Find the power series (expanded about x = 0) for .

We write as (1 + x)(1 − x^{2})^{−½}, and expand the latter term as a binomial series. We have

The coefficient of the general term, x^{2n}, is given by .

The numerator of this expression is a double factorial: (2n − 1)!!.

The denominator may then be written in a similar form by "absorbing" the factors of 2^{n} into n!, giving (2n)!!.

(Alternatively, we could multiply numerator and denominator by 2^{n }n!, and simplify the expression, obtaining .)

Therefore, the power series expansion of = .

Source: To be announced