Taylor Polynomial Error Explorer

A Taylor polynomial of degree n approximates a function near a center a by matching the value and the first n derivatives at that point. When a is 0 it is called a Maclaurin polynomial.

Higher degree adds more terms, which usually improves the fit close to a. The fit is best near the center and degrades as x moves away.

The exact error is the remainder. Lagrange's form gives a value ξ between a and x where the error equals the next derivative divided by a factorial, times a power of (x - a).

Replacing the derivative with its maximum magnitude M on the interval gives an upper bound. The error shrinks like |x - a| raised to the (n+1) power over (n+1) factorial, so it falls fast as the degree grows and as x approaches a.

Centering the expansion near the point you care about keeps |x - a| small, which keeps the error small. The cos preset centered at a near pi/2 shows this effect.

Some series only converge inside a radius. For 1/(1-x) the radius is 1, so the polynomial diverges for |x| at or above 1 no matter how high the degree. The error panel shows this wall clearly as x approaches the singularity.

The top panel draws f(x) and the polynomial P_n(x). The dots mark their values at x0, and the gap between them is the approximation error.

The bottom panel plots the pointwise error across the domain with a dashed line at the Lagrange bound for x0. The bound always sits at or above the true error, which is why the numeric check reads "actual error ≤ bound".