Remarks on some completely monotonic functions

Applying the Euler–Maclaurin summation formula, we obtain upper and lower polynomial bounds for the function , x>0, with coefficients the Bernoulli numbers Bk. This enables us to give simpler proofs of some results of H. Alzer and F. Qi et al., concerning complete monotonicity of certain functions involving the gamma function Γ(x), the psi function ψ(x) and the polygamma functions ψ(n)(x), n=1,2,… .