Nested trig asymptotics

Asked Mar 06 '16 at 13:13

Active Mar 06 '16 at 13:32

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Letting $\ \ \sin^n(x)=\underbrace{\sin\circ \sin\circ\dots\circ \sin(x)}_{n\text{ times}}\ $,

is it true that $\ \ \sin^n(\pi/2)\sim \sqrt{\dfrac{3}{n}}?$

More specifically, is it true that $$\lim_{n\rightarrow\infty}2\sqrt{3 n}-\sum_{x=1}^{n}\sin^x(\pi/2)=2\pi+C$$

where $C\approx \dfrac{1}{2}$?

edited Mar 06 '16 at 13:32

asked Mar 06 '16 at 13:13

martin

1

In the definition of $\sin^n$ it would appear that $n$ is a natural number; how can you use it as an integration variable? – Justpassingby Mar 06 '16 at 13:19
@Justpassingby not sure on standard teerminology - area under discrete curve ... – martin Mar 06 '16 at 13:21
so it's really a $\sum_{k=1}^n \sin^k (\pi/2)$, not $\int_{x=1}^n \sin^x(\pi/2)$? – gt6989b Mar 06 '16 at 13:23
@gt6989b yes, corrected – martin Mar 06 '16 at 13:27
Did you take the problem from here http://jonathan.bergknoff.com/journal/iterated-sine ? – Pierpaolo Vivo Mar 06 '16 at 13:33
@PierpaoloVivo no - just from messing around. - Good find - thanks! :) – martin Mar 06 '16 at 13:35
I note that $$\lim_{n\to\infty}\sin^n(x)=\lim_{n\to\infty}\sin(\sin^n(x))$$ This way, we can see that $\lim_{n\to\infty}\sin^n(x)=0$, regardless of $x$, which seems to fit your first question. – Simply Beautiful Art Mar 07 '16 at 23:34

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