I suppose my question is about limits and infinities. I have been considering the expansion of $(a+bx)^n$ for negative and fractional $n$, and my text book says to always factorise out the $a^n$ like so:
$(a+bx)^n =a^n(1+\frac{b}{a}x)^n= a^n[1+n\frac{b}{a}x+\frac{n(n-1)}{2!}(\frac{b}{a}x)^2+...]$
And then the expansion is valid for $|\frac{bx}{a}|<1$
My question is, why is the $a$ term involved in thevalidity requirement? Why is it incorrect to simply expand the binomial as
$(a+bx)^n = a^n+a^{n-1}n(bx)+\frac{a^{n-2}n(n-1)}{2!}(bx)^2+...]$
and say it is valid for $|bx|<1$?
This is first of all a matter of book-keeping. The $(1+bx/a)^n$ is considered a geometric series $$ (1+y)^n=\sum_{i\ge 0} \binom{n}{i}y^i $$ where $y=bx/a$ is the free variable, and where the radius of convergence is known to be $|y|<1$. So instead of remembering a lot of different subcases of how to approach this, only a single criterion (parameter) comes into play.