Some infinite series summations involving linear recurrence relations of order 2 and 3

Abstract

This paper extends known results of second and third order recursive sequences through extensive formulations of properties of the roots of their characteristic equations, some are old but most are new. They are applied to novel studies of $\sum_{n=0}^{\infty}{\frac{a_{mn}}{10^{n+1}}, \ m=1,2,3}$, including their convergence criteria, and applied to many standard sequences, as particular cases of a generic $\left\{a_n\right\}$. The detailed development of the algebra of the pertinent theorems, and their associated lemmas and corollaries, should open up new vistas for interested number theorists with the concluding results on series values.