Problem

The Taylor series for a function f looks like this: Suppose that f (x), f ′(x),...

The Taylor series for a function f looks like this:

Suppose that f (x), f ′(x), and f ′′(x) are easily computed. Derive an algorithm like Newton’s method that uses three terms in the Taylor series. The algorithm should take as input an approximation to the root and produce as output a better approximation to the root. Show that the method is cubically convergent.

Hint: Use e_n = e_n₊₁ −h and ignore e² _n₊₁ terms as being negligible.

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Solutions For Problems in Chapter 3.2

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