Categories: Mathematical analysis | Mathematical series

Taylor series

As the degree of the taylor series rises, it approaches the correct function. This image shows sin(x) and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

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

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

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

Here, n! is the factorial of n and f ⁽ⁿ⁾(a) denotes the nth derivative of f at the point a. The Taylor series is named for mathematician Brook Taylor, who published the power series formulas in 1715. However, James Gregory had been working with these series long before Taylor, and published several Maclaurin series while Taylor was still very young. Taylor was unaware of Gregory's work.

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.

The importance of such a power series representation is at least fourfold. 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 (often by recasting in the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm). Fourth, algebraic operations can often be done much more readily on the power series representation; for instance the simplest proof of Euler's formula uses the Taylor series expansions for sine, cosine, and exponential functions. This result is of fundamental importance in such fields as harmonic analysis.

The function e^-1/x² is not analytic: the Taylor series is 0, although the function is not.

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

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

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

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be compactly written as

$f(\mathbf{x}) = f(\mathbf{a}) + \nabla f(\mathbf{a})^T (\mathbf{x} - \mathbf{a}) + \frac{1}{2} (\mathbf{x} - \mathbf{a})^T \nabla^2 f(\mathbf{a}) (\mathbf{x} - \mathbf{a})$

where $\nabla f(\mathbf{a})$ is the gradient and $\nabla^2 f(\mathbf{a})$ is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes

$f(\mathbf{x}) = \sum_{|\alpha| \ge 0}^{}{\frac{D^{\alpha}f(\mathbf{a})}{\alpha !}(\mathbf{x}-\mathbf{a})^{\alpha}}$

in full analogy to the single variable case.

List of Taylor series of some common functions

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

Exponential function and natural logarithm:

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

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

Geometric series:

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

Binomial theorem:

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

Trigonometric functions:

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

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

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

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

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

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

Hyperbolic functions:

$\sinh \left(x\right) = \sum^{\infin}_{n=0} \frac{1}{(2n+1)!} x^{2n+1}\quad\mbox{ for all } x$

$\cosh \left(x\right) = \sum^{\infin}_{n=0} \frac{1}{(2n)!} x^{2n}\quad\mbox{ for all } x$

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

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

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

Lambert's W function:

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

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

Calculation of Taylor series

Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the Taylor series as-is and generalize the form of the coefficients, or one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the Taylor series by repeatedly applying Integration by parts.

For example, consider the function

$f(x)=\ln{(1+\cos{x})} \,$

which we want a Taylor series about 0.

We have:

$\ln(1+x) = \sum^{\infin}_{n=1} \frac{(-1)^{n+1}}{n} x^n = x - {x^2\over 2}+{x^3 \over 3} - {x^4 \over 4} + \cdots \quad\mbox{ for } \left| x \right| < 1$

$\cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 -{x^2\over 2!}+{x^4\over 4!}+\cdots \quad\mbox{ for all } x$

We can simply substitute the second series into the first. Doing so,

$(1 -{x^2\over 2!}+{x^4\over 4!}+\cdots )-{1\over 2}(1 -{x^2\over 2!}+{x^4\over 4!}+\cdots )^2+{1\over 3}(1 -{x^2\over 2!}+{x^4\over 4!}+\cdots)^3-{1\over 4}(1 -{x^2\over 2!}+{x^4\over 4!}+\cdots)^4+\cdots$

Expanding by using multinomial coefficients gives the requisite Taylor series.

Or, for example, consider

$g(x)={e^x \over \sin{x}}$

We have

$e^x = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + \cdots$

$\sin{x} = x - {x^3 \over 3!} + {x^5 \over 5!} + \cdots$

Then,

${e^x \over \sin{x}} = { 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + \cdots \over x - {x^3 \over 3!} + {x^5 \over 5!} + \cdots}$

Assume the power series is

$c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots = {1 + x + {x^2 \over 2!} + {x^3 \over 3!} + \cdots \over x - {x^3 \over 3!} + {x^5 \over 5!} + \cdots}$

Then

$=\left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots\right)\left(x - {x^3 \over 3!} + {x^5 \over 5!} -+ \cdots\right) = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + \cdots$

$=c_0x - {c_0 \over 3!}x^3 + {c_0\over 5!}x^5 + c_1x^2 - {c_1 \over 3!}x^4 + {c_1\over 5!}x^6 + c_2 x^3 - {c_2 \over 3!} x^5 + {c_2 \over 5!} x^7 + c_3x^4-{c_3\over 3!}x^6 + \cdots$

$= 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + \cdots$