I am practising for the test of numerical methods and here I stumbled on the exercise I don't know how to solve:

Show that equation: $x-0,4 \cos(x)=7$ has only one solution $x^*\in\mathbb{R}$ and that given iterative method: $x_n=0,4\cos(x_{n-1}) +7$ converges for every $x_0\in\mathbb{R}$ to the solution $x^*$. Estimate $|x_8-x^*|$, for $x_0=7$ as good as you can.

To show that this equation has exactly one solution is very easy but I completely don't know how to approach this iterative method. I was thinking that maybe if I prove $e_n=x_n-x^*\to 0$ with $n$ approaching $+\infty$ then it will be done, but it's not that easy. If I'm not mistaken, if sequence $x_n$ converges then it converges to $x^*$ but I also don't know how to calculate $\lim_{n\to\infty}x_n$

I will be very grateful for any help.


If you take $\frac {dx_{n+1}}{dx_n}$ and find it to be less than $1$ for all $x_n$ you will have established convergence, because the sequence will be Cauchy. Then in the neighborhood of the root, the error will be multiplied by about this factor at each stage, so that gives the rate of convergence.


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