The summation method and application of power series

Abstract

Summation, or the calculation of the sum of a series, is a fundamental operation in mathematics with widespread applications in various fields. It involves adding together a sequence of numbers or terms to obtain a single value. The ability to accurately determine the sum of a series is crucial for solving mathematical problems, analyzing data, evaluating functions, and making predictions. Power series are an integral part of mathematics and have found extensive applications in various fields. The study of power series summation, specifically the derivation and analysis of summation formulas, plays a crucial role in understanding the behavior and properties of these series. This paper aims to discuss the process of finding the sum function of power series by normalization method, definition method and the power series expansion, thus the power series establishes relations with several series which have common sum functions, and the advantages and limitations of each approach will be discussed, along with practical examples to illustrate the application of the sum function of power series in calculating π and integration.