Does the sequence $|a^{-n} J_0(n)|$ converge? I used the approximation $J_0(n) \approx \sqrt{\frac{2}{\pi n}} \cos(n - \pi/4)$ and assume that $a > 1$. Since the sample sequence of $J_0(n)$ is weighted by a decaying exponential $a^{-n}$, I suspect this will converge, but I'm unable to confirm this.
I want to know if
$\sum_{n=1}^\infty a^{-n}|J_0(n)|$
converges for above approximation?