Binomial Expansion to Evaluate the Value of $\frac{1}{\sqrt{3.4}}$

Question

I've found the binomial expansion of $\frac{1}{\sqrt{1+x}}$ to be $1-\frac{1}{2}x+\frac{3}{8}x^2-\frac{5}{16}x^3+...$, but why does the series converge for $x=2.4$? Is it because $x>1$?

So how can I otherwise evaluate the value of $\frac{1}{\sqrt{3.4}}$ using the binomial expansion?

Answer 1

Hint

$$\frac{1}{\sqrt{3.4}}=\frac{1}{\sqrt{4-0.6}}=\frac{1}{2\sqrt{1-0.15}}$$ I am sure that you can take from here.

Answer 2

You can use $\displaystyle\frac{1}{\sqrt{4+x}}=\frac{1}{2}\frac{1}{\sqrt{1+\frac{x}{4}}}=\frac{1}{2}\left(1+\frac{x}{4}\right)^{-1/2}$, so now

you can substitute $\frac{x}{4}$ for $x$ in your series, and then replace $x$ by -.6.