Binomial series
In mathematics, the binomial series is the Taylor series for the function formula_1 given by formula_2 where formula_3 is an arbitrary complex number and . Explicitly, and the binomial series is the power series on the right-hand side of (), expressed in terms of the (generalized) binomial coefficients Special cases. If is a nonnegative integer , then the th term and all later terms in the series are 0, since each contains a factor ; thus in this case the series is finite and gives the algebraic binomial formula. The following variant holds for arbitrary complex , but is especially useful for  handling negative integer exponents in (): in terms of the multiset coefficient or binomial coefficient. To prove it, substitute in () and apply the binomial coefficient identity Convergence. Conditions for convergence. Whether () converges depends on the values of the complex numbers and . More precisely: In particular, if formula_7 is not a non-negative integer, the situation at the boundary of the disk of convergence, is summarized as follows: Identities to be used in the proof. The following hold for any complex number : Unless formula_7 is a nonnegative integer (in which case the binomial coefficients vanish as formula_10 is larger than formula_7), 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 and . Formula () for the generalized binomial coefficient can be rewritten as Proof. To prove (i) and (v), apply the ratio test and use formula () above to show that whenever formula_7 is not a nonnegative integer, the radius of convergence is exactly 1. Part (ii) follows from formula (), by comparison with the -series with formula_15. To prove (iii), first use formula () to obtain and then use (ii) and  formula () again to prove convergence of the right-hand side when formula_16 is assumed. On the other hand, the series does not converge if formula_17 and formula_18, again by formula (). Alternatively, we may observe that for all formula_19, formula_20. Thus, by formula (), for all formula_21. This completes the proof of (iii). Turning to (iv), we use identity () above with formula_22 and formula_23 in place of formula_7, along with formula (), to obtain as formula_26. Assertion (iv) now follows from the asymptotic behavior of the sequence formula_27. (Precisely, formula_28 certainly converges to formula_29 if formula_30 and diverges to formula_31 if formula_32. If  formula_33, then formula_34 converges if and only if the sequence formula_35 converges formula_36, which is certainly true if formula_37 but false if formula_38: in the latter case the sequence is dense formula_36, due to the fact that formula_40 diverges and formula_41 converges to zero). Summation of the binomial series. The usual argument to compute the sum of the binomial series goes as follows. Differentiating term-wise the binomial series within the disk of convergence and using formula (), one has that the sum of the series is an analytic function solving the ordinary differential equation with initial data . The unique solution  of this problem is the function , which is therefore the sum of the binomial series, at least for . The equality extends to whenever the series converges, as a consequence of Abel's theorem and by continuity of . History. The first results concerning binomial series for other than positive-integer 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 where is a fraction. He found that (written in modern terms) the successive coefficients of 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 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. Later, on 1826 Niels Henrik Abel discussed the subject in a paper published on "Crelle's Journal", treating notably questions of convergence.