Using Taylor expansion to bound the error of approximating e^1 by the 4-term polynomial P3(1). Which inequality correctly bounds the remainder R3?

Using Taylor expansion to bound the error of approximating e^1 by the 4-term polynomial P3(1). Which inequality correctly bounds the remainder R3?

• A. R3 ≤ e/6
• B. R3 ≤ e/24
• C. R3 ≤ e/120
• D. R3 ≤ 1/24

Answer: (B)

Taylor's theorem with remainder in Lagrange form says the error in a degree-3 polynomial approximation is R3 = f^{(4)}(ξ) x^4 / 4! for some ξ between 0 and x. Here f(x) = e^x, so f^{(4)}(ξ) = e^{ξ}. With x = 1, we get R3 = e^{ξ} / 4! = e^{ξ} / 24 for some ξ ∈ (0,1). Since e^{ξ} ≤ e on that interval, R3 ≤ e / 24. This is the bound that matches the remainder term's actual form and magnitude. The other options either drop the exponential factor or misplace the factorial, so they don’t reflect the true remainder.

Taylor's theorem with remainder in Lagrange form says the error in a degree-3 polynomial approximation is R3 = f^{(4)}(ξ) x^4 / 4! for some ξ between 0 and x. Here f(x) = e^x, so f^{(4)}(ξ) = e^{ξ}. With x = 1, we get R3 = e^{ξ} / 4! = e^{ξ} / 24 for some ξ ∈ (0,1). Since e^{ξ} ≤ e on that interval, R3 ≤ e / 24. This is the bound that matches the remainder term's actual form and magnitude. The other options either drop the exponential factor or misplace the factorial, so they don’t reflect the true remainder.