Need help understanding the binomial series

1. The problem statement, all variables and given/known data
My math textbook is currently on the Binomial Series now, after completing the Binomial Theorem (no problems with that one). I believe most of my trouble comes from the book’s rather glancing explanation of it, only giving examples of the form ##(1 +/- kx)##. Now have encountered this question:

Q
Obtain the first three terms, in ascending powers of x, of the expansion of ##(8+3x)^\frac{2}{3}##, stating the set of values for which this expansion is valid.

2. Relevant equations

The binomial series is defined as :

##(1+x)^n = 1 + nx + \frac{n(n+1)}{1*2}x^2 + \frac{n(n+1)(n-2)}{1*2*3}x^3 + … ##, provided that |x| < 1.

Expansion of ##(1+kx)^n## is valid for ##\frac{-1}{k} < x < \frac{1}{k}##

3. The attempt at a solution

I am able to find the expansion by following the binomial series definition, but multiplying everything by ##8^n##, where n is 2/3, -1/3, -4/3, and so on.

##(8+3x)^\frac{2}{3} = 8^\frac{2}{3} + (8^\frac{-1}{3})(\frac{2}{3})(3x) + (8^\frac{-4}{3})\frac{(\frac{2}{3})(\frac{-1}{3})}{2}(3x^2)##

Which is simplified to :

## 4 + x – \frac{x^2}{16} ## This is correct, according to the textbook answers.

However, I cannot find the range of x for which it is valid. The textbook answer is :
##\frac{-8}{3} < x < \frac{8}{3}##.

Why is that so? Could I have an explanation?

http://ift.tt/1jTvgsu

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