# convergence of continued nested function

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.

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.

• 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 $$y=x+\sin (x+\sin (x+\sin (x +\cdots )))).$$ – Oliver Jones Jan 15 '16 at 8:45
• @user55622 That isn't a proof of convergence, though. – Jack M Jan 15 '16 at 8:50
• No, of course not. But it's a trick they do with continued fractions. – Oliver Jones Jan 15 '16 at 8:51
• @Jack M But is it periodic, what are the time period, series general term and Fourier coefficients ? – Narasimham Jan 15 '16 at 9:03
• Can it be plotted for successive nestings to show graphic convergence? – Narasimham Feb 3 '16 at 9:15