The sum of the series of reciprocals of the cubic polynomials with one zero and two different positive integer roots

This contribution is a follow-up to five preceding author’s papers and deals with the sum of the series of reciprocals of the cubic polynomials with one zero and two different positive integer roots. We derive the formula for the sum of these series and verify it by some examples using the basic programming language of the computer algebra system Maple 2020


INTRODUCTION
Let us recall some used basic terms concerning infinite series. We say that a series converges to a limit if and only if the sequence of partial sums = 1 + 2 + ⋯ + converges to , i.e. lim →∞ = .
We than say that the series has a sum and write The sum of the reciprocals of some positive integers is generally the sum of unit fractions (see e.g. [9]). For example the sum of the reciprocals is called Apéry's constant (3), and equals approximately 1.202057,  of the factorials is the transcendental number e ≐ 2.718282.
In contrast to these three convergent series, for example, the following two series of the reciprocals diverge:  the series of the reciprocals of positive integers (the harmonic series)  and the series of the reciprocals of all prime numbers Next, we will use harmonic numbers, where the th harmonic number is the sum of the reciprocals of the first positive integers: Basic and as well interesting information about harmonic numbers can be found in [1], [8].
The values of the harmonic numbers ( ) for = 1,2, … ,10 are stated in the following Table  1. In addition to the harmonic numbers, we will also use telescoping series. The telescoping series is any series where nearly every term cancels with a preceding or following term, so its partial sums eventually only have a fixed number of terms after cancellation. Interesting facts about telescoping series, we can found in [2].

TELESCOPING SERIES EXAMPLE
Let us consider the following example, in which we determine the sum of the telescoping series formed by reciprocals of the cubic polynomial with one zero and two different positive integer roots.
Example 1 Using the th partial sum calculate the sum (0, 5, 8) of the series representing the telescoping series formed by reciprocals of the cubic polynomials with the integer roots 1 = 0, 2 = 5, and 3 = 8.
By means of the result of the Maple 2020 command > convert(1/(k*(k-5)*(k-8)),parfrac); we get the partial fraction decomposition of the th term, where ≠ 5 and ≠ 8, The sum in parentheses we express as the reciprocal of the integers: i.e.
The th partial sum (0, 5, 8) of the series (1) is where (0, 5) = ( 1 1 and Therefore, we have Since for arbitrary real it holds we have  The sum of the series (1) we can also compute by means of Maple 2020 this way:

THREE LEMMAS
This paper is a free follow-up to author's papers [3], [4], [5], [6], [7] dealing with the sum of the telescoping series formed by reciprocals of the cubic polynomials with some positive integer roots. Before we derive the main result of this paper, we present three following lemmas: Lemma 1 Let < be positive integers. Then a fraction 1 ( − )( − ) can be rewritten in the form This expression can also be rewritten as a difference Proof. Can be simply made in Maple using the simplify command applied to expression (2).

Lemma 2 Let < be positive integers. Then it holds
where ( ) is the th harmonic number.
Proof. The sum (0, ) of the infinite series in (3) is the limit of the sequence { (0, )} =1 ∞ of the partial sums In fact Hence, we have

Lemma 3 Let < be positive integers. Then it holds
where ( ) is the th harmonic number.
Proof. The sum ( , ) of the infinite series in (4) is the limit of the sequence { ( , )} =1 ∞ of the partial sums In fact Since < , then

THE MAIN RESULT
Now, let us consider the series formed by reciprocals of the cubic polynomial with one zero and two different positive integer roots < , i.e. the series and let us determine its sum (0, , ).
According to Lemma 1 we can the th term of the series (5) write in the form so the sum (0, , ) of the series (5) is where = 10 7 , using the basic programming language of the computer algebra system Maple 2020, and on the other hand the formula (6) for evaluation the sum (0, , ). We compare one hundred pairs of these ways obtained sums (0, , , ) and (0, , ) to verify the formula (6). We use the following simple procedure ts0abpos and the following double repetition statement: The approximate values of the sums (0, , ) rounded to 10 decimals obtained by this procedure are written into the following Table 2. Computation of 100 pairs of the sums (0, , ) and (0, , , 10 7 ) took over 34 minutes. The absolute errors, i.e. the differences | (0, , ) − (0, , , 10 7 )|, are all only about 5 • 10 −15 . CONCLUSIONS We dealt with the sum of the telescoping series formed by reciprocals of the cubic polynomials with one zero and two different positive integer roots 0 < < . We derived that the sum (0, , ) of this series is given by the formula