THE CLOSED FORMS OF CONVERGENT INFINITE SERIES. The papers in this category are concerned with the closed forms of convergent infinite series. Some provide extensions to the methods devised by Leonhard Euler in the eighteenth century, while others detail methods of generalisation, new methods of determination, and series dependent upon logarithmic functions. The links below are to PDF versions of the papers. |
Extension of Euler's First Method. - (M10 Version 1.0.0). In 1730 Leonhard Euler published his solution of "The Basel Problem", which also provided closed form solutions to many other convergent infinite series. The method he devised used as its basis the Sine circular function. This paper extends this method to the Cosine and Tangent circular functions so increasing the number of infinite series for which a closed form can be found. |
Extension of Euler's Second Method. - (M11 Version 1.0.0) - Leonhard Euler subsequently published a simplified solution to "The Basel Problem", again based upon the Sine function. This paper extends this second method using the Sine circular function in conjunction with the Hyperbolic Sine function. |
Generalisation Using Fourier Series Expansion. - (M12 Version 1.0.0) - This paper, using Fourier series expansion, enables the generalisation of the determination of the closed forms of three main convergent infinite series, the Zeta, Eta and Xi series. The method thereby reduces the degree of computation required for even series with very high exponents. |
Determination Using Recursive Integration. - (M13 Version 1.0.0) - A new method of evaluation is introduced here for determination of the closed forms of the three main series discussed above. The method uses recursive integration which again enables generalisation. |
Precise Estimation of Series Sums. - (M14 Version 1.0.0) - For those alternating term infinite series for which a closed form is not currently attainable, this paper provides a new numerical method for estimating the series sum, (~ closed form), to any desired degree of precision, using the minimum amount of computation. |
Some Logarithmic Integrals and Associated Infinite Series. - (M15 Version 1.0.0) - This final paper introduces some logarithmic integrals and the infinite series for which they represent the closed form. The series are based upon Sir Isaac Newton's expansion of Log(1 + x). |
Infinite Series Via Differentials of Series Expansions. - (M16 Version 1.0.0) - This paper determines the series sums of some divergent alternating infinite series produced via multiple differentiation of Sir Isaac Newton's expansion of Log(1 + x). The results subsequently enable a discussion of the nature of infinity. |
Pseudo Closed Forms of the Eta and Zeta Functions for All Exponents, Both Odd and Even. - (M17 Version 1.0.0) - This paper determines the closed forms of both the Eta and Zeta infinite series for all values of the exponents, both odd and even. The method used is recursive integration plus a formula developed by Leonhard Euler for the evaluation of infinite series. |
Closed Forms of the Zeta and Eta Infinite Series for all Exponents in the Form of Modified Gamma Functions. - (M18 Version 1.0.0) - This paper determines the closed forms of the Zeta, (and Eta), infinite series for all values of the exponents, both odd and even, in the form of modified Gamma Functions. The method used is Leonhard Euler's Sum-Product formula for the Zeta Function to determine a relationship between Zeta Functions of different exponents which is then easily converted to the form stated above. |
Determination of the Limiting Divergent Infinite Series and a Review of the Divergency of the Harmonic Series.. - (M21 Version 1.0.0) - This paper determines the limiting divergent infinte series to be the Unitary series, (Zeta(0)), and subsequently performs an in depth review of the divergency of the Harmonic series which suggests that this series may not be divergent, but instead convergent with a finite closed form value. |
Determination of the Closed Form of the Harmonic Series.. - (M22 Version 1.0.0) - Assuming convergence, as suggested in M21, this paper determines an approximate closed form value for the Harmonic infinte series. The method adopted is a semi analytic/empirical method based on the belief that the closed form of all Zeta infinite series can be represented by a linear algorithm. |
The Closed Form of the Harmonic Series - A Second Deliberation.. - (M23 Version 1.0.0) - Following on from M22, this paper presents a second deliberation on the closed form of the Harmonic infinte series. The method adopted is also a semi analytic/empirical one as in M22. Also presented is a dissertation on the true nature of the Euler - Mascheroni Constant. |
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