Question

Expand to 4 terms the following expressions :
$(1+2x{)}^{-\frac{1}{2}}$

Answer 1

Whatever be value of $n$, negative integer or positive or negative fraction,

$(1+x{)}^n=1+\frac{n}{1!}x+\frac{n(n-1)}{2!}{x}^2+\frac{n(n-1)(n-2)}{3!}{x}^3+...+...+\frac{n(n-1)(n-2)...(n-r+1)}{r!}{x}^r+...+$ to $\infty$

where, $|x|<1$ i.e. $-1<x<1.$

Now,

$(1+2x{)}^{-\frac{1}{2}}=1-\frac{1}{2}\times 2x+\frac{-\frac{1}{2}(-\frac{1}{2}-1)}{1.2}(2x{)}^2+\frac{-\frac{1}{2}(-\frac{1}{2}-1)(-\frac{1}{2}-2)}{\mathrm{1.2.3}}(2x{)}^3+.........$

$=1-x+\frac{3}{2}{x}^2-\frac{5}{2}{x}^3+.....$