Final value theorem in number sequences

Abstract

By definition in sequence number theory, the most primitive entity is the natural number sequence. Numbers are always viewed as members of global sequences. Number sequences can be classified into two primitivity classes, namely the primary sequences and the secondary sequences. Many unsolved number theoretic problems are connected with the search for rigorous proofs of finity or infinity of number sequences. Infinities in most sequences in the primary class have already been proved but unsolved problems occur in the secondary class. With the development of generating functions in the z-domain, the possibilities of applying the final value theorem on number sequences look promising but precaustions must be taken in its applications as there are pitfalls. Breakthroughs in proofs of infinities in Mersenne number and Fermat number sequences are found. However, there are sequences in the secondary category which still do not yield to this method of proof. This note explains why.