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 \( b= 1\),
$$
x^{\frac{1}{2}} = 1 + \frac{1}{2}(x-1) + \frac{ \left( \frac{1}{2} \right) \left( \frac{1}{2} - 1 \right) }{2!} (x-1)^2 + \cdots
$$