Binomial series

From Wikipedia, the free encyclopedia

In mathematics, the binomial series generalizes the purely algebraic binomial theorem. It is a special case of a Newton series. The binomial series is the series

$(1 + x)^\alpha = \sum_{k=0}^{\infty} \; {\alpha \choose k} \; x^k$

where

$\alpha \,$ is not necessarily a positive integer

${\alpha \choose k} = \frac{\alpha (\alpha-1) (\alpha-2) \cdots (\alpha-k+1)}{k!}=\frac{(-1)^k}{k!}(-\alpha)_k,$

where $(\bullet)_n\,$ is the Pochhammer symbol, and in particular

${\alpha \choose 0} = 1$

because it is the empty product.

Note:: ${\alpha \choose k}$ is not defined as ${\alpha! \over k!(\alpha-k)!}$ because α is not assumed to be a positive integer.

The results concerning convergence of this series were discovered by Sir Isaac Newton, and it is therefore sometimes referred to as Newton's binomial theorem.

Whether the series converges depends on the values of α and x.

If |x| < 1, the series converges to (1 + x)^α for all α in the real numbers.

If x = 1, the series converges to 2^α for α > −1.

If x = −1, the series converges to 0 for α ≥ 0.

In expositions on calculus, the series is typically constructed by formally deriving a power series for (1 + x)^α, and then proving that the power series converges for some x, namely −1 < x < 1 in this case. Convergence can be proved by the ratio test.

The binomial series generalizes the binomial theorem to non-integer values of α. If α is an integer, then the (α + 1)th term and all later terms in the series are zero, since each one contains a factor equal to (α − α). In that case the summation reduces to the binomial formula.