Abstract
The sum of the series of reciprocals of the quadratic polynomials with purely imaginary conjugate roots
Highlights
AND BASIC NOTIONSIn 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
We solve the problem to determine the values of the sum s(b) of the series k2 + b2 k=1 for
Summary
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. For any sequence {ak} of numbers the associated series is defined as the sum. The sequence of partial sums {sn} associated to a series ∑∞k=1 ak is defined for each n as the sum. Because (see [7]) hyperbolic cosine and hyperbolic sine can be defined in terms of the exponential function we get ex + e−x e2x + 1 cosh x = 2 = 2ex , ex − e−x e2x − 1 sinh x = 2 = 2ex , By means of the gamma function, which is defined via a convergent improper integral
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