Binomial series
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In mathematics, the binomial series generalizes the purely algebraic binomial theorem; it is the series
- <math> (1 + x)^\alpha = \sum_{k=0}^{\infty} \; {\alpha \choose k} \; x^k <math>
in which
- <math> {\alpha \choose k} = \frac{\alpha (\alpha-1) (\alpha-2) \cdots (\alpha-k+1)}{k!}=\frac{(-1)^k}{k!}(-\alpha)_k,<math>
where <math>(\cdot)_n\,<math> is the Pochhammer symbol, and in particular
- <math>{\alpha \choose 0} = 1<math>
because it is the product of no terms at all.
Nota bene: We do not define <math>{\alpha \choose k}<math> to be <math>{\alpha! \over k!(\alpha-k)!}<math> because we do not assume that α is a positive integer.
The results concerning convergence of this series were discovered by Sir Isaac Newton, and therefore one sometimes speaks of 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 2α 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 formula to noninteger 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.nl:Binomiaalreeks