Taylor Series and Error Bound (NOT a test question)

Question

If f(x) = 1 + 3x - 5x² + 4x³ + 2x⁴, which of the following are true statements?

1. The Taylor polynomial of degree 6 about x = 3 for f(x) is equal to f(x).
2. The remainder term R₅(x) for f(x) is equal to zero.
3. The remainder term R₃(x) for f(x) is equal to zero.

	I only
	I and II only
	I and III only
	I, II, and III

What does the remainder term mean and how do I find it? Please help me and thank you in advance!

Answer 1

Taylor expansion T_n(x) for f(x) is a particular approximation of the function f(x)

by a polynomial of degree n.

The remainder, R_n(x), is then defined by

R_n(x) = f(x) - T_n(x)

If f(x) is a polynomial of degree <= n then T_n(x) is exact, that is

T_n(x) = f(x)

R_n(x) = 0

If f(x) is a polynomial of degree > n then T_n(x) is not exact, that is

T_n(x) ≠ f(x)

R_n(x) ≠ 0

In your case, f(x) is a polynomial of degree 4.

Therefore

T₆(x) = f(x)

R₅(x) = 0

R₃(x) ≠ 0