Taylor Series

1. Taylor Series

$$f(x)=f(x_0)+\frac{f'(x_0)}{1!}(x-x_0)+\frac{f”(x_0)}{2!}(x-x_0)^2+…$$

2.

$$\cos x= 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+…$$

3.

$$\sin x= x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+…$$

4.

$$e^x= 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+…$$

5. Geometric series

$$\frac{a}{1-r}=a+ar+ar^2+ar^3+…$$

6.

$$\ln(1+x)=x-\frac1 2x^2+\frac 1 3x^3-\frac1 4x^4+…$$

7.

$$(1+x)^a= 1+ax+\frac{a(a-1)}{2!}x^2+\frac{a(a-1)(a-2)}{3!}x^3…$$

8.

$$f(x+a)=f(x)+f'(x)a+f”(x)a^2+…=e^{a\frac{d}{dx}}f(x)$$