Consider a sequence of x-values:[br][math]\;\;\;[/math]Begin with [math] {\small x_0} [/math], chosen at pleasure.[br][math]\;\;\;[/math]Draw the tangent to the curve at x0:[br] [math]\;\;\;\;\;\; {\small y - f(x_0) = f'(x_0) (x-x_0) }. [/math][br][math]\;\;\;[/math]The x-intercept of this tangent line, is the next number in the sequence:[br] [math]\;\;\;\;\;\; {\small x_1 = x_0+ \frac{- f(x_0)}{ f'(x_0)} }. [/math]

Assuming f(x) has at least one real zero, under what conditions does it converge to a real zero of f(x)?[br][br][br]If I choose x0 blindly, the sequence can be grumpy. How can we help it along?