Negative Binomial Series — Definition, Formula & Examples

The negative binomial series is the power series expansion of

(1 + x)^{-n}

, where

n

is a positive integer or more generally any positive real number. It extends the binomial theorem to negative exponents, producing an infinite series rather than a finite polynomial.

For $|x| < 1$ and $\alpha > 0$ , the expansion $(1 + x)^{-\alpha} = \sum_{k=0}^{\infty} \binom{-\alpha}{k} x^k$ defines the negative binomial series, where the generalized binomial coefficient is $\binom{-\alpha}{k} = \frac{(-\alpha)(-\alpha - 1)\cdots(-\alpha - k + 1)}{k!}$ . The series converges absolutely for $|x| < 1$ .

Key Formula

(1 + x)^{-n} = \sum_{k=0}^{\infty} (-1)^k \binom{n + k - 1}{k} x^k, \quad |x| < 1

Where:

$n$ = A positive integer (or more generally a positive real number) giving the magnitude of the exponent
$x$ = The variable, restricted to |x| < 1 for convergence
$k$ = The summation index running from 0 to infinity

How It Works

To expand

(1 + x)^{-n}

, compute the generalized binomial coefficients

\binom{-n}{k}

for

k = 0, 1, 2, \ldots

and multiply each by

x^k

. These coefficients simplify neatly:

\binom{-n}{k} = (-1)^k \binom{n + k - 1}{k}

. This identity means you can rewrite the series as

(1+x)^{-n} = \sum_{k=0}^{\infty} (-1)^k \binom{n+k-1}{k} x^k

. The alternating signs appear naturally from the negative exponent. You must have

|x| < 1

for the series to converge.

Worked Example

Problem: Expand

(1 + x)^{-2}

as a power series up to the

x^3

term.

Step 1: Apply the formula with

n = 2

. The general term is

(-1)^k \binom{2 + k - 1}{k} x^k = (-1)^k \binom{k+1}{k} x^k = (-1)^k (k+1) x^k

(-1)^k (k+1)\, x^k

Step 2: Write out the first four terms by setting

k = 0, 1, 2, 3

k=0:\; 1, \quad k=1:\; -2x, \quad k=2:\; 3x^2, \quad k=3:\; -4x^3

Step 3: Combine into the partial expansion.

(1+x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \cdots

Answer:

(1+x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \cdots

for

|x| < 1

Why It Matters

Negative binomial expansions appear frequently when integrating rational functions as power series and in generating functions used in combinatorics and probability. In physics and engineering, expanding expressions like

(1 + x)^{-1/2}

is essential for approximations in relativity, optics, and circuit analysis.

Common Mistakes

Mistake: Forgetting the alternating signs and writing all positive coefficients.

Correction: Each term carries a factor of

(-1)^k

. For

(1+x)^{-n}

, the signs alternate starting positive. If you expand

(1 - x)^{-n}

instead, the

(-1)^k

from the coefficient and the

(-1)^k

from

(-x)^k

cancel, giving all positive terms.