Ramanujan Summation, Zeta Function and Divergent Series

This essay explores the interrelationship between Ramanujan summation, the Riemann zeta function, and divergent series. the connection between Ramanujan summation, the Riemann zeta function, and divergent series will be explored. My objective is to illustrate how Ramanujan summation can be interpreted in a way to assign values to divergent series, thus extending classical convergence and gaining better understanding of infinite sums. Originally going back to the development of convergent series the Riemann zeta function is a fundamental element in analytic number theory and produces an architecture for understanding how prime numbers are distributed. Finally, I test the accuracy of the Ramanujan series and the analytical equation by presenting the special divergent series from the sum of all natural numbers. It may not be an intuitive outcome from a more elementary viewpoint, but it has been obtained in the overseas of some complicated mathematics and appears in the application of theory physics, for example; cable theory and Casimir effect computations involving infinite series sums that naturally emerge in quantum field theory.