Proof of finite difference method using Taylor series? - Answers

The finite difference method approximates derivatives using Taylor series expansions. For example, the first derivative ( f'(x) ) can be expressed as ( f'(x) = \frac{f(x+h) - f(x)}{h} + O(h) ), where ( O(h) ) represents the error term. By expanding ( f(x+h) ) using Taylor series, we can isolate and approximate the derivative, demonstrating that the method converges to the true derivative as the step size ( h ) approaches zero. This approach can similarly be applied to higher-order derivatives and different difference schemes.