[Math] Use Newton’s Method to find all solutions of the equation correct to six decimal places. $\ln x=\frac1{x−3}$

Question

When I use desmos for graphing, there are two intersections and the smaller one is about $0.653$.
According to Newton's method, when I plug $1$ into $x_1$, $x_2$ is $0.6$ which is less than $0.653$, which implies that $x_n$ will be less than $0.653$.
Why is this happening?

Answer 1

Nothing about the definition of the sequence $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}, \quad n \ge 0,$$ suggests that it must be monotone, so I am puzzled as to why you believe that if $x_0 = 1$ is your initial guess and $x_1 < x_0$, then $x_2$ would be less than $x_1$. In fact, this is not true: such a recursively defined sequence can be alternating, or it can be partially alternating and partially monotone. Simply continue to calculate the successive iterates of $1$ and you will see the result numerically converge.