Need help with a sum series (Frobenius series)

[xago]

Homework Statement

I've obtained a recurrence relation of:
$a_{n+2}$ = $\frac{(n-1)(n-2)-\frac{2k}{w_{o}^{2}}}{R^{2}(n+1)(n+2)}$$a_{n}$
from a Frobenius series solution problem and I've expanded it to give the series:

f(r) = 1 + $\frac{-\frac{2k}{w_{o}^{2}}}{6R^{2}}$$r^{2}$ + $\frac{-\frac{20k}{w_{o}^{2}}+\frac{4k^{2}}{w_{o}^{4}}}{120R^{4}}$$r^{4}$ + $\frac{-\frac{560k}{w_{o}^{2}}+\frac{152k^{2}}{w_{o}^{4}}-\frac{8k^{3}}{w_{o}^{6}}}{5040R^{6}}$$r^{6}$ ...

I can see that the denominator is 3!$R^{2}$ , 5!$R^{4}$ , 7!$R^{6}$ which is (2n+1)! $R^{2n}$ however I need a bit of help condensing the numerator into a single series term.

 

[member 553083]

The Attempt at a SolutionI think it can be condensed into:-\frac{2k}{w_{o}^{2}} \sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n+1)!} (\frac{k}{w_{o}^{2}})^{n-1}But I'm not sure.