Binomial series
The binomial series is the Taylor series for the function given by , where is an arbitrary complex number. Explicitly,
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Specialized 

and the binomial series is the power series on the right hand side of (1), expressed in terms of the (generalized) binomial coefficients
Special cases
If α is a nonnegative integer n, then the (n + 2)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, which is,
Convergence
Conditions for convergence
Whether (1) converges depends on the values of the complex numbers α and x. More precisely:
 If x < 1, the series converges absolutely for any complex number α.
 If x = 1, the series converges absolutely if and only if either Re(α) > 0 or α = 0.
 If x = 1 and x ≠ −1, the series converges if and only if Re(α) > −1.
 If x = −1, the series converges if and only if either Re(α) > 0 or α = 0.
 If x > 1, the series diverges, unless α is a nonnegative integer (in which case the series is a finite sum).
In particular, if is not a nonnegative integer, the situation at the boundary of the disk of convergence, , is summarized as follows:
 If Re(α) > 0, the series converges absolutely.
 If −1 < Re(α) ≤ 0, the series converges conditionally if x ≠ −1 and diverges if x = −1.
 If Re(α) ≤ −1, the series diverges.
Identities to be used in the proof
The following hold for any complex number α:
Unless is a nonnegative integer (in which case the binomial coefficients vanish as is larger than ), a useful asymptotic relationship for the binomial coefficients is, in Landau notation:
This is essentially equivalent to Euler's definition of the Gamma function:
and implies immediately the coarser bounds
for some positive constants m and M .
Using formula (2), it is easy to prove by induction that
Proof
To prove (i) and (v), apply the ratio test and use formula (2) above to show that whenever is not a nonnegative integer, the radius of convergence is exactly 1. Part (ii) follows from formula (5), by comparison with the pseries
with . To prove (iii), first use formula (3) to obtain
and then use (ii) and formula (5) again to prove convergence of the righthand side when is assumed. On the other hand, the series does not converge if and , again by formula (5). Alternatively, we may observe that for all . Thus, by formula (6), for all . This completes the proof of (iii). Turning to (iv), we use identity (7) above with and in place of , along with formula (4), to obtain
as . Assertion (iv) now follows from the asymptotic behavior of the sequence . (Precisely, certainly converges to if and diverges to if . If , then converges if and only if the sequence converges , which is certainly true if but false if : in the latter case the sequence is dense , due to the fact that diverges and converges to zero).
Summation of the binomial series
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) = αu(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}.
History
The first results concerning binomial series for other than positiveinteger exponents were given by Sir Isaac Newton in the study of areas enclosed under certain curves. John Wallis built upon this work by considering expressions of the form y = (1 − x^{2})^{m} where m is a fraction. He found that (written in modern terms) the successive coefficients c_{k} of (−x^{2})^{k} are to be found by multiplying the preceding coefficient by (as in the case of integer exponents), thereby implicitly giving a formula for these coefficients. He explicitly writes the following instances[1]
The binomial series is therefore sometimes referred to as Newton's binomial theorem. Newton gives no proof and is not explicit about the nature of the series; most likely he verified instances treating the series as (again in modern terminology) formal power series. Later, Niels Henrik Abel discussed the subject in a memoir, treating notably questions of convergence.
References
 The Story of the Binomial Theorem, by J. L. Coolidge, The American Mathematical Monthly 56:3 (1949), pp. 147–157. In fact this source gives all nonconstant terms with a negative sign, which is not correct for the second equation; one must assume this is an error of transcription.