I encounter this series in an old book of infinite series. The series is as followed:
$1+(\dfrac{1}{2})x^2+(\dfrac{1\cdot 3}{2\cdot4})^2x^4+(\dfrac{1\cdot3\cdot5}{2\cdot4\cdot6})^3x^6+(\dfrac{1\cdot3\cdot5\cdot7}{2\cdot4\cdot6\cdot8})^4x^8...$
Judging from the series, it must be an even function. Is it possible to find the closed form function for this series. Also, instead of the power of 2 in coefficients, we may have them in n. Do we have a general formula to generate the functions for power of n?
I try to differentiate this series but it doesn't seem to produce anything fruitful.
Looks like
$$\sum_{n=0}^\infty \bigg(\frac{(2n+1)!!}{(2n)!!}x^2\bigg)^n.$$