Word Count : 250

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Consider the iterative method: [ x_{n+1} = \frac{x_{n-1} f(x_n) - x_n f(x_{n-1})}{f(x_n) - f(x_{n-1})}. ] Let (\alpha) be the simple root of (f(x) = 0), and define the error (e_n = x_n - \alpha). Using the Taylor expansion of (f(x_n)) around (\alpha), since (\alpha) is a simple root: [ f(x_n) = f'(\alpha)e_n + \frac{f''(\alpha)}{2}e_n^2 + O(e_n^3), \quad f(x_{n-1}) = f'(\alpha)e_{n-1} + \frac{f''(\alpha)}{2}e_{n-1}^2 __________ _______ _______ _______ ___ _____ ___ ________.
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