Before revision:  An elementary proof of a constant in the strong discrete Stechkin inequality given by Bennett is provided, and we improve the constants given by Stechkin and Copson. By means of convex optimization, we compute two decimal places of the optimal constant in the classical version of the strong discrete Stechkin inequality .
After revision:  An elementary proof of a constant in the strong discrete Stechkin inequality given by Bennett is provided, and we improve the constants given by Levin and Stechkin and by Copson .