Taylor Series

#Math
Wolfram Article

$\displaystyle f(x)=\sum_{n = 0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^{n}=f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^{2}+\ldots$

• Read as the Taylor expansion of $\displaystyle f(x)$ in $\displaystyle x$ about $\displaystyle a$
• $\displaystyle f^{(n)}$ is the $\displaystyle n$th derivative of the function we are trying to approximate
• $\displaystyle a$ is the value we are inputting into $\displaystyle f(x)$

$\displaystyle f(x_{0}+\Delta x)=\sum_{n = 0}^{\infty}\frac{f^{(n)}(x_{0})}{n!}(\Delta x)^{n}$

• $\displaystyle \Delta x=x-x_{0}$

Types

• $\displaystyle e^{x}=\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}$
  • Derivation:
    • $\displaystyle e^{x}=\sum_{n = 0}^{\infty} \frac{e^{a}}{n!}(x-a)^{n}$
    • Assume $\displaystyle a=0$
    • $\displaystyle e^{x}=\sum_{n = 0}^{\infty} \frac{e^{0}}{n!}(x-0)^{n}$
    • $\displaystyle e^{x}=\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}$
• $\displaystyle (1+x)^{n}\approx 1+nx, \text{ x small}$
• $\displaystyle \sin x=\sum_{n = 0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}x^{2n+1}$
• $\displaystyle \cos x=\sum_{n = 0}^{\infty} \frac{(-1)^{n}}{(2n)!}x^{2n}$
• $\displaystyle \ln(1-x)=-\sum_{n = 1}^{\infty} \frac{x^{n}}{n}$
• $\displaystyle \ln(1+x)=\sum_{n = 1}^{\infty} (-1)^{n+1} \frac{x^{n}}{n}$