So I just finished learning the standard binomial theorem and I've just come across the extended (newtons binomial theorem).
As expected I am completely baffled about how it works I do not understand how the following holds :
Suppose $n \in \mathbb{R}$ then for every $x$ satisfying $|x| < 1$, we have $$ (1+x)^{n} = \sum_{r=0}^{\infty} \dbinom{n}r x^r \space \space \space \space ; \space \space r = 0,1,2...$$ and the extended binomial coefficient is given as $$\dbinom{n}r = \dfrac{n!}{(n-r)! \space r!}$$
There are a lot of things that trouble me here first I've read online that factorial are not defined for negative integer values yet somehow we are using them to compute the coefficients in the expansion example:.$$\begin{align*} (b + a)^n &= b^n\left(1 + \frac{a}{b}\right)^n\\ &= b^n \left(\sum_{k = 0}^{n}{n \choose k}\left(\frac{a}{b}\right)^k\right)\\ &= b^n \left(1 + n \left(\frac{a}{b}\right) + \frac{(n)(n-1)}{2}\left(\frac{a}{b}\right)^2 + \cdots\right) \end{align*}$$
How is it that say $-4!$ is undefined yet in the formula we are using it is it then not the same as negative factorial but instead some kind of entirely different method that happens to work?.
I would like some basic intuition on how this binomial theorem actually works if it is possible as the book i am reading just states the formula without any explanation as to how it works if there is no basic explanation maybe a list of topics I need to learn before to understand it would be helpful.
Thanks In advance.
In the extended binomial theorem, the definition of $^nC_r$ is not as simple as it is for the 'vanilla' binomial theorem. If we define $$ n!=n \cdot(n-1) \cdot (n-2)\cdot\dots\cdot3\cdot 2\cdot 1 $$ then the formula you have provided is indeed meaningless, as $n!$ only makes sense when $n$ is a natural number. The simplest way to address this is to define $^nC_r$ slightly differently when $n$ is a real number: $$ \binom{n}{r}=\frac{n(n-1)(n-2)\cdots(n-r+1)}{r!} \, , $$ meaning that the problem of defining things such as $(-4)!$ does not arise. Notice that this definition is equivalent to the one you gave when $n$ is a natural number. By the way, the numerator can be written as $n^{\underline{r}}$ using falling factorials.