Memo for Taylor series, expanding $f(x)$ at $x=a$.

General form:

\[\begin{equation} f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!}(x - a)^3 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n + \cdots \end{equation}\]

Maclaurin series ($a = 0$):

\[\begin{equation} f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f^{(3)}(a)}{3!}x^3 + \cdots + \frac{f^{(n)}(0)}{n!}x^n + \cdots \end{equation}\]