Taylor's Theorem

analysis

Definition

Taylor's Theorem

Let be -times differentiable on the closed interval between and , and -times differentiable on the interior of . Then the function can be expressed as the sum of its Taylor polynomial of degree and a remainder term:

where is the error term.

The Remainder Term

The remainder term represents the deviation of the Taylor polynomial from the true function values. A common form is the Lagrange remainder:

Definition

Lagrange Remainder

Let be -times differentiable on an interval containing and . The Lagrange remainder is the error term in the Taylor expansion of of order .

There exists a point between and such that:

This form of the remainder is useful for bounding the error of a Taylor polynomial approximation.

Link to original

If is infinitely often continuously differentiable and , then the Taylor series converges to the function itself. This allows for the approximation of complex functions by polynomials near the expansion point .