Taylor series
Approximation error and convergence

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

sin x ≈ x − x 3 3 ! + x 5 5 ! − x 7 7 !. {\displaystyle \sin {x}\approx x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}.\!}

The error in this approximation is no more than | x | / 9!. For a full cycle centered at the origin (−π < x < π) the error is less than 0.08215. 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 ln(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.

The error incurred in approximating a function by its n th-degree Taylor polynomial is called the remainder or residual and is denoted by the function R (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. 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 be equal to the value of the function f (x). For example, the function

f (x) = { e − 1 / x 2 if x ≠ 0 0 if x = 0 {\displaystyle f(x)={\begin{cases}e^{-1/x^{2}}&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}}

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 the zero function, so does not equal its Taylor series around the origin. Thus, f (x) is an example of a non-analytic smooth function.

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, the holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functions, which might have singularities, never converge to a value different from the function itself. The complex function e, however, does not approach 0 when z approaches 0 along the imaginary axis, so it is not continuous in the complex plane and its Taylor series is undefined at 0.

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. 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.

A function cannot be written as a Taylor series centred at 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 can be written as a Laurent series.
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