Taylor’s Formula

If $f(x) = a_0 + a_1 x + a_2 x^2 + \cdots$,
$$a_n = \frac{f^{(n)}(0)}{n!}$$
with another base point
$$f(x) = f(b) + f’(b)(x-b) + \frac{f’’(b)}{2} (x-b)^2 + \frac{ f^{(3)}(b)}{3!}(x-b)^3 + \cdots$$
For example, $\sqrt{x}$ is not appropriate to expand at $0$ because it’s not differentiable at $0$. so use $b= 1$,
$$x^{\frac{1}{2}} = 1 + \frac{1}{2}(x-1) + \frac{ \left( \frac{1}{2} \right) \left( \frac{1}{2} - 1 \right) }{2!} (x-1)^2 + \cdots$$