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:
“History is not what you thought. It is what you can remember. All other history defeats itself.
In Beverly Hills ... they don’t throw their garbage away. They make it into television shows.
Idealism is the despot of thought, just as politics is the despot of will.”
—Mikhail Bakunin (1814–1876)
“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)
“To history therefore I must refer for answer, in which it would be an unhappy passage indeed, which should shew by what fatal indulgence of subordinate views and passions, a contest for an atom had defeated well founded prospects of giving liberty to half the globe.”
—Thomas Jefferson (1743–1826)