How i can write the Taylor series of the centered difference?

Answer 1

To write the Taylor series for a function ( f(x) ) centered at a point ( a ), you can express it as:

[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots ]

For a centered difference approximation of the derivative, you can utilize the Taylor series expansions of ( f(a+h) ) and ( f(a-h) ) around ( a ). By combining these expansions, you can derive the centered difference formula for the first derivative, which typically takes the form:

[ f'(a) \approx \frac{f(a+h) - f(a-h)}{2h} ]

This approximation will lead to a series representation that includes higher-order terms, which can then be analyzed for accuracy.