ON RAMANUJAN'S INEQUALITIES FOR ${\exp}(\hbox{\itshape k})$

Abstract

Ramanujan claimed in his first letter to Hardy (16 January 1913) that $$\frac{1}{2}e^k-\sum_{\nu=0}^{k-1}\frac{k^{\nu}}{\nu !}=\frac{k^k}{k!}\bigg(\frac{1}{3}+\frac{4}{135(k+\theta(k))}\bigg) \qquad{(k=1,2,{\ldots})},$$ 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. $$\frac{1}{2}e^k-\sum_{\nu=0}^{k-1}\frac{k^{\nu}}{\nu !}=\frac{k^k}{k!} \bigg(\frac{1}{3}+\frac{4}{135 k}-\frac{8}{2835(k+\theta^*(k))^2}\bigg) \qquad{(k=1,2,{\ldots})},$$ where $$-\frac{1}{3}<\theta^*(k)\leq -1+\frac{4}{\sqrt{21(368-135e)}}=-0.140\,74{\ldots}\,.$$ Both bounds for $\theta^*(k)$ are sharp.