on the sum $ \sum _{n=1}^{\infty}J_{0} (2\pi nx) $

Question

given the zeroeth order Bessel function.. is then possible to compute the sum

$ \sum _{n=1}^{\infty}J_{0} (2\pi nx) $ for every 'x' positive real number ?

Answer 1

To investigate convergence, perhaps use: $$ J_0(2\pi n x) = \frac{1}{\pi\sqrt{nx}}\sin\left(2\pi n x + \frac{\pi}{4}\right) + O(n^{-3/2})\qquad\text{as } n \to +\infty $$ and investigate convergence of $$ \sum_{n=1}^\infty \frac{1}{\pi\sqrt{nx}}\sin\left(2\pi n x + \frac{\pi}{4}\right) $$