Answer to Question #112490 in Calculus for Leo

Question

Make use of binomial theorem to expand binomial expression & show the range of values of x for which expansion is valid? 1. (2-x)^-2 and 2. (1-x)^1/2

^ = to power of

Answer 1

The binomial expansion of

$(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+....$

For $(1-x)^{\frac12}$

Put $x=-x$ and $n=\frac{1}2$

$(1-x)^\frac12=1-\frac{x}{2}+\frac{\frac12(\frac{-1}{2})}{2!}x^2+....$

$-1<-x<1$

$-1<x<1$

For $(2-x)^{-2}=2^{-2}(1-\frac x 2)^{-2}$

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

$-1<-(\frac{x}{2})<1$

$-2<x<2$