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.