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    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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    https://brainmass.com/math/graphs-and-functions/estimating-roots-newtons-method-rate-convergence-31912

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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.

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