Math Help - Bessel Identity as a power seriers

1. Bessel Identity as a power seriers

Hello,

Im trying to arrange

$J_0(x) + \sum^{infnty}_{n=1} J_n(x) (r^n+{(-1)}^nr^{-n})$

into a power series.

I've attempted but cannot get it into the form required. I've tried subbing in the series representation of J_n(x) into it.

You want a power series with respect to $r$, right? So you are looking in fact for Laurent series.