# Nonlinear Equations : Esimating Roots by Newton's Method and Rate of Convergence

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19. Show that x = tan^-1(x) has a solution alpha. Find an interval [a,b] containing alphasuch ythat for every x E [a,b] the iteration xn+1 = 1 + tan^-1(xn) n>=0 will converge to alpha. Calculate the first few iterates and estimate the rate of convergence.

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Examine the function

And its graph:

This function is a continuous function, since it is the sum of continuous functions.

Now,

And

Thus there exist at least one ...

#### Solution Summary

Newton's Method is used to estimate a root and the rate of convergence is found. The solution is detailed and well presented. The solution was given a rating of "5" by the student who originally posted the question.