public-notes/Taylor Series Proof.md

#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^2x} 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^2x} f(a)} {2}

Take third derivative of function

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

Create general formula for $n$th element of c

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

Create general formula for function as polynomial

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