Let u[n]=1 for n≥ 0 and u[n]=0 for n<0

Let \(u[n]=1\) for \(n\ge 0\) and \(u[n]=0\) for \(n<0\). Sequences are indexed by \(n\in\mathbb{Z}\). Define the causal geometric signals \[ x[n]=\Bigl(\frac{1}{4}\Bigr)^n u[n],\qquad h[n]=\Bigl(\frac{1}{2}\Bigr)^n u[n]. \] Ordinary linear convolution is \(y[n]=\sum_{k\in\mathbb{Z}}x[k]h[n-k]\). Show that \(y[n]=0\) for every \(n<0\). For \(n\ge 0\), evaluate the finite geometric sum and simplify to a closed-form expression in elementary functions of \(n\). Write the resulting formula for \(y[n]\) using \(u[n]\). Determine whether the LTI system with this \(h\) is causal and whether it is BIBO stable, and compute \(\sum_{n=-\infty}^{\infty}|h[n]|\). Evaluate \(y[0]\), \(y[1]\), and \(y[4]\) exactly as rational numbers.

The convolution of two right-sided geometrics is a finite geometric sum whose ratio is the quotient of the two bases.