Construction of Three Quadrature Formulas of Eighth Order and Their Application for Approximating Series

In this paper, three types of three-parameters families of quadrature formulas for the Riemann’s integral on an interval of the real line ( ) ( ) ∫ b a I f f x x = d are carefully studied. This research is a continuation of the results in the [1]-[3]. All these quadrature formulas are not based on the integration of an interpolant as so as the Gregory rule, a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order (see [4]). In some natural restrictions on the parameters we construct the only one quadrature formula of the eight order which belongs to the first, second and third family. For functions whose 8th derivative is either always positive or always negative, we use these quadrature formulas to get good two-sided bound on ( ) I f . Additionally, we apply these quadratures to obtain the approximate sum of slowly convergent series ∑ i i s a ∞ = = 0 , where i a ∈ .

. All these quadrature formulas are not based on the integration of an interpolant as so as the Gregory rule, a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order (see [4]).In some natural restrictions on the parameters we construct the only one quadrature formula of the eight order which belongs to the first, second and third family.For functions whose 8th derivative is either always positive or always negative, we use these quadrature formulas to get good two-sided bound on ( ) I f .Additionally, we apply these quadratures to obtain the approximate sum of slowly convergent series

Introduction
We consider the three-parameters families .These quadratures are linear combinations of the quadrature investigated in papers [1]- [3] respectively.The error estimates are calculated in dependence of the parameters α , β , γ and then in some natural restrictions on them these are investigated the quadrature formulas of the 8th order.The desired conclusions are made by means of properties of Peano kernels using substantially well-known error formulas.We construct the only one quadrature formula of the eight order which belongs to the family , , α β γ


, the only one quadrature formula of the eight order too, which belongs to the family , , α β γ


and the only one quadrature formula of the eight order too, which belongs to the family , , α β γ


. Because of the Peano kernels for these quadratures have different signs, for functions whose 8th derivative is either always positive or always negative we use these quadrature formulas to get good bounds on ( ) I f .So, by suitable choice of parameters one can increase quadrature order from two or four respectively to eight.

The Three-Parameters Family of Quadrature Formulas
, ,


given by ( ) . This family generalizes the family Q δ discussed in [1], here it is enough to put For arbitrary α , β , γ the quadrature formula , , α β γ


is of the second order.The error the range of quadrature formula increases.These triples we can write in the form  is of the fourth order, and moreover  In this case we have The six order Peano kernel ( where ( ) ( ) This kernel is a periodic function with period h and on every interval ( )  is symmetrical respect to its midpoint.So, it is enough to define it on the interval , 2 ( ) The kernel ( ) (see Figure 3).From Peano theorem (see [5]) the error , , , 6 6 6 1 14 5 5 10 . Moreover, using Peano theorem we can prove the following: , , , , , , , a b , and , , , , , , . ( ) . 44100 44100 The eight order Peano kernel where ( ) ( ) . This kernel is a periodic function with period h and on every interval ( )  symmetrical with respect to its midpoint.So us for 8 K  , it is enough to define it on the interval , 2 4).This kernel From the Peano theorem (see [5]) we obtain for any function , a b ξ ∈ .

The Three-Parameter Family of Quadrature Formulas
, , We consider the family of quadrature formulas of the form where is the trapezoidal rule, and α , β , γ are para- . We are proved that 5) are of the six order.If we define the error , where . On the Figure 6 we have graphs of the kernels . For any n the kernel ( ) ( ) where now , 3 4 24 , : , and the derivative ( )

The Three-Parameter Family of Quadrature Formulas
, , We consider the family of quadrature formulas of the form where ( ) is the midpoint rule, and α , β , γ are parameters.Parti- cular cases 1 γ = and 0 γ = are investigated in the paper [3] and . We are proved that where 2 π 108864 0, and arccos 1 2 60835 are of the six order.If we define the error 81 2468413440 425279232 161395255 .

Series Estimation
The sum of a series Therefore, if we have a method of estimating the sum of an infinite series, then this method will enable us to estimate the error of the N-term approximation.One way to estimate the sum of the series is to take into consideration the fact that a series can be viewed as an integral over an infinite domain ( ) ( ) for some function  for all n.Therefore, if for a given series, we know an explicitly integrable function ( ) f x with this property, then we can take the value ( ) I f of the integral as an estimate for s.
Theorem 4. We assume that the function f is such that 1) f is either positive and decreasing, or negative and increasing.

2)
( ) If the pair ( ) , β γ is a root of the polynomial


then the range of quadrature increases as before.We can write these pairs in the form is of the six order but we must restrict the interval for γ .The quadrature nodes belongs to interval [ ] presented on the Figure1.

Figure 1
Figure 1.Graphs of

Figure 2 .
Figure 2. Graph of the kernel

Figure 4 .
Figure 4. Graph of the fragment of the kernel
Figure 6.Graphs of the kernels

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For every m ∈  , the quadrature formula is of the eight order.The Peano kernel for the quadrature formula with respect to its midpoint.The quadrature formula (12) has ( )4 1 m n + + nodes.Because of Peano kernels for quadrature formulas Similarly from the formula (10):

(
error of this estimation can be represented as the sum of , then we get a similar inequality, but with the right-hand side instead of the left-hand side, and vice versa.Proof.First, from the inequalities (19) we have: passing with n to ∞ in the inequality (24) we obtain Passing with n to ∞ we obtain sides of the inequality (25).□ ) )