What is the formula for the Lagrange error bound when approximating a function using a Taylor polynomial?

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ The Lagrange error bound for a Taylor polynomial approximation is given by the formula: $$ R_n(x) = \frac{M}{(n+1)!} |x - c|^{n+1} $$ where $ M $ is an upper bound on the absolute value of the $(n+1)$th derivative of the function on the interval between $ c $ and $ x $. ‖ ‖ step 2 ⋮ Identify the degree $ n $ of the Taylor polynomial and the point $ c $ around which the polynomial is centered. ‖ ‖ step 3 ⋮ Determine the $(n+1)$th derivative of the function, $ f^{(n+1)}(x) $. ‖ ‖ step 4 ⋮ Find an upper bound $ M $ for $ |f^{(n+1)}(x)| $ on the interval between $ c $ and $ x $. ‖ ‖ step 5 ⋮ Substitute $ M $, $ n $, $ x $, and $ c $ into the Lagrange error bound formula to find the error bound $ R_n(x) $. ‖ ∻Answer∻ ⚹ The Lagrange error bound formula is $ R_n(x) = \frac{M}{(n+1)!} |x - c|^{n+1} $. ⚹ ∻Key Concept∻ ⚹Lagrange Error Bound⚹ ∻Explanation∻ ⚹The Lagrange error bound provides an estimate of the error when approximating a function using a Taylor polynomial. It depends on the $(n+1)$th derivative of the function and the distance between the point of approximation and the center of the Taylor series.⚹◊What is the general form of the Lagrange error bound when approximating a function using a Taylor polynomial?⍭ Generate me a similar question◊

Sia