The sum of the telescoping series formed by reciprocals of the cubic polynomials with three different negative integer roots

This paper deals with the sum of a special telescoping series and is a free follow-up to author’s preceding paper. The terms of this series are reciprocals of the cubic polynomial with three different negative integer roots. The main result of the paper is to derive a formula for the sum of this series. This formula uses the limit of the sequence of the partial sums and is expressed by harmonic numbers. After that the main result is verified by some examples using the basic programming language of the computer algebra system Maple 19.


INTRODUCTION
Let us recall some basic terms. The series converges to a limit if and only if the sequence of partial sums = 1 + 2 + ⋯ + converges to , i.e. lim →∞ = .
We 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. The th harmonic number is the sum of the reciprocals of the first natural numbers: (0) being defined as 0. Basic and as well interesting information about harmonic numbers can be found in [1], [2]. For = 1,2, … ,10 we get the following table:  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 can be found in [3]. For example, the series has the general th term, after partial fraction decomposition, in a form After that we arrange the terms of the th partial sum = 1 + 2 + ⋯ + in a form where can be seen what is cancelling. Then we find the limit lim →∞ of the sequence of the partial sums in order to find the sum of the infinite telescoping series. In our case we get

PARTICULAR EXAMPLE
Let us consider a particular exampleto determine the sum of the telescoping series formed by reciprocals of the cubic polynomial with three different specific negative integer roots.
By means of the CAS Maple 19 we get that the partial fraction decomposition of the th term = 1 ( + 2)( + 6)( + 9) has the form The expression in parentheses we now express as the reciprocal of the natural numbers: i.e.

THE SUM OF THE TELESCOPING SERIES FORMED BY RECIPROCALS OF THE CUBIC POLYNOMIAL WITH THREE DIFFERENT NEGATIVE INTEGER ROOTS
This paper is a free follow-up to author's paper [4] dealing with the sum of the telescoping series formed by reciprocals of the quadratic polynomials with two different positive integer roots. Before we derive the main result of this paper, we present two lemmas.
Lemma 1 Let < < are positive natural numbers. Then a fraction 1 ( + )( + )( + ) can be rewritten in the form Since − = ( − ) − ( − ), we can (2) write in the form of the difference Proof. The proof can be done in Maple 19 using the simplify command applied to expressions (2) and (3).

Lemma 2 Let < are positive natural numbers. Then it holds
where ( ) is the th harmonic number.
According to Lemma 1 we can write so the th partial sum is According to Lemma 2 the first partial sum equals ( ) − ( ) and the second one equals ( ) − ( ), so the sum ( , , ) is We have derived the following statement: where > > are negative integers, has the sum where ( ) is the th harmonic number.

NUMERICAL VERIFICATION
We , where = 10 7 , using the basic programming language of the computer algebra system Maple 19, and on the other hand the formula (7) for evaluation the sum ( , , ). We compare 60 pairs of these ways obtained sums ( , , , ) and ( , , ) to verify the formula (7). We use the following simple procedure tsabcneg and the following double repetition statement:  (",a,b,c,t")=",evalf[12](sabct)); print("diff=",evalf[12](abs(sabct-sabc))); end proc: > for i from -1 by -1 to -4 do for j from i-1 by -1 to i-5 do for k from j-1 by -1 to j-5 do tsabcneg(i,j,k,10000000); end do; end do; end do; The approximative values of the sums ( , , ) rounded to 10 decimals obtained by this procedure are written into the table 1:

CONCLUSIONS
We dealt with the sum of the telescoping series formed by reciprocals of the cubic polynomials with three different negative integer roots > > , i.e. with the series where ( ) is the th harmonic number.
We verified this result by computing 60 sums using the computer algebra system Maple 19. This series so belong to special types of infinite series, such as geometric series, which sums are given analytically by means of a simple formula.