What can I do to this expression to lose the summations?

Question

I'm at the end of a past paper question and need to derive this answer:

I am very close and have got to this by doing d/dx to the * equation:

What can I do to get rid of these summation signs and h?

*I have also been told that

Answer 1

It seems that for any fixed $x$, you have the power series in $h$: $$ \sum_{n=-\infty}^\infty h^n\left(J_{n+1}(x)-J_{n-1}(x)+2\frac{\mathrm{d}J_n}{\mathrm{d}x}(x)\right)=0\tag{1} $$ Thus, we have that for each $n$, $$ J_{n+1}(x)-J_{n-1}(x)+2\frac{\mathrm{d}J_n}{\mathrm{d}x}(x)=0\tag{2} $$ which is your desired equation.