Computational Technique and Differential Calculus for the Summation of Geometric Series and Binomial Expansions

Nowadays, the growing complexity of mathematical and computational modelling demands the simplicity of mathematical, combinatorial, and numerical equations for solving today’s scientific problems and challenges. Combinatorics involves integers, factorials, binomial coefficients, binomial series, and numerical computation with geometric series for finding solutions to the problems in computing and computational science. This paper presents computational technique and differential calculus for the summation of geometric series and binomial expansions in an innovative way and also theorems and relationship between the binomial expansions and geometric series. These computational approaches refer to the methodological advances which are useful for researchers who are working in computational science. Computational science is a rapidly growing multi-and inter-disciplinary area where science, engineering, computation, mathematics, and collaboration use advance computing capabilities to understand and solve the most complex real life problems.