Why does the binomial theorem for negative numbers never terminate

Question

Why does $(a+bx)^n$ never terminate if you use negative numbers or fractions for n?

Surely $(a+bx)^{-2}=\frac{1}{a^2+2abx+b^2x^2}$ and not an infinite series?

Thanks

Answer 1

If $n$ is real, there exist coefficients $a_{k}$ such that $$ (1 + x)^{n} = \sum_{k=0}^{\infty} a_{k} x^{k} $$ for all real $x$ in some interval about $0$. (In fact, there is a formula for $a_{k}$, but that's incidental to the question.) The main points are:

• If $n$ is a non-negative integer, only finitely many $a_{k}$ are non-zero.

• If $n$ is not a non-negative integer, infinitely many $a_{k}$ are non-zero. (Compare Captain Lama's and Erick Wong's comments.)

• While it's true that $$ \frac{1}{(1 + x)^{2}} = \frac{1}{1 + 2x + x^{2}},\quad x \neq -1, $$ the right-hand side is not a power series (though it can be written as a power series, compare hunter's and coffeemath's comments.)