Suppose that Newton’s method is applied to find the solution p = 0 of the equation e^x − 1 − x −1/2x^2 = 0. It is known that, starting with any p0 > 0, the sequence {pn} produced by the Newton’s method is monotonically decreasing (i.e., p0 > p1 > p2 > · · ·) and converges to 0. Prove that {pn} converges to 0 linearly with rate 2/3. (hint: use L’Hospital rule repeatedly. )

Suppose that Newton’s method is applied to find the solution p = 0 of the equation e^x − 1 − x −1/2x^2 = 0. It is known that, starting with any p0 > 0, the sequence {pn} produced by the Newton’s method is monotonically decreasing (i.e., p0 > p1 > p2 > · · ·) and converges to 0. Prove that {pn} converges to 0 linearly with rate 2/3. (hint: use L’Hospital rule repeatedly. )

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 67E

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