Binom Katsayılı Geometrik Serilere Farklı Bir Bakış

Abstract

The study of mathematical series has long been a fascinating and essential component of mathematics, providing valuable insights into numerous real-world applications and theoretical concepts. Among the various types of series, the "Geometric Series with Binomial Coefficients" stands out as a particularly intriguing and powerful subject of investigation. A geometric series is a sequence of terms in which each successive term is obtained by multiplying the previous one by a constant factor, known as the common ratio. This classical concept has found extensive applications in fields like finance, physics, engineering, and computer science, making it an indispensable tool for solving a wide array of problems. However, in the context of the "Geometric Series with Binomial Coefficients," we encounter a fascinating twist that elevates the complexity and versatility of the series. Instead of dealing with constant factors as in the traditional geometric series, the coefficients in this new variant are given by the binomial coefficient formula. Binomial coefficients, also known as "n choose k," are fundamental in combinatorial mathematics and represent the number of ways to choose k elements from a set of n elements. This work presents a new approach for the computation to geometric series with binomial coefficients. The geometric series with binomial coefficients is derived from the multiple summations of a geometric series. In this article, several theorems and corollaries are established on the innovative geometric series and its binomial coefficients.