A Supremum of a Trig Recurrence

★ 8/10trigonometry Limits Monotonicity Recursion sequences

Problem

A sequence $\{a_n\}$ satisfies $a_1=1$ , $a_2=6$ , and for $n\geqslant 3$ ,

a_n=\begin{cases}a_{n-1}+\sin a_{n-1},&a_{n-1}\geqslant a_{n-2},\\ a_{n-1}+\cos a_{n-1},&a_{n-1}<a_{n-2}.\end{cases}

Determine, with proof, whether $\{a_n\}$ has a supremum, and if so, find it.

Answer

Solution

Difficulty	8/10
Topics	trigonometry, Limits, Monotonicity, Recursion, sequences

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