In previous studies we applied Lanzcos' τ-method to get polynomial and rational approximations to series of hypergeometric type. It was shown that the approximations could be viewed as a weighted sum of the partial sums of the given series. This we call a summability method. The τ-method starts from the differential equation satisfied by the given function. But in view of the summability feature, the approximations can be found directly from the hypergeometric series. In the present paper, we explore the applicability of the direct summability process for functions of hypergeometric type to functions which are not of hypergeometric type. Our treatment is expository. We treat two examples. The first converges in |z| 0. These approximations are quite remarkable in that they appear to converge in a region where the given series upon which they are based are divergent.