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Taylor Series Proof.md

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