Question

How do you use the binomial theorem to find the Maclaurin series for the function `y=f(x)` ?

Answer 1

Binomial Series

`(1+x)^{alpha}=sum_{n=0}^infty((alpha),(n))x^n`,

where `((alpha),(n))={alpha(alpha-1)(alpha-2)cdot cdots cdot(alpha-n+1)}/{n!}`.

Let us look at this example below.

`1/{sqrt{1+x}}`

by rewriting a bit,

`=(1+x)^{-1/2}`

by Binomial Series,

`=sum_{n=0}^infty((-1/2),(n))x^n`

by writing out the binomial coefficients,

`=sum_{n=0}^infty{(-1/2)(-3/2)(-5/2)cdots(-{2n-1}/2)}/{n!}x^n`

by simplifying the coefficients a bit,

`=sum_{n=0}^infty(-1)^n{1cdot3cdot5cdot cdots cdot(2n-1)}/{2^n n!}x^n`

I hope that this was helpful.