Taylor polynomials

9.10 TAYLOR's FORMULA

If we wish to express f(x) as a power series in x - c, we need two things:

(1)    A sequence of polynomials which approximate f(x) near x = c,

(1)

a₀, a₀ + a₁(x - c), ...,a₀ + a₁(x - c) + ... + a_n(x - c)ⁿ,....

(2)     An estimate for the error E_n between f(x) and the nth polynomial,

(2)

f(x) = a₀ + a₁(x - c) + ... + a_n(x - c)ⁿ + E_n.

In the last section the formula

was used to obtain power series approximations. A much more general formula of this type is Taylor's Formula. In Taylor's Formula the nth polynomial P_n(x) is chosen so that its value and first n derivatives agree with

f(x) at x = c.

The tangent line at x = c,

P₁(x) = f(c) + f'(c)(x - c),

has the same value and first derivative as

f(x) at x = c.

A polynomial of degree two with the same value and first two derivatives as f(x) at c is

P₁(x) and P₂(x) are the first and second Taylor polynomials of f(x) (see Figure 9.10.1).

Figure 9.10.1 First and Second Taylor Polynomials

To continue the procedure we need a formula for the nth derivative of a polynomial.

For each m ≤ n, the mth derivative of P(x) at x = c divided by m! is equal to the coefficient a_m,

This lemma shows us how to find a polynomial P(x) whose value and first n derivatives agree with f(x) at x = c. The mth coefficient of P(x) must be

By Lemma 1, P_n(x) is the unique polynomial of degree n whose value and first n derivatives at x = c agree with f(x),

P_n(c) = f(c), P_n'(c) = f'(c),.... P_n⁽ⁿ⁾(c) = f⁽ⁿ⁾(c).

The difference between f(x) and the nth Taylor polynomial is called the nth Taylor remainder,

R_n(x) = f(x) - P_n(x). Thus

f(x) = f(c) + f'(c)(x - c) +(x - c)² + ... + (x - c)ⁿ + R_n(x).