Singularity of a sum of Bessel functions

Question

I would like to know the nature of $$\sum_{m=1}^{\infty}J_{0}(mx)=\sum_{m=1}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{k}m^{2k}x^{2k}}{2^{2k}(k!)^{2}}$$ as $x\to0^{+}$. For any $x>0$, this series of Bessel functions converges (being a sequence of numbers decaying in magnitude and varying in phase). However, at $x=0$, the series does not converge and I would like to know at what rate it diverges e.g. as $ln(x)$ or $x^{-1}$. Clearly you can't swap any of the limits involved so I'm a bit stuck. Any help would be greatly appreciated. Thanks.

Answer 1

You can use Euler-Maclaurin: $$ \sum_{m>0}J_0(mx)=\int_{m>0}J_0(mx)+\mathcal O(1)=\frac{1}{x}+\mathcal O(1) $$