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

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Answer 1 · 2014-09-28T16:15:34

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

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.

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