Find the fixed point(s) of $g(x) = (1/2)x^2 + (1/2)x$. Does the fixed point iteration(s) converge(s) to the the fixed point(s) if you start with a close enough approximation?

Then choose $x_0 = (1/2)$, apply the fixed point iteration $5$ times and obtain $x_1$ through $x_5$. To which number does the sequence $\{x_n\}$ from 0 to $+\infty$ seem to converge to? According to you guess, find the corresponding error terms $e_0$ through $e_5$.

I got the fixed points $x = 0$ or $x = 1$ but I don't know what to do afterwards.


As you rightly pointed out, the fixed points are the solution of $x=\frac{1}{2}(x^2+x)$ which are $0,1$. Doing the iteration $x_{n+1}=\frac{1}{2}(x_n^2+x_n)$ starting at $x_0=\frac{1}{2}$ gives a sequence of numbers that seems to converge to 0. [$x_1=\frac{3}{8},x_2=\frac{33}{128},...$]

So here the error terms are just $x_i-0=x_i$, so $e_0=x_0=\frac{1}{2},e_1=x_1=\frac{3}{8},...$


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