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Taylor series

In mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval (a − r, a + r) is the power series

<math>

\sum_{n=0}^{\infin} \frac{f^{(n)}(a)}{n!} (x-a)^{n}. <math>

Here, n! is the factorial of n and f ⁽ⁿ⁾(a) denotes the nth derivative of f at the point a. If this series converges for every x in the interval (a − r, a + r) and the sum is equal to f(x), then the function f(x) is called analytic. To check whether the series converges towards f(x), one normally uses estimates for the remainder term of Taylor's theorem. A function is analytic if and only if it can be represented as a power series; the coefficients in that power series are then necessarily the ones given in the above Taylor series formula.

If a = 0, the series is also called a Maclaurin series.

The importance of such a power series representation is threefold. First, differentiation and integration of power series can be performed term by term and is hence particularly easy. Second, an analytic function can be uniquely extended to a holomorphic function defined on an open disk in the complex plane, which makes the whole machinery of complex analysis available. Third, the (truncated) series can be used to compute function values approximately.

Note that there are examples of infinitely often differentiable functions f(x) whose Taylor series converge, but are not equal to f(x). For instance, for the function defined piecewise by saying that f(x) = exp(−1/x²) if x ≠ 0 and f(0) = 0, all the derivatives are zero at x = 0, so the Taylor series of f(x) is zero, and its radius of convergence is infinite, even though the function most definitely is not zero. This particular pathology does not afflict complex-valued functions of a complex variable. Notice that exp(−1/z²) does not approach 0 as z approaches 0 along the imaginary axis.

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) = exp(−1/x²) can be written as a Laurent series.

The is a recent advance in finding Taylor series which are solutions to differential equations. This theorem is an expansion on the Picard iteration.

List of Taylor series

Several important Taylor series expansions follow. All these expansions are also valid for complex arguments x.

Exponential function and natural logarithm:

<math>e^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!}\quad\mbox{ for all } x<math>

<math>\ln(1+x) = \sum^{\infin}_{n=1} \frac{(-1)^{n+1}}n x^n\quad\mbox{ for } \left| x \right| < 1<math>

Geometric series:

<math>\frac{1}{1-x} = \sum^{\infin}_{n=0} x^n\quad\mbox{ for } \left| x \right| < 1<math>

Binomial theorem:

<math>(1+x)^\alpha = \sum^{\infin}_{n=0} {\alpha \choose n} x^n\quad\mbox{ for all } \left| x \right| < 1\quad\mbox{ and all complex } \alpha<math>

Trigonometric functions:

<math>\sin x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1}\quad\mbox{ for all } x<math>

<math>\cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n}\quad\mbox{ for all } x<math>

<math>\tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1}\quad\mbox{ for } \left| x \right| < \frac{\pi}{2}<math>

<math>\sec x = \sum^{\infin}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n}\quad\mbox{ for } \left| x \right| < \frac{\pi}{2}<math>

<math>\arcsin x = \sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\mbox{ for } \left| x \right| < 1<math>

<math>\arctan x = \sum^{\infin}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1}\quad\mbox{ for } \left| x \right| < 1<math>

Hyperbolic functions:

<math>\sinh x = \sum^{\infin}_{n=0} \frac{1}{(2n+1)!} x^{2n+1}\quad\mbox{ for all } x<math>

<math>\cosh x = \sum^{\infin}_{n=0} \frac{1}{(2n)!} x^{2n}\quad\mbox{ for all } x<math>

<math>\tanh x = \sum^{\infin}_{n=1} \frac{B_{2n} 4^n (4^n-1)}{(2n)!} x^{2n-1}\quad\mbox{ for } \left| x \right| < \frac{\pi}{2}<math>

<math>\sinh^{-1} x = \sum^{\infin}_{n=0} \frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\mbox{ for } \left| x \right| < 1<math>

<math>\tanh^{-1} x = \sum^{\infin}_{n=0} \frac{1}{2n+1} x^{2n+1}\quad\mbox{ for } \left| x \right| < 1<math>

Lambert's W function:

<math>W_0(x) = \sum^{\infin}_{n=1} \frac{(-n)^{n-1}}{n!} x^n\quad\mbox{ for } \left| x \right| < \frac{1}{e}<math>

The numbers B_k appearing in the expansions of tan(x) and tanh(x) are the Bernoulli numbers. Tte binomial expansion uses binomial coefficients. The E_k in the expansion of sec(x) are Euler numbers.

Multiple dimensions

The Taylor series may be generalised to functions of more than one variable with

<math>

\sum_{n_1=0}^{\infin} \cdots \sum_{n_d=0}^{\infin} \frac{\partial^{n_1}}{\partial x^{n_1}} \cdots \frac{\partial^{n_d}}{\partial x^{n_d}} \frac{f(a_1,\cdots,a_d)}{n_1!\cdots n_d!} (x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d} <math>

History

The Taylor series is named for mathematician Brook Taylor, who first published the power series formulas in 1715.