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In this paper, three types of three-parameters families of quadrature formulas for the Riemann’s integral on an interval of the real line 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 . Additionally, we apply these quadratures to obtain the approximate sum of slowly convergent series , where .

We consider the three-parameters families

respectively. The error estimates are calculated in dependence of the parameters

We consider family of quadrature formulas

for integral

For arbitrary

If a triple

formula increases. These triples we can write in the form

where

If the pair

quadrature increases as before. We can write these pairs in the form

for

Every quadrature

nodes belongs to interval

and

In this case we have

The six order Peano kernel

kernel is a periodic function with period h and on every interval

its midpoint. So, it is enough to define it on the interval

The kernel

The integral of the six order Peano kernel takes form

(see

From Peano theorem (see [

for any function

Theorem 1. If

if

if

Proof. Assume that

Similarly

because of

The function

formula

The eight order Peano kernel

is a periodic function with period h and on every interval

midpoint. So us for

(see

This kernel

From the Peano theorem (see [

where

We consider the family of quadrature formulas of the form

where

meters. Particular cases

where

where

So, for every

With

The eight order Peano kernel

is a symmetrical function respect to the point

where

and

in the case

if

where

Let

where now

Obviously

Because of Peano kernels for quadrature formulas

Theorem 2. If function

if

if

Proof. Assume that

because of

because of

We consider the family of quadrature formulas of the form

where

cular cases

where

are of the six order. If we define the error

where

So, for every

(see

With

The eight order Peano kernel

where

and

where

and

From the Peano theorem (see [

where

Theorem 3. If function

if

if

Proof. Assume that

because of

because of

The sum of a series

can be approximated by a finite sum

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 conside- ration the fact that a series can be viewed as an integral over an infinite domain

for some function

an explicitly integrable function

Theorem 4. We assume that the function f is such that

1) f is either positive and decreasing, or negative and increasing.

2)

3)

4)

5)

6)

Under this assumptions, if

where

If

Proof. First, from the inequalities (19) we have:

We can rewrite this inequality in an equivalent form:

In this inequality we put:

Because of

than passing with n to

We complete the first part of the proof by adding the term

Let

We rewrite this inequality in an equivalent form:

and put:

because of

We complete the proof by adding the term

We thank the editor and the referee for their comments.