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## Taylor and Maclaurin Series

Definition of Taylor series: $$f(x) = f(a) + f'(a)(x-a) + \frac{f”(a)(x-a)^2}{2!} + \cdots +\frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!} + R_n$$ $$R_n = \frac{f^{(n)}(\xi)(x-a)^n}{n!} \text{ where } a \leq \xi \leq x, \quad \text{ ( Lagrangue’s form )}$$ $$R_n = \frac{f^{(n)}(\xi)(x-\xi)^{n-1}(x-a)}{(n-1)!} \text{ where } a \leq \xi \leq x, \quad \text{ ( Cauch’s form )}$$ This result […]