Lagrange Error Bound

A Taylor polynomial is a guess built from finitely many terms, so it almost never equals the function exactly. You can't compute the leftover error directly because that requires knowing the true value you're trying to approximate in the first place. The Lagrange error bound gets around this by trapping the error: it guarantees the mistake is no bigger than some number, even though it won't tell you the mistake itself.

The bound depends on the next derivative past where you stopped. If you stop a Taylor polynomial at degree n, the error is controlled by the (n+1)-th derivative somewhere between the center and x. Since you usually can't pin down that exact point, you instead find M, the largest possible size that derivative could take on the interval. The bound is then M times the distance from the center raised to the (n+1) power, divided by (n+1) factorial. Bigger derivatives or farther distances mean looser guarantees; factorial growth in the denominator is what makes these bounds shrink fast as degree increases, which is why adding more terms helps so much.

This bound is the bridge between finite polynomials and infinite series. If the error bound shrinks to zero as n grows, the Taylor polynomials are converging to the actual function value, and the full Taylor series equals the function at that point. If the bound doesn't go to zero, you can't conclude convergence to the function from this test alone, though the series itself might still converge elsewhere. This is exactly the question of when a Taylor series represents its function, which is the payoff of the whole unit.