binomial series

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, where Re(α) denotes the real part of α.

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 except when α is a non-negative integer, in which case the series is a finite sum.

In particular, if α is not a non-negative integer, the situation at the boundary of the disk of convergence, |x| = 1, 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:

(

α

0

)

=

1

,

{\displaystyle {\alpha \choose 0}\!=1,}

Unless

α

{\displaystyle \alpha }

is a nonnegative integer (in which case the binomial coefficients vanish as

k

{\displaystyle k}

is larger than

α

{\displaystyle \alpha }

), a useful asymptotic relationship for the binomial coefficients is, in Landau notation:

This is essentially equivalent to Euler's definition of the Gamma function:

Γ

(

z

)

=

lim

k

→

∞

Proof

To prove (i) and (v), apply the ratio test and use formula (2) above to show that whenever

α

{\displaystyle \alpha }

is not a nonnegative integer, the radius of convergence is exactly 1. Part (ii) follows from formula (5), by comparison with the p-series

∑

k

=

1

∞

1

k

p

,

{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{p}}},}

with

p

=

1

+

Re

⁡

(

α

)

{\displaystyle p=1+\operatorname {Re} (\alpha )}

. 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

Re

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 |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) = 0 with initial condition u(0) = 1.

The unique solution of this problem is the function u(x) = (1 + x)α. Indeed, multiplying by the integrating factor (1 + x)−α−1 gives

0

=

(

1

+

x

)

−

α

u

′

(

x

)

−

α

(

1

+

x

)

−

α

−

1

u

(

x

Negative binomial series

Closely related is the negative binomial series defined by the MacLaurin series for the function

g

(

x

)

=

(

1

−

x

)

−

α

{\displaystyle g(x)=(1-x)^{-\alpha }}

, where

α

∈

C

{\displaystyle \alpha \in \mathbb {C} }

and

|

x

|

<

1

{\displaystyle |x|<1}

. Explicitly,

1

(

1

−

x

)

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 y = (1 − x2)m where m is a fraction. He found that (written in modern terms) the successive coefficients ck of (−x2)k are to be found by multiplying the preceding coefficient by ⁠m − (k − 1)/k⁠ (as in the case of integer exponents), thereby implicitly giving a formula for these coefficients. He explicitly writes the following instances

(

1

−

x

2

)

1

/

2

=

1

−

x

2

2

−

x

4

8

−

x

6

16

⋯

{\displaystyle (1-x^{2})^{1/2}=1-{\frac {x^{2}}{2}}-{\frac {x^{4}}{8}}-{\frac {x^{6}}{16}}\cdots }

(

1