Question 1

What is the difference between the Taylor series remainder and the Lagrange remainder?

Accepted Answer

They refer to the same concept. The Lagrange remainder is a specific formula (also called the Lagrange form of the remainder) that gives R_n(x) = f^{(n+1)}(c)(x−a)^{n+1}/(n+1)! for some c between a and x. It is the most commonly used way to express the Taylor series remainder.

Question 2

How do you find the value of c in the Taylor remainder formula?

Accepted Answer

You typically do not find the exact value of c. The theorem only guarantees that such a c exists somewhere between a and x. In practice, you find an upper bound for |f^{(n+1)}(c)| over the interval from a to x, which gives you a worst-case error bound without knowing c precisely.

Question 3

When does the Taylor series remainder go to zero?

Accepted Answer

If the remainder R_n(x) → 0 as n → ∞ for a particular value of x, the Taylor series converges to f(x) at that point. This is how you prove a Taylor series actually represents the original function, not just that it converges to something. Functions like e^x, sin(x), and cos(x) have remainders that go to zero for all x.

Taylor Series Remainder