hi Still Learning,

Welcome to the forum.

Mathematicians say that the formula used here converges because successive iterations lead to values of x that get closer and closer to a solution to the equation.

I have made a graph that shows what is going on.

You put the equation

on a graph (blue in my picture)

and also the line

shown red.

You choose an initial guess (x1) and draw a vertical line until it meets the blue curve. This is at the point (x1,y1).

Then you draw horizontally until you meet the red line. This is the point (y1,y1). But rename it (x2,x2)

Now use x2 as the new value to try for x.

Draw a vertical line until it meets the blue curve. This is the point (x2,y2)

Draw horizontally from here to meet the red line at (y2,y2). Rename this (x3,x3).

and so on.

I have shown these movements as a green path. If this path spirals inwards towards a point, then the iteration is converging.

Where the blue curve and the red line cross, is the solution.

Sometimes the iterations do not move inwards towards the solution because the iteration isn't converging. So you have to find the right iterative equation.

Surprisingly, all it depends on is the gradient of the blue curve at the point it crosses the red line.

Bob