Calculus and Computation for Geometric Series with Binomial Coefficients

In today’s world, 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 expansions, geometric series with binomial coefficients, and computation for finding solutions to the problems in computing and engineering science. This paper presents computational techniques and differential and integral calculus for the summation of geometric series with binomial coefficients in an innovative way. Also, it presents theorems, binomial expansions, and relationship between the binomial expansions and geometric series. These computing techniques 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.