Taylor's Theorem

analysis

Definition

Taylor's Theorem

Let $f$ be $n$-times differentiable on the closed interval $I$ between $x_0$ and $x$, and $(n+1)$-times differentiable on the interior of $I$. Then the function can be expressed as the sum of its Taylor polynomial of degree $n$ and a remainder term:

$$f(x)=\sum _{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0{)}^k+R_n(x)$$

where $R_n(x)$ is the error term.

The Remainder Term

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

Definition

Lagrange Remainder

Let $f$ be $(n+1)$-times differentiable on an interval $I$ containing $x_0$ and $x$. The Lagrange remainder $R_n(x)$ is the error term in the Taylor expansion of $f$ of order $n$.

There exists a point $\xi$ between $x_0$ and $x$ such that:

$$R_n(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-x_0{)}^{n+1}$$

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

Link to original

If $f$ is infinitely often continuously differentiable and ${\lim }_{n\to \infty }{R}_n(x)=0$, then the Taylor series converges to the function itself. This allows for the approximation of complex functions by polynomials near the expansion point $x_0$.