Solve equations using the Newton-Raphson method and other recurrence relations of the form xₙ₊₁ = g(xₙ); understand how such methods can fail

The Newton-Raphson method is a powerful iterative technique for finding approximate solutions to equations of the form f(x) = 0. It uses the formula xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ) to generate a sequence of approximations that, under suitable conditions, converges rapidly to a root. This method is part of the A-Level Edexcel syllabus and is essential for solving equations that cannot be solved algebraically, such as those involving transcendental functions like eˣ + x = 0.

Recurrence relations of the form xₙ₊₁ = g(xₙ) are a broader class of iterative methods, where the next approximation is defined by a function g. The Newton-Raphson method is a special case where g(x) = x - f(x)/f'(x). Understanding how these recurrence relations work, including conditions for convergence and potential failure, is crucial for applying them correctly. These methods are widely used in numerical analysis and have practical applications in engineering, physics, and computer science.

In the context of A-Level Mathematics, you will learn to apply the Newton-Raphson method to find roots of equations, derive recurrence relations, and analyse their convergence. You will also explore scenarios where the method fails, such as when the derivative is zero or the initial guess is poor. This topic builds on your knowledge of differentiation and iteration, and it prepares you for more advanced numerical methods in further study.