On the General Finite Sum of $r^{k}$ and Its Applications Ahmed Ashraf Ali

# On the General Finite Sum of $r^{k}$ and Its ApplicationsAhmed Ashraf Ali

Abstract: This paper analyses the field of mathematics, number theory, around the divergence and convergence of series of varying forms such as P-series, geometric series and even more compound functions including the hyperbolic trigonometric function of sine, cosine and tangent. The purpose is typically to support and embed an additional step forward to great conjectures proposed by early mathematicians around the field, with more precision the paper aims to redefine the Riemann zeta functions by proposing new methods of concluding the finite sum of series, and while the hypotheses of Riemann discusses infinite series it remains possible to, with limits, evaluate the newly defined result to infinity. Mainly, a question that has drawn the attention of early mathematicians was the ordinary power series in which Bernoulli has proposed a theorem of his involving Bernoulli polynomials that was only limited to power series. This paper enhances upon this and suggests with proof new methods and applications. The methods used to investigate this dilemma came from previous research documents published by Bernoulli himself along with Euler and other more recent mathematicians (Details included in the references) and how they were able to approach and influence the problem. Fortunately, the research established has gained extensive success as with reasonable complexity new summations have been defined for the power series along with details on reduction and more composite functions that influence modern arithmetic and number theory. The results propose a new method of looking at functions and analysing their behaviour as series along with comparing them to the infamous zeta function and sequences such as the Fibonacci sequence and the Legendre sequence as well without direct approach. It is with all delight that these results have been previously placed into applications however with yet no direct proposal for publication.

Keywords: Exponential function, Hyperbolic function, Maclaurin Series, Mathematical Induction, Nested series: Power series, P-series, Reduction, Taylor Series, Trigonometric.

Ahmed Ashraf Ali, On the General Finite Sum of $r^{k}$ and Its Applications, International Journal of Advances in Mathematics, Volume 2018, Number 3, Pages 54-65, 2018.