Using the binomial expansion to approximate a square root ... | square root expansion
Herewelookatsomeargumentswhytakingoutafactor$2.5$isaconvenientchoice.Problem:Wewantto(manually)approximate$sqrt{6}$byusingthefirstfewtermsofthebinomialseriesexpansionofegin{align*}sqrt{1-4x}&=sum_{n=0}inftyinom{frac{1}{2}}{n}(-4x)nqquadqquadqquadqquad|x|
Here we look at some arguments why taking out a factor $2.5$ is a convenient choice.
Problem: We want to (manually) approximate $sqrt{6}$ by using the first few terms of the binomial series expansion of egin{align*} sqrt{1-4x}&= sum_{n=0}infty inom{frac{1}{2}}{n}(-4x)nqquadqquadqquadqquad |x|<frac{1}{4}\ &= 1-2x-2x2-4x3+cdots ag{1} end{align*}
In order to apply (1) we are looking for a number $y$ with egin{align*} sqrt{1-4x}&=sqrt{6y2}=ysqrt{6} ag{2}\ color{blue}{sqrt{6}}&color{blue}{=frac{1}{y}sqrt{1-4x}} end{align*}
We see it is convenient to choose $y$ to be a square number which can be easily factored out from the root. We obtain from (2) egin{align*} 1-4x&=6y2\ 4x&=1-6y2\ color{blue}{x}&color{blue}{=frac{1}{4}-frac{3}{2}y2} ag{3} end{align*}
When looking for a nice $y$ which fulfills (3) there are some aspects to consider:
We have to respect...