + 421 ( 368 − 135 e )= − 0.14074….Both bounds for θ * ( k ) are sharp.