Taylor Series - Approximation and Convergence

Approximation and Convergence

Pictured on the right is an accurate approximation of sin(x) around the point x = 0. The pink curve is a polynomial of degree seven:

The error in this approximation is no more than |x|9/9!. In particular, for −1 < x < 1, the error is less than 0.000003.

In contrast, also shown is a picture of the natural logarithm function log(1 + x) and some of its Taylor polynomials around a = 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function. This is similar to Runge's phenomenon.

The error incurred in approximating a function by its nth-degree Taylor polynomial is called the remainder or residual and is denoted by the function R_n(x). Taylor's theorem can be used to obtain a bound on the size of the remainder.

In general, Taylor series need not be convergent at all. And in fact the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions. Even if the Taylor series of a function f does converge, its limit need not in general be equal to the value of the function f(x). For example, the function

is infinitely differentiable at x = 0, and has all derivatives zero there. Consequently, the Taylor series of f(x) about x = 0 is identically zero. However, f(x) is not equal to the zero function, and so it is not equal to its Taylor series around the origin.

In real analysis, this example shows that there are infinitely differentiable functions f(x) whose Taylor series are not equal to f(x) even if they converge. By contrast in complex analysis there are no holomorphic functions f(z) whose Taylor series converges to a value different from f(z). The complex function e−z−2 does not approach 0 as z approaches 0 along the imaginary axis, and its Taylor series is thus not defined there.

More generally, every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma (see also Non-analytic smooth function). As a result, the radius of convergence of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.

Some functions cannot be written as Taylor series because they have a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f(x) = e−x−2 can be written as a Laurent series.