Summation of certain series using the Shannon sampling theorem

In communication theory texts, it is usually observed that if the sampling theorem is uncritically applied to a pure sinusoidal signal sin 2πWt using the Nyquist sampling rate of 2W samples/ s, then all the samples taken at the points {k/2W } are zero and a reconstruction of the sinusoid from its sample values is clearly impossible. The author shows that the suggested expedient of using a slightly higher sampling rate does not suffice to give a convergent sampling expansion for the sinusoid and that the equations used to establish this result give rise to a number of classical series summations usually evaluated by means of contour integration. The Shannon sampling theorem itself can be employed to yield interesting closed-form summations. Some series involving the zeroth-order Bessel function are given as examples of the method