Taylor Approximation Error: Worked Example and Solution

Problem 1: Bound the error of the degree-3 Maclaurin approximation to e^{0.2}.

Answer: $|R_3|\le e^{0.2}(0.2)^4/4!<0.000082$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

1. Taylor's theorem gives $|R_3|\le e^{0.2}(0.2)^4/4!$ on $[0,0.2]$.
2. The bound is approximately $0.00008143<0.000082$.
3. Therefore the result is $|R_3|\le e^{0.2}(0.2)^4/4!<0.000082$.