Let . Consider the sequence of real numbers given by and for . Thus we consider iterations of sine nesting:
(a) Prove that is a convergent sequence.
(b) Determine its leading-order asymptotic behavior for large .
(a) The restriction of to makes it so that . We would like to show that is decreasing and bounded below by zero. We may do it by induction: suppose so that, because for , we have . This is exactly the desired statement. Therefore the sequence is convergent by the monotone convergence theorem, and we may define
Now to get the value of the limit , we write
This equation is solved by , and therefore
independent of the value of .
(b) For large , is small and positive. We postulate an ansatz of algebraic decay,
Now, the recursive definition of gives
and, plugging in the ansatz, we find
The left hand side is expanded according to
Forcing agreement to first order, we have
In order to match powers of , we need and subsequently . Therefore, the leading order asymptotic behavior is
where the notation refers to iterating rather than differentiating or exponentiating.
If we make another ansatz,
we quickly find that it gives us no information. This failure indicates that the subsequent correction to the asymptotic form is not algebraic. In this case, the next term in the expansion turns out to be logarithmic, specifically