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Abstract
Geometric series plays a vital role in the areas of combinatorics, science, economics, and medicine. This paper presents algorithmic and numerical techniques for computing the summation of multiple series of binomial coefficients and the multiple summations of geometric series in an innovative way and also the relation between the binomial expansions and geometric series. These are the methodological advances which are useful for researchers who are working in science, economics, engineering, computation, and management.
