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:
“This is the greatest week in the history of the world since the Creation, because as a result of what happened in this week, the world is bigger, infinitely.”
—Richard M. Nixon (19131995)
“... in America ... children are instructed in the virtues of the system they live under, as though history had achieved a happy ending in American civics.”
—Mary McCarthy (19121989)
“This above all makes history useful and desirable: it unfolds before our eyes a glorious record of exemplary actions.”
—Titus Livius (Livy)