Created At: [[2024-11-11]] ## Basics ### Taylor Series The Taylor Series is the infinite expansion of an **infinitely differentiable function** $f$ at point $a$: $ {\displaystyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}.} $ - $f^{(n)}$ here denotes the $n^\text{th}$ derivative of the function $f$ ### Maclaurin Series When $a = 0$, the Taylor Series becomes a Maclaurin Series: $ f(0)+\frac{f^{\prime}(0)}{1!} x+\frac{f^{\prime \prime}(0)}{2!} x^2+\frac{f^{\prime \prime \prime}(0)}{3!} x^3+\cdots=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n. $