# Find the function that has the following series representation

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.

• The exponents on the coefficients are really annoying. If I am right, by Stirling, the coefficients (with exponents) are asymptotic to $n^{-n/2}$. – Yves Daoust Nov 6 '19 at 17:49
• @Yves Daoust Yeah, you are right, without the exponent things may become much easier. – James Warthington Nov 6 '19 at 17:49

$$\sum_{n=0}^\infty \bigg(\frac{(2n+1)!!}{(2n)!!}x^2\bigg)^n.$$