Use Newton’s method to evaluate $\lim _{k→∞} x_k$, the sequence satisfies $x_{k+1} = x_k - {(−x^ 2 _k + 11x_k − 10)\over −2x_k + 11}$ for $x_0 = 6$.

Question

Use Newton’s method for locating roots of equations to evaluate $\lim _{k→∞} x_k$, where the sequence satisfies $x_{k+1} = x_k - {(−x^ 2 _k + 11x_k − 10)\over −2x_k + 11}$ for $k ≥$ 0 and $x_0 = 6$.

I know that I can just do the iteration a few times and approximate the root, but is that what I need to do in this problem? Is there a better way to do this? Any solutions/hints are greatly appreciated.

Answer 1

This is what Newton's method for finding roots would look like, if you were trying to find a root of $f(x)=-x^2+11x-10$. Since this function's graph is a downward-opening parabola with central axis at $x=11/2$, then when you start at $x_0=6$ you will end up finding the rightmost of the two roots.

So find those two roots, and the answer is the larger of them.