Binomial series ( $(1 + x)^{α}$ )

For

∣ x ∣ < 1

, the binomial series is

(1 + x)^{α} = \sum_{n = 0}^{\infty} (n α) x^{n}, (n α) = \frac{α ( α - 1 ) \dots ( α - n + 1 )}{n !} .

Table of contents
- Square-root binomial series at its endpoints Binomial series
- Negative binomial series Binomial series

Square-root binomial series at its endpoints ( $1 - u = 1 - \sum_{n \geq 1} b_{n} u^{n}, - 1 \leq u \leq 1$ )

Write

a_{n} = (n 2 n) / 4^{n}

,

a_{0} = 1

. The binomial series for

1 - u

has positive numbers

b_{n} = a_{n - 1} - a_{n} = (n 2 n) / [4^{n} (2 n - 1)]

as the magnitudes of its nonconstant coefficients. The product

a_{n} = \prod_{j = 1}^{n} [1 - 1/ (2 j)]

tends to zero, since

ln a_{n} \leq - \frac{1}{2} \sum_{j = 1}^{n} 1/ j \to - \infty

. Thus

\sum b_{n} = 1

by telescoping. The Weierstrass M-test gives uniform convergence and absolute convergence on

[- 1, 1]

. Continuity extends the interior identity to both endpoints. Its radius of convergence is one; outside this interval its terms do not tend to zero.

Negative binomial series ( $\sum_{k \geq 0} (k L + k) z^{k} = (1 - z)^{- L - 1}$ )

For a nonnegative integer

L

and

∣ z ∣ < 1

, this specialization of the binomial series sums the normalizing coefficients of a negative binomial distribution.

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Binomial series by

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The binomial series is a way to express the expansion of a binomial expression raised to a power. Specifically, it provides the expansion of the expression \((a + b)^n\) for any real (or complex) number \(n\).

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