On the divergence of a certain series

We investigate for which real numbers α the series (4) converges, and prove that, even though it converges almost everywhere in the sense of Lebesgue to a periodic, with a period 1, odd function in L2([0,1]), it is divergent at uncountably many points, the set of which is dense in [0,1]. Finally, we find the Fourier expansion of the function defined by the series (4).