So I have got the expansion of $$ (4-5x)^.5 = 2 + (5/4)x + (25/64)x^2 $$
I am told to use $ x = 1/10 $ to find an approximation of $ \sqrt2 $. I can do this, giving $ 181/128 $, however the last part asks: "Explain why substituting $ x = 1/10 $ into this binomial expansion leads to a valid approximation. "
The answer is said to be because $ |x| < 4/5 $. Why?
Best Answer
The actual explanation has something to do with the convergence of Taylor (or Maclaurin) series but it involves quite a high-level calculus.
A not-so-satisfying answer is that you have to put the LHS into the form $k(1+ax)$ and the approximation is valid for $\lvert ax \rvert <1$