Before Cauchy, the indiscriminate uses of infinite series like finite sum had resulted in many paradoxical situations. The method formulated by Cauchy was so natural and efficacies in its use, that mathematicians were inclined to believe that the problem of sum of an infinite series has fully and finally been resolved. The researchers like Abel, Dedekind, Weierstrass and Dirichlet have further strengthened his idea on the uses of infinite series. This book contains the investigations on some properties of a subset of absolutely convergent sequence space, some general properties of the space of entire double sequences and analytic double sequences, the sequence spaces using modulus function defined on semi-normed spaces, on indexed summability methods and on Absolute Banach summability of a Fourier series.