Consider a function $f(x)$ that is defined as$$f(x)=x+\sin(x+\sin(x+\sin(x+\cdots)))$$Now, say we wanted to find the value of $f(1)$ to $5$ decimal places. My question is, how would I go about doing this?
Here is what I have tried:
Note that we have $$f(1)=1+\sin(1+\sin(1+\sin(1+\cdots)))=1+\sin(f(1))$$So we need to find a number such that$$y=1+\sin(y)$$or$$\sin(y)-y+1=0$$however I do not know how to solve it from here. So my question is: How should I go about evaluating $f(1)$ to $5$ decimal places?
For context I have been finding the exact values of functions at certain points such as$$g(e)\text{ for }g(x)=\ln(x+\ln(x+\ln(x+\ln(x+\cdots))))$$and$$h(1)\text{ for }h(x)=\ln(x+1/(x+\ln(x+1/(x+\ln(\cdots)))))$$and decided to be evil and gave myself this to evaluate to only $5$ decimal places.
Newton's method is perfect for finding approximations to roots of equations. Here we can find that $y\approx 1.93456321$ is a solution of $\sin y - y + 1 = 0$.