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 History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

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. In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... A mathematician is a person whose area of study and research is mathematics. ... Brook Taylor (August 18, 1685 - December 29, 1731) was an English mathematician. ... Events September 1 - King Louis XIV of France dies after a reign of 72 years, leaving the throne of his exhausted and indebted country to his great-grandson Louis XV. Regent for the new, five years old monarch is Philippe dOrléans, nephew of Louis XIV. September - First of the... James Gregory (November 1638 – October 1675), was a Scottish mathematician and astronomer. ...

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. In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

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. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. ... In the mathematical subfield of numerical analysis the Clenshaw algorithm is a recursive method to evaluate polynomials in Chebyshev form. ... This article is about the Eulers formula in complex analysis. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

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. In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

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. In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... In mathematics, a Laurent series is an infinite 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. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

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

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

where is the gradient and is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change. ... In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ... The notion of multi-indices simplifies formula used in the calculus of several variables, partial differential equations and the theory of distributions by generalising the concept of an integer index to an array of indices. ...

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: The exponential function is one of the most important functions in mathematics. ... The natural logarithm is the logarithm to the base e, where e is approximately equal to 2. ...

Geometric series: In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

Binomial theorem: In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

Trigonometric functions: In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Hyperbolic functions: In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...

Lambert's W function: In mathematics, Lamberts W function, named after Johann Heinrich Lambert, also called the Omega function, is the inverse function of f(w) = w·ew for complex numbers w; where ew is the exponential function. ...

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. In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... In mathematics, in particular in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here m! denotes the factorial of m). ... The Euler numbers are a sequence En of integers defined by the following Taylor series expansion: (Note that e, the base of the natural logarithm, is also occasionally called Eulers number, as is the Euler characteristic. ...

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. In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

For example, consider the function

which we want a Taylor series about 0.

We have:

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

Expanding by using multinomial coefficients gives the requisite Taylor series. In mathematics, the multinomial formula is an expression of a power of a sum in terms of powers of the addends. ...

Or, for example, consider

We have

Then,

Assume the power series is

Then

Comparing coefficients yields the Taylor series for the function.