How do you use the binomial theorem to find the Maclaurin series for the function #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.