Taylor series

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Taylor series

"Series expansion" redirects here. For other notions of the term, see 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 is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named in honor of English mathematician Brook Taylor. If a = 0 in the Taylor series formula (see below), the series is also called a Maclaurin series, named after Scottish mathematician Colin Maclaurin.

Definition and examples

The Taylor series of a function f that is infinitely differentiable in a neighbourhood of a real (or complex) number a, is the power series

$f(a)+\frac{f'(a)}{1!}(x-a)^1+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots$

which in a more compact form can be written

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

where n! is the factorial of n and f ⁽ⁿ⁾(a) denotes the nth derivative of f at the point a; the zeroth derivative of f is defined to be f itself and

(x ? a) 0

is defined to be 1.

For example, the Taylor series of the exponential function

e x

at

a = 0

(that is, the Maclaurin series) is

$1 + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \qquad = \qquad 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots\ .$

The above expansion obtains because the derivative of

e x

is also

e x

and

e 0

equals 1. This leaves the terms

(x ? 0) n

in the numerator and n! in the denominator for each term in the infinite sum.

The function f need not in general be equal to its Taylor series, but often it is. If the function f is equal to its Taylor series in a neighbourhood of a, it is said to be analytic in this neighborhood. If f is equal to its Taylor series everywhere it is called entire. The exponential function

e x

and the trigonometric functions sine and cosine are examples of such functions. Examples of functions that are not entire include the logarithm, the trigonometric function tangent, and its inverse arctan. For these functions the Taylor series do not even converge if x is far from a.

A Taylor series can be used to calculate the value of an entire function in every point, if the value of the function, and of all of its derivatives, is known at a single point. Uses of the Taylor series for entire functions include:

The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function. These approximations are good if sufficiently many terms are included.
The series representation simplifies many mathematical proofs.

Pictured above are increasingly accurate approximations of sin(x) around the point a = 0. The yellow curve is a polynomial of degree seven:

$\sin\left( x \right) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}.$

The error in this approximation is no more than $\tfrac{|x|^9}{9!}$ . In particular, for

| x | < 1

, the error is less than 0.000003.

Properties

The function e^?1/x² is not analytic at x = 0: the Taylor series is identically 0, although the function is not.

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 said to be analytic in the interval (a ? r, a + r). If this is true for any r then the function is said to be an entire function. 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 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.

Note that there are examples of infinitely 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) = e^?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 everywhere, and its radius of convergence is infinite, even though the function most definitely is not zero everywhere. This particular pathology does not afflict complex-valued functions of a complex variable. Notice that e^?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) = e^?1/x² can be written as a Laurent series.

The Parker-Sochacki method is a recent advance in finding Taylor series which are solutions to differential equations. This algorithm is an extension of the Picard iteration.

List of Taylor series of some common functions

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.

The cosine function.

An 8th degree approximation of the cosine function in the complex plane.

The two above curves put together.

Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments

x

.

Square root:

$\sqrt{x+1} = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)n!^24^n}x^n \quad\mbox{ for } |x|<1$

Exponential function and natural logarithm:

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

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

Geometric series:

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

Binomial theorem:

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

where ${\alpha\choose n} = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}$

Trigonometric functions:

$\sin x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1}\quad = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\mbox{ for all } x$

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

$\tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1}\quad = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots \mbox{ for } \left| x \right| < \frac{\pi}{2}$

where the Bs are Bernoulli numbers.

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

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

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

Hyperbolic functions:

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

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

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

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

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

Lambert's W function:

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

The numbers B_k appearing in the summation 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. Particularly convenient is the use of computer algebra systems to calculate Taylor series.

First example

Consider the function

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

for which we want a Taylor series at 0.

We have for the natural logarithm

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

and for the cosine function

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

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

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

Expanding by using multinomial coefficients gives the required Taylor series. Note that cosine and therefore

f

are even functions, meaning that

f (x) = f ( ? x)

, hence the coefficients of the odd powers

x

,

x 3

,

x 5

,

x 7

and so on have to be zero and don't need to be calculated. The first few terms of the series are

$\ln(1+\cos x)=\ln 2-{x^2\over 4}-{x^4\over 96}-{x^6\over 1440} -{17x^8\over 322560}-{31x^{10}\over 7257600}-\cdots$

The general coefficient can be represented using Faà di Bruno's formula. However, this representation does not seem to be particularly illuminating and is therefore omitted here.

Second example

Suppose we want the Taylor series at 0 of the function

$g(x)=\frac{e^x}{\cos x}\,.$

We have for the exponential function

$e^x = \sum^\infty_{n=0} {x^n\over n!} =1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} +\cdots$

and, as in the first example,

$\cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots$

Assume the power series is

${e^x \over \cos x} = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots$

Then multiplication with the denominator and substitution of the series of the cosine yields

$\begin{align} e^x &= (c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots)\cos x\\ &=\left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4x^4 + \cdots\right)\left(1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots\right)\\ &=c_0 - {c_0 \over 2}x^2 + {c_0 \over 4!}x^4 + c_1x - {c_1 \over 2}x^3 + {c_1 \over 4!}x^5 + c_2x^2 - {c_2 \over 2}x^4 + {c_2 \over 4!}x^6 + c_3x^3 - {c_3 \over 2}x^5 + {c_3 \over 4!}x^7 +\cdots \end{align}$

Collecting the terms up to fourth order yields

$=c_0 + c_1x + \left(c_2 - {c_0 \over 2}\right)x^2 + \left(c_3 - {c_1 \over 2}\right)x^3+\left(c_4+{c_0 \over 4!}-{c_2\over 2}\right)x^4 + \cdots$

Comparing coefficients with the above series of the exponential function yields the desired Taylor series

$\frac{e^x}{\cos x}=1 + x + x^2 + {2x^3 \over 3} + {x^4 \over 2} + \cdots$

Taylor series as definitions

Classically, the above functions are defined by some property that holds for them. For example, the exponential function is defined as the function that is equal to its own derivative. However, in computable analysis, functions must be defined by algorithms rather than properties, so the above Taylor expansions are used as primary definitions rather than derived results. This is also likely to be the case in software implementations of the functions.

Using Taylor series, one may define analytical functions of matrices and operators, such as matrix exponential or matrix logarithm.

Taylor series for several variables

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

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

For example, for a function that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is:

$f(x,y) \,$

$\approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) \,$

$+ \frac{1}{2}\left[ f_{xx}(a,b)(x-a)^2 + 2f_{xy}(a,b)(x-a)(y-b) + f_{yy}(a,b)(y-b)^2 \right].$

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

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

where $\nabla f(\mathbf{a})$ is the gradient and $\nabla^2 f(\mathbf{a})$ is the Hessian matrix (not to be confused with the Laplacian, which sometimes has the same notation). Applying the multi-index notation the Taylor series for several variables becomes

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

in full analogy to the single variable case.

Source-Wikipedia???.

Community Contributions - Articles by goIITians

Taylor series

Definition and examples

Properties

List of Taylor series of some common functions

Calculation of Taylor series

First example

Second example

Taylor series as definitions

Taylor series for several variables