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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
Download as PDF; Printable version; ... In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence ... The generalized binomial theorem gives
The Gaussian binomial coefficient, written as () or [], is a polynomial in q with integer coefficients, whose value when q is set to a prime power counts the number of subspaces of dimension k in a vector space of dimension n over , a finite field with q elements; i.e. it is the number of points in the finite Grassmannian (,).
Composed in 1669, [4] during the mid-part of that year probably, [5] from ideas Newton had acquired during the period 1665–1666. [4] Newton wrote And whatever the common Analysis performs by Means of Equations of a finite number of Terms (provided that can be done) this new method can always perform the same by means of infinite Equations.
These "generalized binomial coefficients" appear in Newton's generalized binomial theorem. For each k, the polynomial () can be characterized as the unique degree k polynomial p(t) satisfying p(0) = p(1) = ⋯ = p(k − 1) = 0 and p(k) = 1. Its coefficients are expressible in terms of Stirling numbers of the first kind:
Download as PDF; Printable version; ... From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Binomial theorem#Newton's generalized binomial theorem;
The binomial series is therefore sometimes referred to as Newton's binomial theorem. Newton gives no proof and is not explicit about the nature of the series. Later, on 1826 Niels Henrik Abel discussed the subject in a paper published on Crelle's Journal, treating notably questions of convergence. [4]
Newton's introduction of the notions "fluent" and "fluxion" in his 1736 book. A fluent is a time-varying quantity or variable. [1] The term was used by Isaac Newton in his early calculus to describe his form of a function. [2] The concept was introduced by Newton in 1665 and detailed in his mathematical treatise, Method of Fluxions. [3]