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Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series. In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper ...
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 is credited with the generalised binomial theorem, valid for any exponent. He discovered Newton's identities , Newton's method , classified cubic plane curves ( polynomials of degree three in two variables ), made substantial contributions to the theory of finite differences , with Newton regarded as "the single most significant ...
In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence written in the form = = ... The generalized binomial theorem gives
Newton would begin his mathematical training as the chosen heir of Isaac Barrow in Cambridge. His aptitude was recognized early and he quickly learned the current theories. By 1664 Newton had made his first important contribution by advancing the binomial theorem, which he had extended to include fractional and negative exponents.
Newton [4] The explication was written to remedy apparent weaknesses in the logarithmic series [ 6 ] [infinite series for log ( 1 + x ) {\displaystyle \log(1+x)} ] , [ 7 ] that had become republished due to Nicolaus Mercator , [ 6 ] [ 8 ] or through the encouragement of Isaac Barrow in 1669, to ascertain the knowing of the prior authorship ...
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:
1665 - Isaac Newton discovers the generalized binomial theorem and develops his version of infinitesimal calculus, 1667 - James Gregory publishes Vera circuli et hyperbolae quadratura, 1668 - Nicholas Mercator publishes Logarithmotechnia, 1668 - James Gregory computes the integral of the secant function,