AI Tutor
If you repeatedly press the cos⁑\cos button on your calculator starting from any number, the display always marches straight toward 0.739085...0.739085....
What does this mysterious path look like when we draw it on a graph, and why do some formulas spiral in while others step cleanly like stairs?

The Two Lines of Iteration

An iteration is the process of repeatedly feeding the output of a function back in as its next input, written as xn+1=g(xn)x_{n+1} = g(x_n).
A fixed point is a value Ξ±\alpha where the input equals the output, meaning g(Ξ±)=Ξ±g(\alpha) = \alpha, which marks the intersection of y=g(x)y = g(x) and the line y=xy = x.
To move from any estimate xnx_n to the next, we go vertically to the curve y=g(x)y = g(x) to evaluate the value, then horizontally to the line y=xy = x to turn that yy-value into our next xx-value.
What happens if the slope of our curve stays positive between 00 and 11?

The Staircase Pattern

When the derivative satisfies 0<gβ€²(x)<10 < g'(x) < 1 near the root, each new value stays on the same side of the root as the previous value, producing a staircase diagram.
Let's trace g(x)=x+2g(x) = \sqrt{x + 2} starting at x0=0x_0 = 0 to find the positive root of x2βˆ’xβˆ’2=0x^2 - x - 2 = 0.
First, x1=g(0)=2β‰ˆ1.414x_1 = g(0) = \sqrt{2} \approx 1.414. Next, x2=g(1.414)=3.414β‰ˆ1.848x_2 = g(1.414) = \sqrt{3.414} \approx 1.848.