Taylor Series Proof

Math Calculus

Represent function using power series:

$$f(x)=\sum _{n=0}^{\infty }{c}_n(x-a{)}^n$$

Find $c_0$

$$c_0=f(a)$$

Take derivative of function

$$\frac{d}{dx}f(x)=\sum _{n=0}^{\infty }{c}_{n+1}(n+1)(x-a{)}^n$$

Find $c_1$

$$c_1=\frac{d}{dx}f(a)$$

Take second derivative of function

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

Find $c_2$

$$c_2=\frac{\frac{d^2}{d^{2}x}f(a)}{2}$$

Take third derivative of function

$$\frac{d^3}{d^{3}x}f(x)=\sum _{n=0}^{\infty }{c}_{n+3}(n+1)(n+2)(n+3)(x-a{)}^n$$

Find $c_3$

$$c_3=\frac{\frac{d^3}{d^{3}x}f(a)}{6}$$

Create general formula for $n$th element of $c$

$$c_n=\frac{\frac{d^n}{d^{n}x}f(a)}{n!}$$

Create general formula for function as polynomial

$$f(x)=\sum _{n=0}^{\infty }\frac{\frac{d^n}{d^{n}x}f(a)}{n!}(x-a{)}^n$$