Binomial expansion for any index

Expanding (1 + x)^n and (a + bx)^n when n is negative or fractional — the series form, when it's valid, and combining it with partial fractions.

Pure · Sequences & series

Binomial expansion for any index

Expanding (1 + x)^n and (a + bx)^n when n is negative or fractional — the series form, when it's valid, and combining it with partial fractions.

Why it works

When

n

is a positive integer,

(1 + x)^n

has finitely many terms. When

n

is negative or fractional, the factorial/

\binom{n}{r}

shortcut breaks (you can't take

5!

of a half), but the pattern still works as an infinite series:

(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots

Each coefficient just keeps multiplying by the next "

(n - k)

" and dividing by the next integer — so it never terminates.

Validity. Because it's an infinite sum, it only equals

(1+x)^n

when the terms shrink — that needs

|x| < 1

. Always state this condition.

For $(a + bx)^n$ , factor out $a^n$ first — the series above only starts "

1 + \dots

", so you must make the bracket begin with

1

(a + bx)^n = a^n\left(1 + \tfrac{b}{a}x\right)^n,

which is valid when

\left|\tfrac{b}{a}x\right| < 1

, i.e.

|x| < \tfrac{a}{|b|}

.

With partial fractions. A rational function can be split into partial fractions, each of the form

\dfrac{A}{(1 + cx)}=A(1+cx)^{-1}

, and each expanded as a binomial series — then add the series term by term. The overall expansion is valid on the strictest of the individual ranges.