Can I get help in this question, I just want to understand it. $$ x_{n+1}=-\sin(x_n)^3-\sin(x_n)^4 , ~~x_0=-0.6 $$
- To find the convergence's order.
- Is the series a special case of fixed point iteration? (if yes find an equation that enables to find the series's root).
- Find an interval on the real line in which the series converges foe every starting point in this interval and prove this formally. What is the value that the series converges to?
I know the definitions...but couldn't manage to connect them to this series..because of sin..
You should know that $|\sin x|\le \min(1,|x|)$ so that $$ |x_{n+1}|\le 2|x_n|^3. $$ All the other answers easily follow from this.
Note that $|x_k|\le 2$, $k>0$, so that the following plot suggests that there are no other fixed points than the obvious one.