On some quadrature rules with Gregory end corrections

Abstract

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, <i>Quadrature rules with Gregory-Laplace end corrections</i>, Journal of Computational and Applied Mathematics 36 (1991), 251–253] and [W. Solak, <i>A remark on power series estimation via boundary corrections with parameter</i>, Opuscula Mathematica 19 (1999), 75–80].