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

Sep 28, 2014

Binomial Series

${\left(1 + x\right)}^{\alpha} = {\sum}_{n = 0}^{\infty} \left(\begin{matrix}\alpha \\ n\end{matrix}\right) {x}^{n}$,

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

Let us look at this example below.

$\frac{1}{\sqrt{1 + x}}$

by rewriting a bit,

$= {\left(1 + x\right)}^{- \frac{1}{2}}$

by Binomial Series,

$= {\sum}_{n = 0}^{\infty} \left(\begin{matrix}- \frac{1}{2} \\ n\end{matrix}\right) {x}^{n}$

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.