Integral estimates for convergent positive series (Q1766691)

Let \(f:[1,\infty) \to (0,\infty)\) be a convex differentiable function such that the series \(\sum_{n=1}^{\infty} f(n)\) converges. It is shown that there exists a unique sequence \((\theta_n)_{n\geq 1}\) with \(1/2<\theta_n< 1\) and \[ \sum_{k=n+1}^{\infty} f(k) =\int_{n+\theta_n}^{\infty} f(t) \,dt \] for every \(n\geq 1\). This sequence need not be monotone but under further assumptions it is strictly decreasing and convergent to \(1/2\). A detailed study of \((\theta_n)_{n\geq 1}\), various examples and an iterative method for the estimation of \[ \sum_{k=n+1}^{\infty} \Biggl( \int_{k-1/2}^{k+1/2} f(t) \,dt - f(k)\Biggr) \] are also presented.