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The **binomial series** is the Taylor series for the function given by , where is an arbitrary 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

## Contents

## Special cases[edit]

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[edit]

### Conditions for convergence[edit]

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 non-negative integer (in which case the series is a finite sum).

In particular, if is not a non-negative 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[edit]

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[edit]

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 p-series

with . To prove (iii), first use formula (3) to obtain

and then use (ii) and formula (5) again to prove convergence of the right-hand 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[edit]

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*) = *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[edit]

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 *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.^{[citation needed]} Later, Niels Henrik Abel discussed the subject in a memoir, treating notably questions of convergence.

## See also[edit]

## References[edit]

**^**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 non-constant terms with a negative sign, which is not correct for the second equation; one must assume this is an error of transcription.