Convergence, monotonicity, and inequalities of sequences involving continued powers (Q2852510)

Let NEWLINE\[CARRIAGE_RETURNNEWLINET_{n,t}(a) = \underbrace{\sqrt{[t]a- \sqrt{[t]a - \sqrt{[t]a - \dots - \sqrt{[t]a}}}}}_n,CARRIAGE_RETURNNEWLINE\]NEWLINE and NEWLINE\[CARRIAGE_RETURNNEWLINE g_{n,t}(a)=\frac{a-T_{n+1,t}(a)}{a-T_{n,t}(a)}, CARRIAGE_RETURNNEWLINE\]NEWLINE where \(n \in \mathbb{N}\). This formula is defined when either \(a>1\) and \(t>1\), or \(0<a<1\) and \(0<t<1\), or \(a<-1\) and \(t\) odd, or \(-1<a<0\) and \(\tfrac{1}{t}\) odd. The convergence, monotonicity and inequalities of the sequences \(\{T_{n,t}(a)\}\) and \(\{g_{n,t}(a)\}\) are discussed in the paper. The bibliography contains 7 sources.