Isomorphism Theorems of a Series Sum and the Improper Integral

The discrete and continuous dependencies’ relationship question has been investigated. An algorithm for determining the final and total series sums through the equivalence ratio of the series common term an and the an-model function improper integral mean value within the change unit interval based on the extended integral Cauchy convergence criterion has been developed. Examples of determining for the statistical sum in the Boltzmann distribution, for the first time directly expressed through an-model function. This eliminates the need for calculations to accumulate the sum of the series up to a value that is specified by a certain accuracy of this sum. In addition, it allows in this case to vary the energy variation interval with any given accuracy. The conducted studies allow solving both theoretical and practical problems of physics and materials science, directly using the Boltzmann distribution (energy spectrum) to calculate the entropy, which determines the loss of thermal energy in technological processes.