I am aiming to prove that the Bessel function of the First Kind, $J_\nu (x)$, satisfies the Bessel Equation by direct substitution for any value of parameter $\nu$.
$J_\nu(x) = \sum_{m=0}^\infty \dfrac{(-1)^m}{m!\Gamma(m + \nu + 1)}\left(\dfrac x2\right)^{2m+\nu}$
It seems to be a simple matter of differentiating the Function twice and substititing into the Bessel Equation, but I can't seem to get it to solve to 0 as it ought to do if the Function is a solution.
Also, apologies for any janky use of the MathJax tools, first time posting and on a mobile, at the moment.