Inequalities approach in determination of convergence of recurrence sequences

• Main Authors: Albert Adu-Sackey, Francis T. Oduro, Gabriel Obed Fosu
• Format: Article
• Language: English
• Published: Ptolemy Scientific Research Press 2021-02-01
• Series: Open Journal of Mathematical Sciences
• Subjects: monotonic sequence; mathematical induction; boundedness; recursive sequence; convergent sequence.
• Online Access: https://pisrt.org/psr-press/journals/oms-vol-5-2021/inequalities-approach-in-determination-of-convergence-of-recurrence-sequences/

The paper proves convergence for three uniquely defined recursive sequences, namely, arithmetico-geometric sequence, the Newton-Raphson recursive sequence, and the nested/composite recursive sequence. The three main hurdles for this prove processes are boundedness, monotonicity, and convergence. Oftentimes, these processes lie in the predominant use of prove by mathematical induction and also require some bit of creativity and inspiration drawn from the convergence monotone theorem. However, these techniques are not adopted here, rather, as a novelty, extensive use of basic manipulation of inequalities and useful equations are applied in illustrating convergence for these sequences. Moreover, we established a mathematical expression for the limit of the nested recurrence sequence in terms of its leading term which yields favorable results.