The sum of the series of reciprocals of the quadratic polynomials with double positive integer root

This contribution, which is a follow-up to author's papers [3] and [4], deals with the series of reciprocals of the quadratic polynomials with double positive integer root. The formula for the sum of this kind of series expressed by means of harmonic numbers are derived and verified by several examples evaluated using the basic programming language of the computer algebra system Maple 16. There is stated another formula using generalized harmonic numbers, too. This contribution can be an inspiration for teachers of mathematics who are teaching the topic Infinite series or as a subject matter for work with talented students.


INTRODUCTION AND BASIC NOTIONS
Let us recall the basic terms.For any sequence of numbers the associated series is defined as the sum The sequence of partial sums associated to a series is defined for each as the sum The series converges to a limit if and only if the sequence converges to s, i.e. .We say that the series has a sum and write .
The -th harmonic number is the sum of the reciprocals of the first natural numbers: The generalized harmonic number of order in power is the sum where are harmonic numbers.Every generalized harmonic number of order in power can be written as a function of generalized harmonic number of order in power using formula (see [6]) whence From formula (1), where and , we get the following table 1.
Tab.1: Some harmonic and generalized harmonic numbers

THE SUM OF THE SERIES OF RECIPROCALS OF THE QUADRATIC POLYNOMIALS WITH DOUBLE POSITIVE INTEGER ROOT
We deal with the problem to determine the sum of the series for positive integers , i.The partial sums , i.e. generalized harmonic numbers are also determined by the formula (see [5]) This surprising identity was derived by the contemporary brilliant amateur French mathematician Benoit Cloitre (see [1]).
Tab. 2 Some approximative values of the sums 3.156731 Computation of 16 couples of the sums and took over 16 hours.The relative errors, i.e. the ratios , range between for and for .

CONCLUSIONS
We dealt with the sum of the series of reciprocals of the quadratic polynomials with double positive integer root , i.e. with the series We derived that the sum of this series is given by the formula We verified this main result by computing 16 sums by using the CAS Maple 16.
Two another ways how to calculate the sum is using the value of generalized harmonic number of order in power and the improper integral or the short formula with the value of the generalized harmonic number The series of reciprocals of the quadratic polynomials with double positive integer root so belong to special types of infinite series, such as geometric and telescoping series, which sums are given analytically by means of a formula which can be expressed in closed form.
e. to determine the sum of the series the sum of the series the sum of the series etc.Clearly, we get the formula where is the th partial sum of the series , and also the formula A problem to determine the sum is so called Basel problem.This problem was posed by Pietro Mengoli (1625-1686) in 1644.In 1689 Jacob Bernoulli (1654-1705) proved that the series converges and its sum is less than 2. In 1737 Leonhard Euler (1707-1783) showed his famous result .This sum presents the value of the Riemann zeta function The values of the -th partial sum correspond to the values , so their first ten values are presented in the third row of the table 1.Some another values of the -th sums , computed by CAS Maple 16, are , , , , , whereas the series converges to the number .

) we get Theorem 1 Theorem 2 Example 1
The series where is integer, has the sum Remark In[2] it is stated the equality which can be proved using a geometric sum-type expansion of the denominator and evaluation of the subsequent integrals by means of the integration by parts and L'Hôpital's Rule.Using formula (6) we get The series where is integer, has the sum Evaluate the sum of the series by formula i) (9), ii) (11), and iii) (8) and compare the obtained results.