I was given a problem that asked if it was possible to find the roots of $f(x) = sin(x) + 1$ using Newton's method, which by using an online calculator I had no problem showing. The next part asked what the order of convergence is and why.
Given that $f(r) = 0$, we want to find how many derivatives we need before we have a nonzero result when evaluated at $x = r$. In other words, $f(r) = f'(r) = ... = f^{(p)}(r) = 0$ and $f^{(p+1)}(r) \neq 0$. If this is the case, then the order of convergence is $p$. So in this particular problem $p = 1$, therefore, the order of convergence is 1 (linear).
Is this correct? Are there better ways of finding/understanding orders of convergence?