On some quadrature rules with Laplace end corrections (Q424154)

A quadrature formula obtained by adding some correction terms to a midpoint quadrature rule is called a quadrature formula of Laplace type. In this paper, the authors investigate a particular case of such a quadrature formula, studied previously by \textit{W. Solak} and \textit{Z. Szydełko} [J. Comput. Appl. Math. 36, No. 2, 251--253 (1991; Zbl 0738.65009)]. Then they apply this quadrature rule (of order six) in approximating the sum of a slowly converging series \(\sum_{i=0}^\infty f(i+1/2)\), when \(f\in C^6\) has the sixth derivative of constant sign on an interval \([m,\infty)\), \(m\in \mathbb N\), and \(\int_m^\infty f(x) dx\) is known. Numerical examples showing the efficiency of the method are considered.