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1&2 finite differences of sequence

Source: France 1986 P4

May 19, 2021
algebraSequencesFinite Differences

Problem Statement

For every sequence {an} (n∈N)\{a_n\}~(n\in\mathbb N) we define the sequences {Δan}\{\Delta a_n\} and {Δ2an}\{\Delta^2a_n\} by the following formulas: \begin{align*}\Delta a_n&=a_{n+1}-a_n,\\\Delta^2a_n&=\Delta a_{n+1}-\Delta a_n.\end{align*}Further, for all n∈Nn\in\mathbb N for which Δan2≠0\Delta a_n^2\ne0, define an′=an−(Δan)2Δ2an.a_n'=a_n-\frac{(\Delta a_n)^2}{\Delta^2a_n}. (a) For which sequences {an}\{a_n\} is the sequence {Δ2an}\{\Delta^2a_n\} constant? (b) Find all sequences {an}\{a_n\}, for which the numbers an′a_n' are defined for all n∈Nn\in\mathbb N and for which the sequence {an′}\{a_n'\} is constant. (c) Assume that the sequence {an}\{a_n\} converges to a=0a=0, and an≠aa_n\ne a for all n∈Nn\in\mathbb N and the sequence {an+1−aan−a}\{\tfrac{a_{n+1}-a}{a_n-a}\} converges to λ≠1\lambda\ne1. i. Prove that λ∈[−1,1)\lambda\in[-1,1). ii. Prove that there exists n0∈Nn_0\in\mathbb N such that for all integers n≥n0n\ge n_0 we have Δ2an≠0\Delta^2a_n\ne0. iii. Let λ≠0\lambda\ne0. For which k∈Zk\in\mathbb Z is the sequence {an′an+k}\{\tfrac{a_n'}{a_{n+k}}\} not convergent? iv. Let λ=0\lambda=0. Prove that the sequences {an′/an}\{a_n'/a_n\} and {an′/an+1}\{a_n'/a_{n+1}\} converge to 00. Find an example of {an}\{a_n\} for which the sequence {an′/an+2}\{a_n'/a_{n+2}\} has a non-zero limit. (d) What happens with part (c) if we remove the condition a=0a=0?
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