Binomial series

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In mathematics, the binomial series generalizes the purely algebraic formula of the binomial theorem to complex values of α. It is also 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 α is a complex number and

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

is the (generalized) binomial coefficient (if α is a nonnegative 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): thus, in that case, the summation reduces to the algebraic binomial formula).

As in the case of natural numbers, the following relations hold for any complex number α :

{\alpha \choose 0} = 1,
{\alpha \choose k+1} = {\alpha\choose k}\,\frac{\alpha-k}{k+1}, \qquad\qquad(1)
{\alpha \choose k-1} + {\alpha\choose k} = {\alpha+1 \choose k}. \qquad\qquad(2)

When α is not a natural number a useful asymptotics for the binomial coefficients is, in Landau notation:

{\alpha \choose k} = \frac{(-1)^k} {\Gamma(-\alpha)k^ {1+\alpha} } \,(1+o(1))=\; O\left(\frac {1} {k^{1+Re(\alpha)}}\right), \qquad\qquad(3)

as k\to\infty\;, which is essentially equivalent to Euler's definition of the Gamma function given by the formula

\Gamma(z) = \lim_{k \to \infty} \frac{k! \; k^z}{z \; (z+1)\cdots(z+k)}. \,\qquad

The first results concerning convergence of the binomial series were discovered by Sir Isaac Newton, and it is therefore sometimes referred to as Newton's binomial theorem. Later, Niels Henrik Abel also treated the subject in a memoir.

Whether the series converges depends on the values of the complex numbers α and x. More precisely:

(i) If |x| < 1, the series converges for any complex number α.

(ii) If |x| = 1, the series converges absolutely if and only if either Re(α) > 0 or α = 0.

(iii) If |x| = 1 and x ≠ −1, the series converges if and only if Re(α) > −1.

Sketch of a proof of these facts. (i) in fact, the radius of convergence is exactly 1 whenever α is not a natural number : in this case the binomial coefficients never vanish, and the radius of convergence can be easily computed by the ratio test, starting from formula (1) above. The absolute convergence (ii) follows from formula (3), just by comparison with the test series

\sum_{k=1}^{\infty} \; \frac {1} {k^s }, \qquad

with s=1+Re(α). To prove the convergence result (iii), first write the following algebraic identity, which is a plain consequence of formula (2)

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

then use (ii) and formula (3) again to prove convergence of the right-hand side when Re(α)> −1 is assumed. On the other hand the series does not converge if |x| = 1 and Re(α) ≤ −1, for in that case there holds, for all k

\left|{\alpha \choose k}\; x^k \right| \geq 1. \qquad

The usual argument to compute the sum of the binomial series goes as follows. Differentiating term-wise the binomial series within the convergence disk |x| < 1 and using formula (1), one has that the sum of the series is an analytic function solving the ordinary differential equation (1 + x)u'(x) = α u(x) with initial data u(0) = 1. The unique solution of this problem is the function u(x) = (1 + x)α, which is therefore the sum of the binomial series, at least for |x| < 1. The equality extends to |x| = 1 whenever the series converges, as a consequence of Abel's theorem and by continuity of (1 + x)α.

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