History
The first results concerning binomial series for other than positive-integer exponents were given by Sir Isaac Newton in the study of areas enclosed under certain curves. Extending work by John Wallis who calculated such areas for y = (1 − x2)n with n = 0, 1, 2, 3, ... he considered fractional exponents. He found for such exponent m that (in modern formulation) the successive coefficients ck of (−x2)k are to be found by multiplying the preceding coefficient by (as in the case of integer exponents), thereby implicitly giving a formula for these coefficients. He explicitly writes the following instances
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; most likely he verified instances treating the series as (again in modern terminology) formal power series. Later, Niels Henrik Abel treated the subject in a memoir, treating notably questions of convergence.
Read more about this topic: Binomial Series
Famous quotes containing the word history:
“America is the only nation in history which, miraculously, has gone directly from barbarism to degeneration without the usual interval of civilization.”
—Attributed to Georges Clemenceau (1841–1929)
“The only history is a mere question of one’s struggle inside oneself. But that is the joy of it. One need neither discover Americas nor conquer nations, and yet one has as great a work as Columbus or Alexander, to do.”
—D.H. (David Herbert)
“It gives me the greatest pleasure to say, as I do from the bottom of my heart, that never in the history of the country, in any crisis and under any conditions, have our Jewish fellow citizens failed to live up to the highest standards of citizenship and patriotism.”
—William Howard Taft (1857–1930)