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 ?
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 ?
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) $