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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 ?

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It seems for $x=1$ the series diverges to infinity. But probably for most $x$ is converges conditionally. – GEdgar Sep 11 '12 at 12:11

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 $$ \sum_{n=1}^\infty \frac{1}{\pi\sqrt{nx}}\sin\left(2\pi n x + \frac{\pi}{4}\right) $$

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