Expansion of Elemetnary Functions into a Power Series Using Algebraic Methods

Abstract

As is well known, power series-based methods are one of the pillars of mathematical analysis. For analytic functions, the power series can be easily obtained from Taylor's theorem in the form of Maclaurin series by calculating the derivatives of these functions at x=0. At the same time, it would be useful to be able to expand functions into series without using the concept of a derivative. Such expansions, in particular, would be helpful in calculating limits, which are fundamental to mathematical analysis. In the work of L. P. Myronenko and O. A. Rubtsova (2013), the idea was proposed that the properties of elementary functions can be used for series expansion, and the first terms of their power series expansions were obtained. In the work of V. I. Romanenko and A. V. Romanenko (2024), a generalization of this approach allowed for obtaining the series terms for sine, cosine, and exponential functions in a general form. In the article presented to the reader, the main results of the cited works are provided in a simplified form, supplemented by deriving a formula for the power series expansion of the tangent function. The presentation is structured in such a way that the core idea and the mathematical derivations are accessible to readers with knowledge of the school curriculum. The possibility of an algebraic approach to function series expansion might explain how the Indian mathematician Madhava (14th–15th century) managed to derive the first few terms of the series expansion for sine, cosine, and arctangent long before function analyses methods in the modern form.