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1 2009

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On some quadrature rules with Gregory end corrections

Bożek B., Solak W., Szydełko Z.

Opuscula Mathematica

2009

Vol. 29, no. 2

117-129

How can one compute the sum of an infinite series s := a1 + a2 + ź ź ź ? 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 a1 +a2 +ź ź ź+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 ƒ(x) to the value n: an = ƒ(n). In such situations, we can get more accurate estimates if instead of using the upper bounds on the remainder infinite sum R = ƒ(n + 1) + ƒ(n + 2) + . . ., we approximate this remainder by the corresponding integral I of ƒ(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 [3] and [2].