is contractive?
Posted: Tue Nov 10, 2015 11:30 am
A sequence \(\{{\alpha_{n}}\}_{n=1}^{\infty}\) is contractive iff there exists a constant \(c\), with \(0<c<1\), such that, for all \(n\in\mathbb{N}\), holds:
\[|a_{n+2}-a_{n+1}|\leqslant c\,|a_{n+1}-a_{n}|\]
Examine if the sequence
\[a_{n}=({\underbrace{\sin\circ\sin\circ\ldots\circ\sin}_{n-{\rm{times}}}})({n})\,,\;n\in\mathbb{N}\,,\]
is contractive.
\[|a_{n+2}-a_{n+1}|\leqslant c\,|a_{n+1}-a_{n}|\]
Examine if the sequence
\[a_{n}=({\underbrace{\sin\circ\sin\circ\ldots\circ\sin}_{n-{\rm{times}}}})({n})\,,\;n\in\mathbb{N}\,,\]
is contractive.