  Using a method of factorization and by introducing a generalized discrete
Dirichlet's Laplacian matrix $(-\Delta_{\Lambda})$, we establish an extended
improved discrete Hardy's inequality and Rellich inequality in one dimension.
We prove that the discrete Copson inequality (E.T. Copson, \emph{Notes on a
series of positive terms}, J. London Math. Soc., 2 (1927), 9-12.) in
one-dimension admits an improvement. We also prove that the improved Copson's
weights are optimal (in fact \emph{critical}). It is shown that improvement of
the Knopp inequalities (Knopp in J. London Math. Soc. 3(1928), 205-211 and
5(1930), 13-21) lies on improvement of the Rellich inequalities. Further, an
improvement of the generalized Hardy's inequality (Hardy in Messanger of Math.
54(1925), 150-156) in a special case is obtained.
