On the paradoxical behavior of divergent series

This paper presents some considerations on summation method on infinite divergent series. Irrefutably divergent series don’t have a sum in the traditional logic of the term. However there are extensions, where transformed definitions apportion changed values to the same divergent series and they rarely have agreeable properties. Particularly, at beginning and manipulating these instinctively, it easily comes across nasty paradoxes. In this article for any odd prime p, the divergent series  and on using this representation for further leads to a paradoxical bewildering novel formula which evidently contradicts the basic principles of arithmetic and the definition of a divergent series identical to Ramanujan paradox. Illustrations to support this illogicality result are discussed analytically and demonstrated graphically.