On some quadrature rules with Gregory end corrections

How can one compute the sum of an infinite series \(s := a_1 + a_2 + \ldots\)? If the series converges fast, i.e., if the term \(a_n\) tends to \(0\) fast, then we can use the known bounds on this convergence to estimate the desired sum by a finite sum \(a_1 + a_2 + \ldots + a_n\). However, the series often converges slowly. This is the case, e.g., for the series \(a_n = n^{-t}\) that defines the Riemann zeta-function. In such cases, to compute \(s\) with a reasonable accuracy, we need unrealistically large values \(n\), and thus, a large amount of computation. Usually, the \(n\)-th term of the series can be obtained by applying a smooth function \(f(x)\) to the value \(n\): \(a_n = f(n)\). In such situations, we can get more accurate estimates if instead of using the upper bounds on the remainder infinite sum \(R = f(n + 1) + f(n + 2) + \ldots\), we approximate this remainder by the corresponding integral \(I\) of \(f(x)\) (from \(x = n + 1\) to infinity), and find good bounds on the difference \(I - R\). First, we derive sixth order quadrature formulas for functions whose 6th derivative is either always positive or always negative and then we use these quadrature formulas to get good bounds on \(I - R\), and thus good approximations for the sum \(s\) of the infinite series. Several examples (including the Riemann zeta-function) show the efficiency of this new method. This paper continues the results from [W. Solak, Z. Szydełko, Quadrature rules with Gregory-Laplace end corrections, Journal of Computational and Applied Mathematics 36 (1991), 251–253] and [W. Solak, A remark on power series estimation via boundary corrections with parameter, Opuscula Mathematica 19 (1999), 75–80]