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Does $$\sin (x + \sin (x + \sin (x + \sin (x + ... ) ) ))) $$ converge to any limit? If so, to what?

Recursive plotting appears at times to come to same/similar profiles.

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Define the sequence

$$x_{n+1} = x_0 + \sin x_n$$

For any $x_0$, the function $x_0+\sin x$ is $1$-lipschitzian and maps the interval $[x_0 - 1, x_0 + 1]$ into itself. Therefore it has a unique fixed point in that interval, and the sequence $x_n$ converges to it according to this modification of the Banach fixed point theorem.

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  • $\begingroup$ Here's another way to look at it. Given $x$, let $y$ be the solution to $y=x+\sin y$. Replacing $y$ on the right hand side by $x+\sin y$ we get $$y=x+\sin (x+\sin y).$$ Continuing in this way we get $$y=x+\sin (x+\sin (x+\sin (x +\cdots )))).$$ $\endgroup$ – Oliver Jones Jan 15 '16 at 8:45
  • $\begingroup$ @user55622 That isn't a proof of convergence, though. $\endgroup$ – Jack M Jan 15 '16 at 8:50
  • $\begingroup$ No, of course not. But it's a trick they do with continued fractions. $\endgroup$ – Oliver Jones Jan 15 '16 at 8:51
  • $\begingroup$ @Jack M But is it periodic, what are the time period, series general term and Fourier coefficients ? $\endgroup$ – Narasimham Jan 15 '16 at 9:03
  • $\begingroup$ Can it be plotted for successive nestings to show graphic convergence? $\endgroup$ – Narasimham Feb 3 '16 at 9:15

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