A Conceptual doubt on Binomial Theorem Fractional index.

Question

I am coming across this problem on Binomial Theorem Fractional index.

In the book the author states that :–

By actual evolution, we have

$(1+x)^{1/2}=\sqrt{1+x}=1+\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3-...............;$

and by actual division,

$(1-x)^{-2}=\frac{1}{(1-x)^2}=1+2x+3x^2+4x^3+................;$

Here, I don't really get the idea of how one can get an infinite series out of these kinds of terms (terms having fractional index).

I know what "actual division" means and that it gives this infinite series but I have never heard of "actual evolution".

So, I have 2 main questions to ask.

Is there a more convincing way to look at this equation?

And,

What is "actual evolution" and how do you use it?

Answer 1

The first few terms in the binomial theorem are \begin{eqnarray*} (1+x)^n =1+ nx +\frac{n(n-1)}{2} x^2+\frac{n(n-1)(n-2)}{6}x^3+\cdots. \end{eqnarray*} Substitute $n=1/2$ will rapidly give you the equation stated in your question. I guess "evolution" should be "evaluation" & this is a typo?