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t. e. 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.
v. t. e. In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function.
z is a complex variable. The radius of convergence r is a nonnegative real number or such that the series converges if. and diverges if. Some may prefer an alternative definition, as existence is obvious: On the boundary, that is, where | z − a | = r, the behavior of the power series may be complicated, and the series may converge for some ...
Linear approximation. Tangent line at ( a, f ( a )) In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function ). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.
Power series. In mathematics, a power series (in one variable) is an infinite series of the form. where an represents the coefficient of the n th term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions.
Series expansion. An animation showing the cosine function being approximated by successive truncations of its Maclaurin series. In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating a function that cannot be expressed by just elementary ...
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 18th century. The partial sum formed by the first n + 1 terms of a Taylor ...
The derivative of an odd function is even. The integral of an odd function from − A to + A is zero (where A is finite, and the function has no vertical asymptotes between − A and A ). For an odd function that is integrable over a symmetric interval, e.g. [ − A , A ] {\displaystyle [-A,A]}