Taylor's Theorem with Remainder

Real Analysis · Axiom Academy

LESSON Taylor's Theorem with Remainder Approximating functions with polynomials — and pinning down the exact error so the approximation becomes rigorous. Let f be n+1 times differentiable on an interval containing a and x . Then there is a point between a and x for which f splits exactly into a degree- n polynomial plus a remainder: The polynomial part is the Taylor polynomial T_n(x) ; the last term R_n(x) is the remainder — the error you make by stopping at degree n . The most-used form writes the remainder using the (n+1) -th derivative, evaluated at an unknown point strictly between a and x : It looks exactly like the next term of the Taylor series — but with the derivative evaluated at instead of at a . The case n=0 is the Mean Value Theorem: . We never learn the exact — but we don't need to. As long as we can bound f^ (n+1) on the interval, that bound passes straight through to an error estimate. The payoff of the Lagrange form is a concrete bound on the worst-case error. Replace the unknown by its largest possible size M on the interval: This is a guarantee : no matter where actually lands, the true error can never exceed this envelope. Use it backwards to decide how many terms you need for a target accuracy. 4. Higher-Degree Approximations As the degree n climbs, T_n hugs the true curve over a wider stretch and the remainder R_n collapses toward zero near the center a . Tangent-line approximation. Error O(|x-a|^2) . Adds curvature. Error O(|x-a|^3) .

This is the written version of the interactive lesson above. See the full Real Analysis course.