Consider the sequence
$$x\left[ n \right]$$= $${a^n}u\left[ n \right] + {b^{\partial n}}u\left[ n \right]$$ , where u[n] denotes the unit step sequence and 0<$$\left| a \right| < \left| b \right| < 1.$$
The region of convergence (ROC) of the z-transform of $$\left[ n \right]$$ is

A discrete-time signal$$x\left[ n \right]\, = \delta \left[ {n - 3} \right]\, + 2\delta \left[ {n - 5} \right]$$ has z-transform x(z). If Y (z)=X (-z) is the z-transform of another signal y$$\left[ n \right]$$, then

Two casual discrete-time signals $$x\left[ n \right]$$ and $$y\left[ n \right]$$ =$$\sum\limits_{m = 0}^n x \left[ m \right]$$. If the z-transform of y$$\left[ n \right]$$=$${2 \over {z{{(z - 1)}^2}}}$$ , the value of $$x\left[ 2 \right]$$ is _____________________

The sequence x $$\left[ n \right]$$ = $${0.5^n}$$ u[n], where u$$\left[ n \right]$$ is the unit step sequence, is convolved with itself to obtain y $$\left[ n \right]$$ . Then $$\sum\limits_{n = \infty }^{ + \infty } y \left[ n \right]$$ is ____________.