In mathematics, the binomial series is the Taylor series of the function (1 + x)^{ α} at x = 0, where α is a complex number. Explicitly,
and the binomial series is the power series on the right hand side of (1), expressed in terms of the (generalized) binomial coefficients
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If α is a nonnegative integer n, then the (n + 1)th term and all later terms in the series are 0, since each contains a factor (n − n); thus in this case the series is finite and gives the algebraic binomial formula.
The following variant holds for arbitrary complex β, but is especially useful for handling negative integer exponents in (1):
To prove it, substitute x = −z in (1) and apply a binomial coefficient identity.
Whether (1) 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.
The following hold for any complex number α:
When α is not a nonnegative integer, a useful asymptotic relationship for the binomial coefficients is, in Landau notation:
as k → ∞. This is essentially equivalent to Euler's definition of the Gamma function:
and implies immediately the coarser asymptotics
as k → ∞, which is sufficient for our needs. The simple bound (3) may also be obtained by means of elementary inequalities (see the addendum below).
To prove (i), apply the ratio test and use formula (1) above to show that whenever α is not a nonnegative integer, the radius of convergence is exactly 1. The absolute convergence (ii) follows from formula (3), by comparison with
with s = 1 + Re(α). To prove (iii), first use formula (2) to obtain
and then use (ii) and formula (3) again to prove convergence of the righthand side when Re(α) > −1 is assumed. On the other hand, the series does not converge if x = 1 and Re(α) ≤ −1, because in that case, for all k,
The usual argument to compute the sum of the binomial series goes as follows. Differentiating termwise 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}.
The first results concerning convergence of the binomial series were discovered by Sir Isaac Newton, so it is therefore sometimes referred to as Newton's binomial theorem. Later, Niels Henrik Abel treated the subject in a memoir.
In order to keep the whole discussion within elementary methods, one may derive the asymptotics (3) proving the inequality
with
as follows. By the inequality of arithmetic and geometric means
Using the expansion
the latter arithmetic mean writes
To estimate its kth power we then use the inequality
that holds true for any real number r as soon as 1 + r/k ≥ 0. Moreover, we have elementary bounds for the sums:
Thus,
with
proving the claim.
