In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing . This implies that the function is analytic at every point of the interval (or disk). Definition The Taylor series of a real or complex-valued function that is infinitely differentiable at a real or complex number is the power series : where denotes the factorial of . In the more compact sigma notation, this can be written as : where denotes the th derivative of evaluated at the point . (The derivative of order zero of is defined to be itself and and are both defined to be 1.) When , the series is also called a Maclaurin series. Examples The Taylor series of any polynomial is the polynomial itself. The Maclaurin series of is the geometric series So, by substituting for , the Taylor series of at is By integrating the above Maclaurin series, we find the Maclaurin series of , where denotes the natural logarithm: The corresponding Taylor series of at is and more generally, the corresponding Taylor series of at an arbitrary nonzero point is: The Maclaurin series of the exponential function is The above