The Asymptotic Behavior of a Family of Sequences via Tauberian Theorems (original) (raw)

The asymptotic behavior of a family of sequences

Pacific Journal of Mathematics, 1987

A class of sequences defined by nonlinear recurrences involving the greatest integer fuuction is studied, a typical member of the class being a(0) = 1, a(n) = cz([n/2]) + a(ln/3]) + n(~n/6]) for n r 1. For this sequence, it is shown that lim a(n)/n as n + M exists and equals 12/oog 432). More generally, for any sequence defined by a(0) = 1, a(n) = i qa(ln/m,J) fern 2 1, i-l where the r, > 0 and the m, are integers 2 2, the asymptotic behavior of a(n) is detennhed.

EFFECTIVE ASYMPTOTICS FOR SOME NONLINEAR RECURRENCES AND ALMOST DOUBLY-EXPONENTIAL SEQUENCES

We develop a technique to compute asymptotic expansions for recurrent sequences of the form a n+1 = f (an), where f (x) = x − ax α + bx β + o(x β ) as x → 0, for some real numbers α, β, a, and b satisfying a > 0, 1 < α < β. We prove a result which summarizes the present stage of our investigation, generalizing the expansions in [Amer. Math Monthly, Problem E 3034[1984, 58], Solution [1986, 739]]. One can apply our technique, for instance, to obtain the formula:

On certain positive integer sequences

2004

A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.

Analysis of Limiting Ratios of Special Sequences

Mathematics and Statistics, 2022

In this paper, we have determined the limit of ratio of (n+1)th term to the nth term of famous sequences in mathematics like Fibonacci Sequence, Fibonacci – Like Sequence, Pell’s Sequence, Generalized Fibonacci Sequence, Padovan Sequence, Generalized Padovan Sequence, Narayana Sequence, Generalized Narayana Sequence, Generalized Recurrence Relations of Fibonacci – Type sequence, Polygonal Numbers, Catalan Sequence, Cayley numbers, Harmonic Numbers and Partition Numbers. We define this ratio as limiting ratio of the corresponding sequence. Sixteen different classes of special sequences are considered in this paper and we have determined the limiting ratios for each one of them. In particular, we have shown that the limiting ratios of Fibonacci sequence and Fibonacci – Like sequence is the fascinating real number called Golden Ratio which is 1.618 approximately. We have shown that the limiting ratio of Pell’s sequence is a real number called Silver Ratio and the limiting ratios for generalized Fibonacci sequence are metallic ratios. We have also obtained the limiting ratios of Padovan and generalized Padovan sequence. The limiting ratio of Narayana sequence happens to be a number called super Golden Ratio which is 1.4655 approximately. We have shown that the limiting ratios of Generalized Narayana sequence are the numbers known as super Metallic Ratios. We have also shown that the limiting ratio of generalized recurrence relation of Fibonacci type is 2 and that of Polygonal numbers and Harmonic numbers are 1. We have proved that the limiting ratio of the famous Catalan sequence and Cayley numbers are 4. Finally, assuming Rademacher’s Formula, we have shown that the limiting ratio of Partition numbers is the natural logarithmic base e. We have proved fourteen theorems to derive limiting ratios of various well known sequences in this paper. From these limiting ratio values, we can understand the asymptotic behavior of the terms of all these amusing sequences of numbers in mathematics. The limiting ratio values also provide an opportunity to apply in lots of counting and practical problems.

On the dynamics of certain recurrence relations

The Ramanujan Journal, 2007

In previous analyses the remarkable AGM continued fraction of Ramanujandenoted R 1 (a, b)-was proven to converge for almost all complex parameter pairs (a, b). It was conjectured that R 1 diverges if and only if (0 = a = be iφ with cos 2 φ = 1) or (a 2 = b 2 ∈ (−∞, 0)). In the present treatment we resolve this conjecture to the positive, thus establishing the precise convergence domain for R 1 . This is accomplished by analyzing, using various special functions, the dynamics of sequences such as (t n ) satisfying a recurrence t n = (t n−1 + (n − 1)κ n−1 t n−2 )/n, where κ n := a 2 , b 2 as n be even, odd respectively. As a byproduct, we are able to give, in some cases, exact expressions for the n-th convergent to the fraction R 1 , thus establishing some precise convergence rates. It is of interest that this final resolution of convergence depends on rather intricate theorems for complex-matrix products, which theorems evidently being extensible to more general continued fractions.

On the recursive sequence

Computers & Mathematics with Applications, 2008

The aim of this work is to investigate the global stability, periodic nature, oscillation and the boundedness of solutions of the difference equation x n+1 = A k i=l x n−2i−1 B + C k−1 i=l x n−2i , n = 0, 1,. .. where A, B, C are nonnegative real numbers and l, k are nonnegative integers, l < k. We discuss the existence of unbounded solutions under certain conditions when l = 0.

On the Speed of Convergence of the Sequences

2013

The study of any nontrivial convergence of a sequence of real numbers conducts to the problem of finding the limit but also to the problem of the speed of this convergence. This speed of convergence is characterized by the first iterated limit (that supposes the existence of a function of natural variable that tends to zero for n tending to infinity, a function that appears as the first term of a sequences of such functions). A significant characterization of the first iterated limit may be given by a two sided estimate, from which the first iterated limit results immediately.