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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.
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.
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending ...
Polynomial expansion. In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by ...
In the case of a smooth function, the n th-order approximation is a polynomial of degree n, which is obtained by truncating the Taylor series to this degree. The formal usage of order of approximation corresponds to the omission of some terms of the series used in the expansion. This affects accuracy.
Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula. which using factorial notation can be compactly expressed as.
Trinomial expansion. In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by. where n is a nonnegative integer and the sum is taken over all combinations of nonnegative indices i, j, and k such that i + j + k = n. [1] The trinomial coefficients are given by.
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 called the center of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions.