Taylor Series

Wolfram Article

$f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n=f(a)+f^{\prime }(a)(x-a)+\frac{1}{2}{f}^{\prime \prime }(a)(x-a{)}^2+\dots$

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

$f(x_0+\Delta x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(x_0)}{n!}(\Delta x{)}^n$

• $\Delta x=x-x_0$

Types

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