Question

How do you use the binomial series to expand `1/sqrt(4+x)`?

Answer 1

The answer is `=1/2-x/16+(3x^2)/256-(5x^3)/2048+......`

Explanation:

The binomial theorem is
`(a+b)^n=((n),(0))a^n+((n),(1))a^(n-1)b+((n),(3))a^(n-2)b^2+((n),(4))a^(n-3)b^3+......`

Where `((n),(p))=(n!)/((n-p)!p!)`

Let's rewrite `1/(sqrt(4+x))`

as `1/(2sqrt(1+x/4))=1/2(1+x/4)^(-1/2)`

Now we can use the binomial theorem,

`1/2(1+x/4)^(-1/2)`

`=1/2(1-1/2*x/4+1/2*3/2*1/2*x^2/16-1/2*3/2*5/2*1/6*x^3/64)`

`=1/2-x/16+(3x^2)/256-(5x^3)/2048+......`