The sum of the series of reciprocals of the quadratic polynomials with purely imaginary conjugate roots

This contribution, which is a follow-up to author's papers dealing with the sums of the series of reciprocals of quadratic polynomials with different positive integer roots, different negative integer roots, and one negative and one positive integer root, deals with the sum of the series of reciprocals of the quadratic polynomials with purely imaginary conjugate roots. We derive the formula for the sum of these series and verify it by some examples evaluated using the basic programming language of the computer algebra system Maple 16. 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
In the papers [6], [5] and [4] author dealt with the sums of the series of reciprocals of quadratic polynomials with different positive integer roots, different negative integer roots, and one negative and one positive integer root.This contribution is focused on the sum of the series of reciprocals of the quadratic polynomials with purely imaginary conjugate roots.
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 ∑   ∞ =1 is defined for each  as the sum   = �   =  1 +  2 + ⋯ +   .

𝑛 𝑘=1
The series ∑ ∞ =1 k k a converges to a limit  if and only if the sequence {  } converges to s, i.e. lim →∞   = .We say that the series ∑ The sum of the reciprocals of some positive integers is generally the sum of unit fractions.
For example the sum of the reciprocals of the square numbers (the Basel problem) is 6 2 π : The th harmonic number   is the sum of the reciprocals of the first  natural numbers: .
The hyperbolic cotangent is defined as a ratio of hyperbolic cosine and hyperbolic sine is defined so called digamma function (see [1]) Γ() .

THE SUM OF THE SERIES OF RECIPROCALS OF THE QUADRATIC POLYNOMIALS WITH REAL ROOTS
As regards the sum of the series of reciprocals of the quadratic polynomials with different positive integer roots  and,, i.e. with the series in the paper [6] it was derived that the sum (, ) is given by the following formula using the In the paper [5] it was shown that the sum of the series of reciprocals of the quadratic polynomials with different negative integer roots  and,, i.e. with the series is given by the simple formula The sum of the series of reciprocals of the quadratic polynomials with integer roots,  > 0 was derived in the paper [4].This sum is given by the formula

THE ASSIGNMENT OF THE SOLVED PROBLEM
Now, we deal with the problem to determine the sum ) (b s of the series where, i.e. if the quadratic polynomial in the denominator has conjugate purely imaginary , where is the imaginary unit.So, we can write For example, we want to determine the sum, corresponding with the complex conjugates roots , of the series

THE SUM OF THE SERIES OF RECIPROCALS OF THE QUADRATIC POLYNOMIALS WITH PURELY IMAGINARY CONJUGATE ROOTS
Through the Weierstrass product (see [3]) for the hyperbolic sine function , and by the logarithmic derivation of both its sides we get the equality (see web pages [1] and [2]) and after multiplication by the fraction /2 we get the equality

By the relation (1) we get
Theorem 1 The series where 0 > b , has the sum Example 1 Evaluate the sum (4) of the series i) by formula (7), ii) in the CAS Maple 16 by means of the partial sum using the first one million terms.
Compare the obtained results.

Solution i)
The series has by formula (7) from Theorem 1, where, the sum ii) By using the CAS Maple 16 and evaluation the approximate value  10 6 (4) of the sum ) 4 ( s we get the following sequence of commands and results: So we can state that both these obtained results, using ten first decimals, are almost the same and they differ from each other only about 10 −6 .Let us note that in the Maple notation the symbol means the imaginary unit and the function Ψ denotes the digamma function.

RESULTS AND DISCUSSION -NUMERICAL VERIFICATION
We solve the problem to determine the values of the sum ) (b s of the series for.We use on the one hand an approximative direct evaluation of the sum , and obtained by formula (7), by the procedure sumb and by direct computation and rounded to 9 decimals, are written into the following table 1  for.

CONCLUSIONS
We dealt with the sum of the series of reciprocals of the quadratic polynomials with purely imaginary conjugate roots bi ± , i.e. with the series . 1 We verified this main result by computing 10 sums by using the CAS Maple 16.The series of reciprocals of the quadratic polynomials with purely imaginary conjugate roots 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.