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:
“The thing that struck me forcefully was the feeling of great age about the place. Standing on that old parade ground, which is now a cricket field, I could feel the dead generations crowding me. Here was the oldest settlement of freedmen in the Western world, no doubt. Men who had thrown off the bands of slavery by their own courage and ingenuity. The courage and daring of the Maroons strike like a purple beam across the history of Jamaica.”
—Zora Neale Hurston (1891–1960)
“There is no example in history of a revolutionary movement involving such gigantic masses being so bloodless.”
—Leon Trotsky (1879–1940)
“All history attests that man has subjected woman to his will, used her as a means to promote his selfish gratification, to minister to his sensual pleasures, to be instrumental in promoting his comfort; but never has he desired to elevate her to that rank she was created to fill. He has done all he could to debase and enslave her mind; and now he looks triumphantly on the ruin he has wrought, and say, the being he has thus deeply injured is his inferior.”
—Sarah M. Grimke (1792–1873)