PROOF OF TAYLOR'S FORMULA

Let F(x) = R_n(x), G(x) = (x - c)ⁿ⁺¹.

Then

F(x) = f(x) - P_n(x). f(x)

and the nth Taylor polynomial P_n(x) have the same value and first n derivatives at x = c. Therefore

F(c) = F'(c) = F"(c) = ... = F⁽ⁿ⁾(c) = 0.

We also see that

G(c) = G(c) = G"(c) = ... = G⁽ⁿ⁾(c) = 0.

Using the Generalized Mean Value Theorem n + 1 times, we have

for some t₀ strictly between c and x;

for some t₁ strictly between c and t₀;

for some t_n strictly between c and t_n-1.

It follows that

Either

x < t₀ < t₁ < ... < t_n < c or x > t₀ > t₁ > ... > t_n > c,

so t_n is strictly between c and x. The (n + l)st derivatives of F(t) and G(t) are

f⁽ⁿ⁺¹⁾(t) = f⁽ⁿ⁺¹⁾(t) - 0, G⁽ⁿ⁺¹⁾(t) = (n + 1)!

Substituting, we have

and Taylor's Formula follows at once.